Abstract
This is part two of a twopart paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truthpredicates and principles about the interaction of partial ground and truth. We show that our theory is a prooftheoretically conservative extension of the ramified theory of positive truth up to š _{0} and thus is consistent. We argue that this theory provides a natural solution to Fineās āpuzzle of groundā about the interaction of truth and ground. Finally, we show that if we apply the truthpredicate to sentences involving our groundpredicate, we run into paradoxes similar to the semantic paradoxes: we get groundtheoretical paradoxes of selfreference.
Introduction
This is part two of a twopart paper in which we develop axiomatic theories of partial ground.^{Footnote 1} Partial ground, remember, is the relation of one truth holding (either wholly or partially) in virtue of another, cf. [5, 7].^{Footnote 2} Partial ground in this sense is standardly taken to be irreflexive, meaning that no truth partially grounds itself, and transitive, meaning that partial grounds are inherited through partial grounding. In other words, partial ground is a strict partial order on the truths.^{Footnote 3} Consequently, the relation of partial ground induces a hierarchy of grounds on the truths, in which the partial grounds of a truth rank āstrictly belon the truth itself. In the first part of the paper, we axiomatized this hierarchy over the truths of arithmetic using a (binary) ground predicate instead of an operator to formalize the relation of partial ground.^{Footnote 4}
When axiomatizing the grounding hierarchy, we explicitly restricted ourselves to applications of the ground predicate to arithmetical truths (i.e. true sentences in the language of arithmetic), leaving aside applications of the ground predicate to sentences involving the truth predicate and applications of the truth predicate to sentences involving the ground predicate. In the present paper, we lift these restrictions.
First, we turn our attention to applications of the ground predicate to sentences involving the truth predicate. A plausible principle about the interaction of truth and partial ground states that the truth of any true sentence grounds the truth of the sentence which states that the sentence in question is true. This principle traces back to Aristotle and may be put a bit more explicitly as follows:

(Aristotelian Principle): If Ļ is a true sentence, then the truth of \(Tr(\ulcorner \varphi \urcorner )\) holds either wholly or partially in virtue of the truth of Ļ.^{Footnote 5}
Unfortunately, as Fine effectively shows in [6], the Aristotelian Principle is inconsistent with standardly accepted principles about the interaction of partial ground and the logical operators: it allows us to derive that the truths of some true sentences partially ground themselvesāin direct contradiction to the irreflexivity of partial ground. To give a quick example, take the sentence āx T r(x), which says that there is at least one true sentence. This sentence is itself (provably) true, and so we have that āx T r(x) partially grounds \(Tr(\ulcorner \exists xTr(x)\urcorner )\) by the Aristotelian Principle. By the standardly accepted principle of partial ground that an existential truth is partially grounded in all of its true instances, we get furthermore that \(Tr(\ulcorner \exists xTr(x)\urcorner )\) partially grounds āx T r(x). But by the transitivity of partial ground this immediately gives us that āx T r(x) partially grounds itself. This is (an instance of) what in [6] Fine calls the puzzle of ground.
In this paper, we develop, in quite some formal detail, what Fine [6, p. 108ā10] calls a āpredicativistā solution to the puzzle of ground. What makes our solution predicativist is our appeal to Tarskiās hierarchy of objectlanguage, metalanguage, metametalanguage, and so on to rule out the problematic cases of apparently selfgrounding sentences. We show that by observing Tarskiās distinction between object and metalanguage, we can formulate a consistent axiomatic theory of partial ground that proves (a predicativist version of) the Aristotelian principle while retaining the irreflexivity of ground. Formally, we obtain this theory using the method typing, familiar from theories of truth, where applications of a truthpredicate to sentences involving the same truthpredicate are ruled out.^{Footnote 6} To the best of our knowledge, this is the first fully worked out proposal for a predicativist solution to the puzzle of ground in the literature.^{Footnote 7}
Then, we turn our attention to applications of the truth predicate to sentences containing the ground predicate. We show that we cannot consistently add axioms to our axiomatic theory of ground that would allow us to prove all the instances of the Tscheme \(Tr(\ulcorner \varphi \urcorner )\leftrightarrow \varphi \) over formulas Ļ involving the groundpredicate; for if we were do so, weād get what we call groundtheoretic paradoxes of selfreference. We will show that these paradoxes are closely related to the wellknown semantic paradoxes of selfreference, such as the infamous liar paradox, and furthermore that the paradoxes are genuinely different from Fineās puzzle of ground. Fineās puzzle arises from the fact that intuitively plausible principles about partial ground and truth entail that the truths of some sentences partially ground themselves. The groundtheoretic paradoxes of selfreference, in contrast, concern an entirely different aspect of partial ground: the problem is, as weāll argue, that the ground predicate behaves too much like a truthpredicate.^{Footnote 8}
Here is the plan for the paper: SectionĀ 1.1 contains a prĆ©cis of the first part of the paper. Then, in SectionĀ 2, weāll discuss the interaction of truth and partial ground on a general level and introduce the Aristotelian Principle into our predicational setting for axiomatic theories of partial ground. In SectionĀ 3, weāll develop an axiomatic predicational theory that proves (a predicative/typed version of) the Aristotelian principle. We then prove that the theory is a conservative extension of the ramified theory of truth (and is thus, in particular, consistent) and we construct a model. Then, in SectionĀ 4, we will show that if weāre not careful, partial ground, just like truth, can give rise to paradoxes of selfreference. In SectionĀ 5, we conclude with some general observations and a map of possible responses to the groundtheoretic paradoxes of selfreference.
PrĆ©cis of āAxiomatic Theories of Partial Ground I. The Base Theoryā
In the following sections, we assume that the reader is familiar with the techniques and results obtained in the first part of the paper. Those who have these techniques and results sufficiently present in their mind can safely skip this section. But for the rest, letās briefly refresh our memory.
In the first part of the paper, we develop an axiomatic theory of partial ground using a binary ground predicate rather than an operator to express partial ground. We call such a theory a predicational theory of partial ground, in contrast to operational theories. At the outset of the paper, we provide technical and philosophical reasons for using a predicational approach (SectionĀ 2).
In the following sections (SectionsĀ 3 andĀ 4), we develop our theory in formal detail and investigated its properties. On the syntax side, we start from the language \(\mathcal {L}\) of Peano arithmetic. The language \(\mathcal {L}_{Tr}\) is defined as \(\mathcal {L}\cup \{Tr\}\), where T r is the unary truthpredicate, and the language \(\mathcal {L}_{Tr}^{\lhd }\) is \(\mathcal {L}_{Tr}\cup \{\lhd \}\), where \(\lhd \) is the binary (partial) groundpredicate. We formulate our theory in \(\mathcal {L}_{Tr}^{\lhd }\).
On the theory side, we starte from the theory PAG of Peano arithmetic (PA) with all the instances of the induction scheme
over the language \(\mathcal {L}_{Tr}^{\lhd }\). To formulate our predicational theory of ground, we additionally rely on standard syntax theory developed in PA. In particular, we use a standard coding function # to map every expression (variable, term, formula, etc.) Ļ to a unique code # Ļ from the natural numbers. If Ļ is an expression, then we also write \(\ulcorner \sigma \urcorner \) for the numeral intended to denote # Ļ. We use function terms Ā¬Ģ£, ā§Ģ£, ā§Ģ£, āĢ£, āĢ£, = Ģ£ and \(\lhd \)Ģ£ to represent the corresponding syntactic operations.^{Footnote 9}
Now our predicational theory of ground PG has all the axioms of PAG, plus the following axioms:^{Footnote 10}
Basic Ground Axioms:Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Basic Truth Axioms:

G_{1} \(\forall x\neg (x\lhd x)\)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā T_{1}āsāt(T r(s =Ģ£ \(t)\leftrightarrow s^{\circ }=t^{\circ })\)

G_{2} \(\forall x\forall y\forall z(x\lhd y\land y\lhd z\to x\lhd z)\)Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā T\(_{2} \forall s\forall t(Tr(s\underset {.}{\neq }t)\leftrightarrow s^{\circ }\neq t^{\circ })\)

G_{3} \(\forall x\forall y(x\lhd y\to Tr(x)\land Tr(y))\) Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā T_{3}āx(T r(x) ā S e n t(x))

Upward Directed Axioms:
U_{1} \(\forall x (Tr(x)\to x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\)

U_{2} \(\forall x\forall y ((Tr(x)\to x\lhd x\underset {.}{\lor }y) \land (Tr(y)\to y\lhd x\underset {.}{\lor }y))\)

U_{3} \(\forall x\forall y (Tr(x)\land Tr(y)\to (x\lhd x\underset {.}{\land } y)\land (y\lhd x\underset {.}{\land } y))\)

U_{4} \(\forall x\forall y (Tr(\underset {.}{\neg } x)\land Tr(\underset {.}{\neg } y)\to (\underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (\underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)

U_{5} \(\forall x\forall y ((Tr(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }(x\underset {.}{\land }y)) \land (Tr(\underset {.}{\neg }y)\to \underset {.}{\neg }y\lhd \underset {.}{\neg }(x\underset {.}{\land }y)))\)

U_{6} \(\forall x \forall t\forall v(Tr(x(t/v))\to x(t/v)\lhd \underset {.}{\exists } v x)\)

U_{7} \(\forall x\forall v (\forall tTr(\underset {.}{\neg } x(t/v))\to \forall t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\exists }v x))\)

U_{8} \(\forall x \forall v(\forall t(Tr(x(t/v))\to \forall t(x(t/v)\lhd \underset {.}{\forall }v x))\)

U_{9} \(\forall x \forall t\forall v(Tr(\underset {.}{\neg }x(t/v))\to \underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\forall } v x))\)

Downward Directed Axioms:
D _{1} \(\forall x(Tr(\underset {.}{\neg }\underset {.}{\neg }x)\to x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\)

D _{2} \(\forall x\forall y (Tr(x\underset {.}{\lor }y)\to (Tr(x)\to x\lhd x\underset {.}{\lor }y )\land (Tr(y)\to y\lhd x\underset {.}{\lor }y))\)

D _{3} \(\forall x\forall y (Tr(x\underset {.}{\land }y)\to (x\lhd x\underset {.}{\land } y)\land (y\lhd x\underset {.}{\land } y))\)

D _{4} \(\forall x\forall y (Tr(\underset {.}{\neg } (x\underset {.}{\land } y))\to (Tr(\underset {.}{\neg }x)\to \underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (Tr(\underset {.}{\neg }y)\to \underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)

D _{5} \(\forall x\forall y (Tr(\underset {.}{\neg } (x\underset {.}{\lor } y))\to (\underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (\underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)

D _{6} \(\forall x (Tr(\underset {.}{\exists } v x(v))\to \exists t(x(t/v)\lhd \underset {.}{\exists } v x))\)

D _{7} \(\forall x \forall v(Tr(\underset {.}{\neg }\underset {.}{\exists }v x)\to \forall t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\exists }v x))\)

D _{8} \(\forall x \forall v(Tr(\underset {.}{\forall }v x\to \forall t(x(t/v)\lhd \underset {.}{\forall }v x))\)

D _{9} \(\forall x \forall v(Tr(\underset {.}{\neg }\underset {.}{\forall } v x)\to \exists t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\forall } v x))\)
In the remaining sections of part one of the paper, we show that PG is a prooftheoretically conservative extension of the theory PT of positive truth (Theorem 1) and provide a model construction (Theorem 2) by extending the standard model of PT to a canonical model of PG. As an immediate corollary of the conservativity result, we infer that PG is consistent (Corollary 3) and that PG he same arithmetic statements as the theory ACA of arithmetic comprehension (Corollary 4).
The Aristotelian Principle and Typed Truth
Aristotle has the following to say about the interaction of truth and explanation:
It is not because we think truly that you are pale, that you are pale; but because you are pale we who say this have the truth.
(Metaphysics 1051b6ā9)
In the context of partial ground, this motivates the following principle: If Ļ is true, then it is natural to say that Ļ is true in virtue of what it says being the case, and if Ļ is false, then Ļ is false in virtue of what it says not being the case. We will thus call the corresponding informal groundtheoretic principles the Aristotelian principles about truth and falsehood respectively.
If we wish to formalize the Aristotelian principles in the context of our predicational theory of ground, we have to make a couple of adjustments. First, we have to assume that we have a GĆ¶delnumbering for the language \(\mathcal {L}_{Tr}\). In particular, we now assume that we have a name \(\ulcorner \varphi \urcorner \) for every sentence \(\varphi \in \mathcal {L}_{Tr}\). Second, we have to adjust our basic truth axioms. The axioms T _{3} and G _{3} of PG together ensure that both the truth predicate and the ground predicate only apply to sentences of \(\mathcal {L}\). To formalize the Aristotelian principles, however, we need to relax this requirement: we need for the truth predicate and the ground predicate to apply to sentences of \(\mathcal {L}_{Tr}\) that are not already in \(\mathcal {L}\). To allow for this, we adjust the axiom T _{3} to the axiom:
which weāll label T\(_{3}^{\ast }\). Together with G _{3}, this then entails:
as well. Weāll refer to the theory that results from replacing T_{3} in PG with T\(_{3}^{\ast }\) as PGT. Thus, in PGT, we are not only talking about the truths of arithmetic, but also about the truth of the truths of arithmetic.
With these adjustments in place, we can schematically express the Aristotelian principle about truth by saying that for all sentences Ļ that:
For the Aristotelian principle about falsehood, we get:
for all sentences Ļ. Using the same strategy as in the previous paper, we can translate these schemata into the quantified axioms:

(AP _{ T }) \(\forall x(Tr(x)\to x\lhd \underset {.}{Tr}(\dot {x})),\) and

(AP _{ F }) \(\forall x(Tr(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr}(\dot {x})),\)
where \(\underset {.}{Tr}\) represents the function that maps the code # t of a term t to the code # T r(t) of the application T r(t) of the truth predicate to the term t. Thus, we have arrived at a quantified axiomatization of the Aristotelian principles.^{Footnote 11}
Unfortunately, as [6] shows, the Aristotelian principles are groundtheoretically inconsistent:^{Footnote 12}
Lemma 1 (Puzzle of Ground)
A P _{ T } and A P _{ F } are inconsistent over PGT.
Proof
To show that \(AP_{T}\vdash _{PGT}\bot \) consider the following derivation:

1.
\(\overline {0}=\overline {0}\) (Arithmetic)

2.
\(Tr(\ulcorner \overline {0}=\overline {0}\urcorner )\) (Tscheme over \(\mathcal {L}\))

3.
\(\ulcorner \overline {0}=\overline {0}\urcorner \lhd \ulcorner Tr(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner \) (2., AP _{ T })

4.
\(Tr(\ulcorner Tr(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner )\) (3., G _{3})

5.
\(\ulcorner Tr(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner \lhd \exists x Tr(x)\) (4., U _{6})

6.
\(Tr(\ulcorner \exists x Tr(x)\urcorner )\) (5., G _{3})

7.
\(\ulcorner \exists x Tr(x)\urcorner \lhd \ulcorner Tr(\ulcorner \exists x Tr(x)\urcorner )\urcorner \) (6., AP _{ T })

8.
\(Tr(\ulcorner Tr(\ulcorner \exists x Tr(x)\urcorner )\urcorner )\) (7., G _{3})

9.
\(\ulcorner Tr(\ulcorner \exists x Tr(x)\urcorner )\urcorner \lhd \ulcorner \exists x Tr(x)\urcorner \) (8., U _{6})

10.
\(\ulcorner \exists x Tr(x)\urcorner \lhd \ulcorner \exists x Tr(x)\urcorner \) (7.,9., G _{2})

11.
\(\neg (\ulcorner \exists x Tr(x)\urcorner \lhd \ulcorner \exists x Tr(x)\urcorner )\) (G _{1})

12.
ā„ (10.,11., ā„Intro)
To show that \(AP_{F}\vdash _{PGT}\bot ,\) we can perform an analogous derivation, which is left to the interested reader. ā”
We are left with a groundtheoretic puzzle about truth.^{Footnote 13} All the principles involved in the proof of Lemma 1 are intuitively plausible: the basic ground axioms G _{2} and G _{3} directly arise from the definition of partial ground, the upward directed axiom U _{6} about the existential quantifier is plausible in light of the usual semantics for firstorder logic, and the Aristotelian principle for truth is plausible from considerations about truth. (The principles required to show that AP _{ F } is groundtheoretically inconsistent are equally plausible.) So what are we to do? In this section, we will propose a solution to the puzzle of ground that preserves the intuition behind all of these principles. We will achieve this by typing our truth predicateāa move familiar from typed theories of truth.
In typed theories of truth, no applications of the truth predicate to sentences containing the same truth predicate are provable. Thus, typed theories of truth respect Tarskiās distinction between objectlanguage and metalanguage [20]. Tarski motivates this distinction from the semantic paradoxes, such as the infamous liar paradox. This paradox results when we apply the Tscheme to the liar sentence which intuitively āsays of itselfā that it is not true. More specifically, Tarski observed that, in a sufficiently strong background theory, such as PA, if we allow applications of the truth predicate to sentences with the same truth predicate in them, we get a sentence Ī» that is provably equivalent to its own falsehood:
This follows from the socalled diagonal lemma, which is provable in PA in the context of an appropriate GĆ¶delnumbering for \(\mathcal {L}_{Tr}\):^{Footnote 14}
Lemma 2
For every sentence \(\varphi (x)\in \mathcal {L}_{Tr}\) with exactly one free variable x,there is a sentence \(\delta \in \mathcal {L}_{Tr}\) such that
The existence of the liar sentence Ī», then, follows by a simple application of the diagonal lemma to the formula \(\neg Tr(x)\in \mathcal {L}_{Tr}\). It is wellknown that the existence of a liar sentence is inconsistent with the Tscheme over the language \(\mathcal {L}_{Tr}\).
A common intuitive response to the liar paradox is that it somehow arises from the selfreference involved.^{Footnote 15} On this informal view, the problem is that the liar sentence āsays something of itself,ā namely that it is not true.^{Footnote 16} Thus, so the intuitive response, we should put restrictions on our language that prevent selfreference. Tarski makes this response precise by introducing the distinction between objectlanguage and metalanguage. To illustrate the distinction, consider the truths of arithmetic. According to Tarski, if we wish to talk about numbers and their properties, we can do so in the language \(\mathcal {L}\) of PAāour objectlanguage for arithmetic. But if we wish talk about the truth of sentences in \(\mathcal {L}\), we have to do so in the language \(\mathcal {L}_{Tr}\)āour metalanguage for the truths of arithmetic.^{Footnote 17} Moreover, if we wish to talk about the truth of the truths of arithmetic, i.e. the truths of sentences in \(\mathcal {L}_{Tr}\) containing the truth predicate, we need to do so in yet another meta metalanguage, which has a distinct truthpredicate for the sentences of \(\mathcal {L}_{Tr}\). And so on. In contrast, Tarski calls a language that can talk about the truths of its own sentences, i.e. a language that has both names for all of its sentences and a truth predicate that applies to these names, semantically closed. Thus, a semantically closed language is its own metalanguage, as it were, and thus we get selfreferential paradoxes. Tarski shows that if we obey the distinction between objectlanguage and metalanguage, we can formulate a consistent theory of truth: In an appropriate metalanguage, which is not semantically closed, we can consistently affirm the Tscheme for the sentences of the objectlanguage and we never can prove problematic sentences, such as the liar. The liar paradox, on the other hand, shows that if we work in a semantically closed language, disaster ensues: If we have a GĆ¶delnumbering for the terms of \(\mathcal {L}_{Tr}\) within \(\mathcal {L}_{Tr}\) and at the same time affirm the Tscheme over the sentences of \(\mathcal {L}_{Tr},\) i.e. if we use \(\mathcal {L}_{Tr}\) as its own metalanguage, we get semantic paradoxes, like the liar paradox. Thus, so Tarski argues, when we wish to talk about truth, we should not never do so in a semantically closed language, but always in an appropriate metalanguage. Intuitively, the picture is that semantic truths, such as truths about the truths of arithmetic, are on a āhigher levelā than nonsemantic truths, such as the ordinary truths of arithmetic. Moreover, this can be iterated: the truths about truths about the truths of arithmetic are on yet a āhigher levelā than the truths about the truths of arithmetic and so on. What emerges is Tarskiās hierarchy of truths. Following Tarski, if we work in a semantically closed language, we mix the levels of the hierarchy of truthsāand ultimately this is the source of the semantic paradoxes.^{Footnote 18}
Our original, unmodified predicational theory of ground PG respects Tarskiās distinction between object and metalanguage: We have formulated PG in the language \(\mathcal {L}_{Tr}^{\lhd }\) in the context of a coding for the language \(\mathcal {L}\) of PA. In particular, we have assumed that we have a name \(\ulcorner \varphi \urcorner \) for every sentence \(\varphi \in \mathcal {L},\) but not that we have names \(\ulcorner Tr(t)\urcorner \) for sentences of the form \(Tr(t)\in \mathcal {L}_{Tr}\) and so on. Moreover, as we have said before, by the axioms T _{3} and G _{3}, we have ensured that both the truth predicate and the ground predicate only apply to the sentences of \(\mathcal {L}\). Thus, we have used the language \(\mathcal {L}_{Tr}^{\lhd }\) as an appropriate metalanguage for our objectlanguage \(\mathcal {L}\)āin compliance with Tarskiās distinction. When we move to the modified theory P G T, however, we no longer conform with Tarskiās distinction. Since PGT is formulated in \(\mathcal {L}_{Tr}^{\lhd }\) in the context of a GĆ¶del numbering for \(\mathcal {L}_{Tr}\), PGT is formulated in a semantically closed language. Now, the truth predicate may apply to sentences with the same truth predicate in them: Since in PGT we work in the context of a coding for \(\mathcal {L}_{Tr},\) we have names for all the sentences of \(\mathcal {L}_{Tr}^{\lhd }\) within \(\mathcal {L}_{Tr}^{\lhd }\) itself. Moreover, by the axiom T\(_{3}^{\ast }\) we have allowed for these terms to occur truly in the context of the truth predicate and the ground predicate. In other words, when we formulated P G T, we have used \(\mathcal {L}_{Tr}^{\lhd }\) as its own metalanguageāwe talked about the truths of \(\mathcal {L}_{Tr}^{\lhd }\) within \(\mathcal {L}_{Tr}^{\lhd }\) itself.
Based on this observation, we argue that the semantic closure of \(\mathcal {L}_{Tr}^{\lhd }\) is (at least part of) the reason for why the puzzle of ground arises. Note that the semantic closure of \(\mathcal {L}_{Tr}^{\lhd }\) is required for the proof of Lemma 1. In the third step of the derivation, we applied the ground predicate to the truth predicate in \(\ulcorner \overline {0}=\overline {0}\urcorner \lhd \ulcorner Tr(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner .\) Moreover, in the fourth step, we inferred \(Tr(\ulcorner Tr(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner )\) from this and thus applied the truth predicate to a sentence containing the same truth predicate. The main difference between the liar paradox and the puzzle of ground is that, in the case of the liar, we get a truththeoretic inconsistency, i.e. an inconsistency with plausible principles for truth, while in the case of the puzzle, we get a groundtheoretic inconsistency, i.e. an inconsistency with plausible principles for partial ground. Still, the problematic sentences in both cases are quite similar. In both cases some intuitive form of selfreference is involved: while the liar sentence Ī» intuitively says something of itself, the principles of partial ground entail that truth of āx T r(x) partially grounds itself. Thus, we can say that the selfreference in the case of the liar is semantic, while the selfreference in the case of the puzzle of ground is groundtheoretic.^{Footnote 19}
In analogy to typed theories of truth, we propose a typed solution to the puzzle of ground. To formulate this solution, we will move to a slightly different framework, where instead of a single truth predicate T r, we have a family T r _{1},T r _{2},ā¦ of typed truth predicates. These truth predicates intuitively express truth on the first, second, ā¦ level of Tarskiās hierarchy. In the remainder of this section, we will develop a consistent theory of partial ground and typed truth using these typed truth predicates. This theory will contain typed versions of the axioms of PG plus typed versions of the Aristotelian principles. Much like in the case of typed theories of truth, this will mean that no sentence is provable in which the truth predicate is applied to a sentence containing the same truth predicate. We will show that this restriction is sufficient to obtain a consistent theory of partial ground and typed truth.
Axiomatic Theories of Partial Ground and Typed Truth
Typed theories of truth aim to axiomatize Tarskiās hierarchy of truths.^{Footnote 20} For this purpose, in typed theories of truth, we have different truth predicates for the different levels of the hierarchy. Correspondingly, we get a hierarchy of languages with a different language for every level of the hierarchy. To illustrate, we start with \(\mathcal {L}_{0}=_{def}\mathcal {L}\)āthe language of PA. The truth predicate for sentences of arithmetic is, then, T r _{1} and the language \(\mathcal {L}_{1}\) extends \(\mathcal {L}_{0}\) with T r _{1}. The truth predicate for sentences of \(\mathcal {L}_{1}\), in turn, is T r _{2} and the language \(\mathcal {L}_{2}\) extends \(\mathcal {L}_{1}\) with T r _{2}. And so on. Thus, typed theories of truth are formulated using a hierarchical family of truth predicates T r _{1},T r _{2},ā¦ that intuitively correspond to truth on the different levels of Tarskiās hierarchy.
Language and Background Theory
We will now formally define a hierarchy of languages, such that on every level we can talk about the truth of sentences on the lower levels. For reasons of generality, we will define this hierarchy in such a way that it includes even infinitary levels. Specifically, we assume that for every ordinal 0 < Ī± < š _{0}, we have a different truth predicate T _{ Ī± } that intuitively expresses truth at the level Ī±: we have \(Tr_{1}, \mathellipsis , Tr_{\omega }, \mathellipsis , Tr_{\omega ^{\omega }},\mathellipsis , Tr_{\omega ^{\omega ^{\omega }}}, \mathellipsis ,\) where for Ī±ā Ī² < š _{0}, we have T r _{ Ī± }ā T r _{ Ī² }.^{Footnote 21} For all ordinals 1 ā¤ Ī± ā¤ š _{0}, we define the language \(\mathcal {L}_{<\alpha }\) as the language \(\mathcal {L}\) of PA extended with all the truth predicates T _{ Ī² } for 0 < Ī² < Ī±:
Then we set:
for all ordinals 0 ā¤ Ī± < š _{0}. Thus, the language \(\mathcal {L}_{0}\) is \(\mathcal {L},\) the language \(\mathcal {L}_{1}\) is \(\mathcal {L}\cup \{Tr_{1}\}\), and so on. Intuitively, for an ordinal 1 ā¤ Ī± ā¤ š _{0}, the language \(\mathcal {L}_{<\alpha }\) talks about the truths at the levels strictly below Ī± and \(\mathcal {L}_{\alpha }\) talks about the truths at all levels up to and including Ī±. When we are operating on the ordinal level Ī±, the language \(\mathcal {L}_{<\alpha }\) is our intended objectlanguage, i.e. we wish to talk about grounding between the truths of sentences in \(\mathcal {L}_{<\alpha }\). For most informal purposes, however, we already stop at the level of \(\mathcal {L}_{<2}=\mathcal {L}\cup \{Tr_{1}\}\). The reason for this is that \(\mathcal {L}_{<2}\) is the first language in which grounding between arithmetic truths and truths involving a truth predicate occurs. For all ordinals 1 ā¤ Ī± ā¤ š _{0}, the language \(\mathcal {L}_{<\alpha }^{\lhd }\) is \(\mathcal {L}_{<\alpha }\) extended with our binary ground predicate \(\lhd \):
And we set:
for all ordinals 0 ā¤ Ī± < š _{0}. When we are operating on the ordinal level Ī±, weāll use \(\mathcal {L}_{\alpha }^{\lhd }\) as our metalanguage for the objectlanguage language \(\mathcal {L}_{<\alpha }\). Again, for expository purposes, weāll usually stop at \(\mathcal {L}_{2}^{\lhd }=\mathcal {L}_{2}\cup \{\lhd \},\) which is the first language in which we can talk about grounding in \(\mathcal {L}_{<2}=\mathcal {L}_{1}\).
For an ordinal 0 < Ī± < š _{0}, the theory P A T _{<Ī± } is the result of extending PA with all the instances of the induction scheme over the language \(\mathcal {L}_{<\alpha }\) and the theory P A G _{<Ī± } is the result of extending P A T _{<Ī± } with all the missing instances of the induction scheme over \(\mathcal {L}_{\alpha }^{\lhd }\). For 0 ā¤ Ī± < š _{0}, the theory P A T _{ Ī± }, then, is P A T _{<Ī±+1} and similarly P A G _{ Ī± } is P A G _{<Ī±+1}. Thus, P A T _{0} is PAT and P A G _{0} is PAG. In P A T _{ Ī± }, we can develop a syntax theory for the languages \(\mathcal {L}_{<\alpha }\) analogously to the way we developed the syntax theory in the first part of this paper. When we work on an ordinal level Ī±, we assume that in \(\mathcal {L}_{\alpha }^{\lhd },\) via some appropriate GĆ¶del coding, we have names \(\ulcorner \varphi \urcorner \) for all formulas \(\varphi \in \mathcal {L}_{<\alpha }\).^{Footnote 22} Moreover, we assume that for every 0 < Ī² < Ī±, we have a function symbol \(\underset {.}{Tr_{\beta }}\) that represents the function which maps the code # t of a term t to the code # T r _{ Ī² }(t) of the formula \(Tr_{\beta }(t)\in \mathcal {L}_{<\alpha }\). And we abbreviate the formula that allows us to (strongly) represent the (set of codes of) sentences in \(\mathcal {L}_{<\alpha }\) by S e n t _{<Ī± }.
Axioms for Partial Ground and Typed Truth
With the syntax in place, we extend our theory PG to account for partial ground between truths on the same level of Tarskiās hierarchy. Weāll define this extension from the perspective of some ordinal level 0 < Ī± < š _{0}. Thus, we wish to talk about ground between truths on all the ordinal levels 0 < Ī² < Ī±. To achieve this, we have to modify the basic ground axiom G _{3} and the basic truth axioms T _{1}, T _{2}, and T _{3}. The axiom G _{3} splits up in the following pair, for all ordinals 0 < Ī² < Ī±:

(G\(_{3a}^{\beta }\)) \(\forall x\forall y (x\lhd y\to (Sent_{<\beta }(x)\to Tr_{\beta }(x)))\)

(G\(_{3b}^{\beta }\)) \(\forall x\forall y (x\lhd y\to (Sent_{<\beta }(y)\to Tr_{\beta }(y)))\)
These axioms formalize the factivity of ground in a typed context. In particular, the axiom G\(_{3a}^{\beta }\) says that if the truth of some sentence grounds the truth of another, and if the sentence is below the level Ī² in the hierarchy, then it is true at level Ī² of the hierarchy. The axiom G\(_{3b}^{\beta },\) on the other hand, says the same thing the other way around: if the truth of some sentence is grounded in the truth of another, and if the former sentence is below level Ī², then it is true at level Ī². To illustrate, if we let Ī± = 2, we get the following axiom pair:

(G\(_{3a}^{1}\)) \(\forall x\forall y(x\lhd y\to (Sent_{<1}(x)\to Tr_{1}(x)))\)

(G\(_{3b}^{1}\)) \(\forall x\forall y(x\lhd y\to (Sent_{<1}(y)\to Tr_{1}(y)))\)
Thus, for sentences \(\varphi ,\psi \in \mathcal {L}_{<1},\) the axioms G\(_{3a}^{1}\) and G\(_{3b}^{1}\) together say that the truth of Ļ can only ground the truth of Ļ, if Ļ and Ļ are both true at level oneāi.e. if they are truths of arithmetic.
Next, in the typed versions of T _{1} and T _{2}, we wish to make sure that true equations are true at every level below Ī±. Thus, for all ordinals 0 < Ī² < Ī±, we postulate:

(T\(_{1}^{\beta }\)) \(\forall s\forall t(Tr_{\beta }(s\underset {.}{=}t)\leftrightarrow s^{\circ }=t^{\circ })\)

(T\(_{2}^{\beta }\)) \(\forall s\forall t(Tr_{\beta }(s\underset {.}{\neq }t)\leftrightarrow s^{\circ }\neq t^{\circ })\)
Thus, we get for example \(Tr_{1}(\ulcorner \overline {0}=\overline {0}\urcorner ), Tr_{2}(\ulcorner \overline {0}=\overline {0}\urcorner ), \mathellipsis \) and so on, for all ordinal levels below Ī±.
In the case of T _{3}, we wish to make sure that a truth predicate T r _{ Ī² }, for an ordinal 0 < Ī² < Ī±, only applies to sentences on levels below Ī²āin compliance with Tarskiās distinction. Thus, we postulate for all 0 < Ī² < Ī±:

(T\(_{3}^{\beta }\)) āx(T r _{ Ī² }(x) ā S e n t _{<Ī² }(x))
Thus, for example, we get āx(T r _{1}(x) ā S e n t _{<1}(x)), which intuitively says that the predicate T r _{1} only applies to sentences of arithmetic.
Another adjustment is needed: Note that now G\(_{3a/b}^{\beta }\) and T\(_{3}^{\beta }\) do not entail anymore that the ground predicate applies only to sentences. To ensure this, we postulate the following final basic ground axiom:

(G\(_{4}^{\alpha }\)) \(\forall x\forall y(x\lhd y\to Sent_{<\alpha }(x)\land Sent_{<\alpha }(y))\)
Thus, on the level Ī± = 2, we get that \(\forall x\forall y(x\lhd y\to Sent_{<2}(x)\land Sent_{<2}(y))\). In words: if \(x\lhd y,\) then both x and y are sentences of \(\mathcal {L}_{<2},\) which is just the language of arithmetic \(\mathcal {L}\) extended with the truth predicate T r _{1}. Taken together, all of these modified axioms entail that our new theory respects Tarskiās distinction between object and metalanguage. Finally, we have to modify our upwards and downwards directed grounding axioms to apply on all levels below Ī±. We achieve this by postulating that the axioms apply on all of these levels. Take the axioms U _{1} and D _{1} for example. They become the new typed set of axioms for all ordinals 0 < Ī² < Ī±:

(U\(_{1}^{\beta }\)) \(\forall x (Tr_{\beta }(x)\to x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\)

(D\(_{1}^{\beta }\)) \(\forall x(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\neg }x)\to x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\)
Thus, on every level 0 < Ī² < Ī±, if a sentence is true on that level, then the sentence grounds its doublenegation and if a double negation is true on the level, it is grounded by the sentence it is a double negation of.
Putting all of this together, we get:
Definition 1
For all ordinals 0 ā¤ Ī± ā¤ š _{0}, the predicational theory P G _{<Ī± } of ground up to Ī±, consists of the axioms of P A G _{<Ī± } plus the following axioms for all 0 < Ī² < Ī±:
Typed Ground Axioms: Typed Truth Axioms:
G_{1}
\(\forall x\neg (x\lhd x)\) T\(_{1}^{\beta } \forall s\forall t(Tr_{\beta }(s\underset {.}{=}t)\leftrightarrow s^{\circ }=t^{\circ })\)
G_{2}
\(\forall x\forall y\forall z(x\lhd y\land y\lhd z\to x\lhd z)\) T\(_{2}^{\beta } \forall s\forall t(Tr_{\beta }(s\underset {.}{\neq }t)\leftrightarrow s^{\circ }\neq t^{\circ })\)
G\(_{3a}^{\beta }\)
\(\forall x\forall y (x\lhd y\to (Sent_{<\beta }(x)\to Tr_{\beta }(x)))\) T\(_{3}^{\beta } \forall x(Tr_{\beta }(x)\to Sent_{<\beta }(x))\)
G\(_{3b}^{\beta }\)
\(\forall x\forall y (x\lhd y\to (Sent_{<\beta }(y)\to Tr_{\beta }(y)))\)
G\(_{4}^{\alpha }\)
\(\forall x\forall y(x\lhd y\to Sent_{<\alpha }(x)\land Sent_{<\alpha }(y))\)
Typed Upward Directed Axioms:

U\(_{1}^{\beta }\)

\(\forall x (Tr_{\beta }(x)\to x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\)

U\(_{2}^{\beta }\)

\(\forall x\forall y (Tr_{\beta }(x)\to x\lhd x\underset {.}{\lor }y \land Tr_{\beta }(y)\to y\lhd x\underset {.}{\lor }y)\)

U\(_{3}^{\beta }\)

\(\forall x\forall y (Tr_{\beta }(x)\land Tr_{\beta }(y)\to (x\lhd x\underset {.}{\land } y)\land (y\lhd x\underset {.}{\land } y))\)

U\(_{4}^{\beta }\)

\(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg } x)\land Tr_{\beta }(\underset {.}{\neg } y)\to (\underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (\underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)

U\(_{5}^{\beta }\)

\(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }(x\underset {.}{\land }y) \land Tr_{\beta }(\underset {.}{\neg }y)\to \underset {.}{\neg }y\lhd \underset {.}{\neg }(x\underset {.}{\land }y))\)

U\(_{6}^{\beta }\)

\(\forall x \forall t\forall v(Tr_{\beta }(x(t/v))\to x(t/v)\lhd \underset {.}{\exists } v x)\)

U\(_{7}^{\beta }\)

\(\forall x\forall v (\forall tTr_{\beta }(\underset {.}{\neg } x(t/v))\to \forall t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\exists }v x))\)

U\(_{8}^{\beta }\)

\(\forall x \forall v(\forall t(Tr_{\beta }(x(t/v))\to \forall t(x(t/v)\lhd \underset {.}{\forall }v x))\)

U\(_{9}^{\beta }\)

\(\forall x \forall t\forall v(Tr_{\beta }(\underset {.}{\neg }x(t/v))\to \underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\forall } v x))\)

Typed Downward Directed Axioms:
D_{1} \(\forall x(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\neg }x)\to x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\)

D_{2} \(\forall x\forall y (Tr_{\beta }(x\underset {.}{\lor }y)\to (Tr_{\beta }(x)\to x\lhd x\underset {.}{\lor }y )\land (Tr_{\beta }(y)\to y\lhd x\underset {.}{\lor }y))\)

D_{3} \(\forall x\forall y (Tr_{\beta }(x\underset {.}{\land }y)\to (x\lhd x\underset {.}{\land } y)\land (y\lhd x\underset {.}{\land } y))\)

D_{4} \(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg } (x\underset {.}{\land } y))\to (Tr_{\beta }(\underset {.}{\neg }x)\to \underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (Tr_{\beta }(\underset {.}{\neg }y)\to \underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)

D_{5} \(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg } (x\underset {.}{\lor } y))\to (\underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (\underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)

D_{6} \(\forall x (Tr_{\beta }(\underset {.}{\exists } v x(v))\to \exists t(x(t/v)\lhd \underset {.}{\exists } v x))\)

D_{7} \(\forall x \forall v(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\exists }v x)\to \forall t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\exists }v x))\)

D_{8} \(\forall x \forall v(Tr_{\beta }(\underset {.}{\forall }v x\to \forall t(x(t/v)\lhd \underset {.}{\forall }v x))\)

D_{9} \(\forall x \forall v(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\forall } v x)\to \exists t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\forall } v x))\)
For 0 ā¤ Ī± < š _{0}, we define P G _{ Ī± } as P G _{<Ī±+1}.
To illustrate what P G _{ Ī± } looks like for different Ī±ās, letās consider at a few examples. First, note that P G _{0} is PAG. Next, note P G _{1} is a functional analog of our original theory P G, where the truthpredicate has been ārenamedā T r _{1}. In particular, we get that P G _{1} proves the theory PT of positive truth.
Since for all 1 < Ī± < š _{0}, P G _{ Ī± } contains P G _{1}, we can say that P G _{ Ī± } essentially is (in the precise sense sketched above) an extension of PG. For Ī± bigger than one, P G _{ Ī± } essentially consists of Ī±many copies of P G, one for every \(\mathcal {L}_{\beta }\) and truth predicate T r _{ Ī² }, where 1 < Ī² < Ī±. What is new in those theories is that now (names of) sentences involving the truth predicate may occur in the context of the ground predicate and other truth predicatesāas long as we respect the typing restriction that for all 0 < Ī² ā¤ Ī±, if \(Tr_{\beta }(\ulcorner \varphi \urcorner ),\) then \(Sent_{<\beta }(\ulcorner \varphi \urcorner )\). For example, in P G _{2}, we get the following instance of U\(_{1}^{2}\):
Indeed, using G\(_{3b}^{2}\) we can infer from this that:
which together with the previous formula gives us:
The other direction:
can be shown analogously using D\(_{1}^{2}\) and G\(_{3a}^{2}\). Generalizing this idea, we get more substantial truththeoretic theorems in P G _{2}, such as:
for example. But so far, P G _{2} does not allow us to prove any theorems of the form \(Tr_{2}(\ulcorner Tr_{1}(\ulcorner \varphi \urcorner )\urcorner ),\) where \(\varphi \in \mathcal {L}\). In other words, we canāt prove the truth of any sentence involving a truth predicateāeven if they respect the typing restrictions. Thus, P G _{2} is not really a theory of truth at level 2 yetāit canāt even show that \(Tr_{2}(\ulcorner Tr_{1}(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner )\). Moreover, in P G _{2}, we canāt prove any theorems of the form \(\ulcorner \varphi \urcorner \lhd \ulcorner Tr_{1}(\ulcorner \varphi \urcorner )\urcorner \) or the like, where the ground predicate applies to sentence involving a truth predicate. This doesnāt change on any level Ī± > 2. To get a more substantial theory of ground and partial truth, we need to say something about the grounds of truths involving the truth predicate: we need typed versions of the Aristotelian principles.
Typing the Aristotelian principles for use on an ordinal level Ī± is pretty straightforward. We get the following axioms for every Ī³ < Ī±:

(APU\(_{T}^{\gamma }\)) \(\forall x(Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)

(APU\(_{F}^{\gamma }\)) \(\forall x(Tr_{\gamma }(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
The axiom APU\(_{T}^{1},\) for example, allows us to derive that \(\ulcorner \overline {0}=\overline {0}\urcorner \lhd \ulcorner Tr_{1}(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner \) using the fact that by axiom T\(_{1}^{1}\) we have \(Tr_{1}(\ulcorner \overline {0}=\overline {0}\urcorner )\). The axioms APU\(_{T/F}^{\beta }\) are upwards directed axioms. For analogous reasons as in the case of the other ground axioms, we also need downward directed axioms for the Aristotelian principles. Again, straightforwardly, we get for all Ī³ < Ī² ā¤ Ī±:

(APD\(_{T}^{\beta ,\gamma }\)) \(\forall x(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)

(APD\(_{F}^{\beta ,\gamma }\)) \(\forall x(Tr_{\beta }(\underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
If we add the upward and downward directed versions of the Aristotelian principles to the previous theory, we arrive at our theory of ground and typed truth:
Definition 2
For every ordinal 0 ā¤ Ī± < š _{0}, the theory P G A _{ Ī± } of partial ground with the Aristotelian principles up to Ī± consists of the axioms of P G _{ Ī± } plus the following axioms for all Ī³ < Ī² ā¤ Ī±:
Upward Directed Aristotelian Principles:

(APU\(_{T}^{\gamma }\)) \(\forall x(Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)

(APU\(_{F}^{\gamma }\)) \(\forall x(Tr_{\gamma }(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
Downward Directed Aristotelian Principles:

(APD\(_{T}^{\beta ,\gamma }\)) \(\forall x(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)

(APD\(_{F}^{\beta ,\gamma }\)) \(\forall x(Tr_{\beta }(\underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
The theory P G A _{<Ī± } is defined as \(\bigcup _{\beta <\alpha }PGA_{\alpha },\) for all 0 < Ī± ā¤ š _{0}.
To see how P G A _{ Ī± } works for different 0 ā¤ Ī± < š _{0}, letās consider again a few examples. First note that P G A _{0} is P G _{0} which is just PAG. Similarly, P G A _{1} is P G _{1}. Things get interesting at the level P G A _{2}. Here we get:
by instantiating the axiom \(AP{U_{T}^{1}}\) with the term \(\ulcorner \overline {0}=\overline {0}\urcorner \). Moreover, by instantiating the axiom T\(_{1}^{1}\) with the same term, we have:
So putting the two together, we get:
Now, using the instance:
of the axiom G\(_{3b}^{2}\), and since:
is derivable in PA, we can infer:
So, in P G A _{2}, we can indeed derive applications of the truth predicate to sentences with a truth predicate in them. Moreover, by putting \(AP{U_{T}^{1}}\):
and \(APD_{T}^{2,1}\):
together, we can actually prove:
using the axioms G\(_{3a/b}^{1}\) and T\(_{1/2}^{1}\).^{Footnote 23} Thus, P G A _{2} proves intuitive truths at level two, such as \(Tr_{2}(\ulcorner Tr_{1}(\ulcorner \overline {0}=\overline {0}\urcorner )\urcorner )\), as well as quite substantial truththeoretic principles, such as \(\forall x(Tr_{2}(\underset {.}{Tr_{1}}(\dot {x}))\leftrightarrow Tr_{1}(x))\). In other words, P G _{2} proves something that looks like a substantial theory of truth at level two of Tarskiās hierarchy. In the next section, we will show that for 0 < Ī± < š _{0}, P G A _{ Ī± } proves the theory P R T _{ Ī± } of positive ramified truth up to Ī±. Indeed, we can show that P G A _{ Ī± } is a prooftheoretically conservative extension of P R T _{ Ī± }.
Conservativity and Models
The theory P T _{<Ī± } of positive ramified truth up to an ordinal level 1 ā¤ Ī± ā¤ š _{0} is formulated in the language \(\mathcal {L}_{<\alpha }\) and it is the result of modifying the theory of typed truth with the typed versions of its axioms in a similar way as we developed P G _{<Ī± }:
Definition 3 (āPositive Ramified Truthā)
For all ordinals 1 ā¤ Ī± ā¤ _{0}, the theory P R T _{<Ī± } of positive ramified truth up to Ī± consists of the axioms of P A T _{<Ī± } plus the following axioms for all Ī³ < Ī² < Ī±:

Typed Truth Axioms:
T\(_{1}^{\beta }\)
\(\forall s\forall t(Tr_{\beta }(s\underset {.}{=}t)\leftrightarrow s^{\circ }=t^{\circ })\)
T\(_{2}^{\beta }\)
\(\forall s\forall t(Tr_{\beta }(s\underset {.}{\neq }t)\leftrightarrow s^{\circ }\neq t^{\circ })\)
T\(_{3}^{\beta }\)
āx(T r _{ Ī² }(x) ā S e n t _{<Ī² }(x))

Positive Ramified Truth Axioms:
RP\(^{\beta }_{1}\)
\(\forall x(Tr_{\beta }(x)\leftrightarrow Tr_{\beta }(\underset {.}{\neg }\underset {.}{\neg }x))\)
RP\(^{\beta }_{2}\)
\(\forall x\forall y (Tr_{\beta }(x\underset {.}{\land }y)\leftrightarrow Tr_{\beta }(x)\land Tr_{\beta }(y))\)
RP\(^{\beta }_{3}\)
\(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg }(x\underset {.}{\land }y))\leftrightarrow Tr_{\beta }(\underset {.}{\neg }x\lor T_{\beta }\underset {.}{\neg }y))\)
RP\(^{\beta }_{4}\)
\(\forall x\forall y (Tr_{\beta }(x)\underset {.}{\lor }Tr_{\beta }(y)\leftrightarrow Tr_{\beta }(x)\lor Tr_{\beta }(y))\)
RP\(^{\beta }_{5}\)
\(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg }(x\underset {.}{\lor }y))\leftrightarrow Tr_{\beta }(\underset {.}{\neg }x)\land Tr_{\beta }(\underset {.}{\neg }y))\)
RP\(^{\beta }_{6}\)
\(\forall x \forall v(Tr_{\beta }(\underset {.}{\forall }vx) \leftrightarrow \forall t Tr_{\beta }(x(t/v)))\)
RP\(^{\beta }_{7}\)
\(\forall x \forall v(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\forall }vx) \leftrightarrow \exists t Tr_{\beta }(\underset {.}{\neg } x(t/v)))\)
RP\(^{\beta }_{8}\)
\(\forall x \forall v(Tr_{\beta }(\underset {.}{\exists }vx) \leftrightarrow \exists t Tr_{\beta }(x(t/v)))\)
RP\(^{\beta }_{9}\)
\(\forall x \forall v(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\exists }vx) \leftrightarrow \forall t Tr_{\beta }(\underset {.}{\neg }x(t/v)))\)
RP\(_{10}^{\beta }\)
\(\forall x(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\gamma }(x))\)
RP\(_{11}^{\beta }\)
\(\forall x(Tr_{\beta }(\underset {.}{\neg }\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\gamma }(\underset {.}{\neg }x))\)
RP\(_{12}^{\gamma ,\beta }\)
\(\forall x(Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(x)))\)
RP\(_{13}^{\gamma ,\beta }\)
\(\forall x(Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{\neg }\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(\underset {.}{\neg }x)))\)
The theory P R T _{ Ī± }, for 0 ā¤ Ī± < š _{0}, is defined as P R T _{<Ī±+1}.
Note that the theory P R T _{1} is a functional analog of PT in the same way that P G _{1} is a functional analog of PG. The theory P R T _{<Ī± }, for 1 < Ī± ā¤ š _{0}, however, is a much stronger theory of truth than PTāit formalizes the Tarskian hierarchy up to the level Ī±.^{Footnote 24} For example, P G T _{2} contains the axioms:
and
just like P G A _{2}. Indeed, we get:
Proposition 1
For all ordinals 1 ā¤ Ī± ā¤ š _{0},the theory P G A _{<Ī± } proves the theory P R T _{<Ī± } : \(PGA_{<\alpha }\vdash PRT_{<\alpha }\) .
Proof
In large parts, the proof is analogous to the proof of Proposition 4.6 of the first part of this paper: we carry out the same argument for all Ī² < Ī±. The interesting cases are the new axioms \(RP_{1013}^{\beta },\) for Ī² < Ī±, which can be shown from the typed Aristotelian principles \(APU^{\gamma }_{T/F}\) and \(APD_{T/F}^{\beta ,\gamma },\) for Ī³ < Ī² ā¤ Ī±, the typed truth axiom \(T_{3}^{\beta },\) and the typed ground axioms \(G_{3a/b}^{\beta },\) for Ī² < Ī±, and \(G_{4}^{\alpha }\). Here we only show how to derive \(RP_{10}^{\beta }\) and \(RP_{11}^{\beta },\) as the other axioms are analogous:

\(\vdash _{PGA_{<\alpha }}\forall x(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\gamma }(x))\) for Ī³ < Ī² < Ī±
Let x be a fresh variable for āIntro. We now prove both directions of the biconditional \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\gamma }(x)\).(\(\Rightarrow \)): Assume (ā) \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(x))\) for a āIntro. By \(T_{3}^{\beta },\) we can derive \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}(x))\). Using PA and \(T_{3}^{\gamma },\) we can derive (āā) S e n t _{<Ī³ }(x) from this. Moreover, using (ā) and \(APD_{T}^{\beta ,\gamma }\):
$$\forall x(Tr_{\beta}(\underset{.}{Tr_{\gamma}}(\dot{x}))\to x\lhd\underset{.}{Tr_{\gamma}}(\dot{x})),$$we can derive \(x\lhd \underset {.}{Tr_{\gamma }}(\dot {x})\). Using (āā) and \(G_{4}^{\gamma }\):
$$\forall x\forall y(x\lhd y\to (Sent_{<\gamma}(x)\to Tr_{\gamma}(x))),$$we can in turn derive: T r _{ Ī³ }(x). Thus, we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(x))\to Tr_{\gamma }(x)\) by āIntro.
(\(\Leftarrow \)): Assume (ā”) T r _{ Ī³ }(x) for another āIntro. Using PA and \(G_{4}^{\gamma },\) we get S e n t _{<Ī³ }(x). From this and PA, we can derive for all Ī³ < Ī² < Ī± that (ā”ā”) \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\). Moreover, using (ā”) and \(APU_{T}^{\gamma }\):
$$\forall x(Tr_{\gamma}(x)\to x\lhd\underset{.}{Tr_{\gamma}}(\dot{x})),$$we get \(x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\). From this, using (ā”ā”) and \(G_{3b}^{\beta }\), we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(x))\) and thus \(Tr_{\gamma }(x)\to Tr_{\beta }(\underset {.}{Tr_{\gamma }}(x))\) by āIntro.
Putting both ā\(\Rightarrow \)ā and ā\(\Leftarrow \)ā together, we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\gamma }(x)\) by \(\leftrightarrow \)Intro. And, since x was a fresh variable, we can derive:
$$\forall x (Tr_{\beta}(\underset{.}{Tr_{\gamma}}\dot{x})\leftrightarrow Tr_{\gamma}(x))$$by āIntro as desired.

\(\vdash _{PGA_{<\alpha }}\forall x(Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(x)))\) for Ī³ < Ī² < Ī±.
Let x be a fresh variable for āIntro. Assume S e n t _{<Ī³ }(x) for a āIntro. We now prove both directions of the biconditional \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(x)\).(\(\Rightarrow \)): Assume (ā ) \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\) for āIntro. From this, T\(_{3}^{\beta },\) and PA, we can derive \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\) and thus also (ā ā ) S e n t _{<Ī² }(x). As before, we get \(x\lhd \underset {.}{Tr_{\gamma }}\dot {x}\) using \(APD_{T}^{\beta ,\gamma }\). Using (ā ā ) and \(G_{3a}^{\beta },\) we can derive T r _{ Ī² }(x). Thus, we have \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\to Tr_{\beta }(x)\) by āIntro.(\(\Leftarrow \)): Assume T r _{ Ī² }(x) for yet another āIntro. From this and \(APU_{T}^{\beta },\) we get \(x\lhd Tr_{\beta }(x)\). Since we have assumed S e n t _{<Ī³ }(x), we get T r _{ Ī³ }(x) from this and \(G_{3a}^{\gamma }\). From this and \(APU_{T}^{\gamma },\) we get (Ā§) \(x\lhd \underset {.}{Tr_{\gamma }}(\dot {x})\). But now since Ī³ < Ī², we can show in PA that \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\). But from this and (Ā§), we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\). So, we have \(Tr_{\beta }(x)\to Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\) by āIntro.
Now putting both ā\(\Rightarrow \)ā and ā\(\Leftarrow \)ā together, we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\leftrightarrow Tr_{\beta }(x)\) by \(\leftrightarrow \)Intro and so \(Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\leftrightarrow Tr_{\beta }(x))\) by āIntro. Finally, since x was a fresh variable, we have:
$$\forall x(Sent_{<\gamma}(x)\to (Tr_{\beta}(\underset{.}{Tr_{\gamma}}(\dot{x}))\leftrightarrow Tr_{\beta}(x))),$$as desired.
This has the immediate consequence that for all ordinals Ī² < Ī± < š _{0}, the theory P G A _{<Ī± } proves the following typed version of the Tscheme for all languages \(\mathcal {L}_{<\beta }\):
Lemma 3
For all ordinals 0 < Ī³ ā¤ Ī² < Ī± < š _{0} and for all sentences \(\varphi \in \mathcal {L}_{<\gamma }:\)
\(\vdash _{PGA_{<\alpha }} \forall t_{1}, \mathellipsis , \forall t_{n}(Tr_{\beta }(\ulcorner \varphi (\dot {t_{1}},\mathellipsis ,\dot (t_{n}))\urcorner )\leftrightarrow \varphi (t_{1},\mathellipsis , t_{n})).\)
Next, we will now show that for all ordinals 1 ā¤ Ī± < š _{0}, the theory P G A _{ Ī± } is a prooftheoretically conservative extension of the theory P R T _{ Ī± }. But first, we need to introduce some more technical preliminaries: It is wellknown that we can extend the technique of GĆ¶del numbering to get terms for all ordinals below š _{0} [17, p. 17ā42]. Letās denote the set of all ordinals below š _{0} by \(On_{<\epsilon _{0}}\). We can adjust our coding function \(\#:\mathcal {L}\to \mathbb {N}\) such that we injectively assign every ordinal \(\alpha \in On_{<\epsilon _{0}}\) a unique code \(\#\alpha \in \mathbb {N}\) that is different form all the codes # Ļ of the other expressions Ļ of \(\mathcal {L}\). For all \(\alpha \in On_{<\epsilon _{0}}\), we define the term \(\ulcorner \alpha \urcorner \) to be \(\overline {\#\alpha }\), i.e. our term for Ī± is the numeral of the code # Ī± of Ī±. Moreover, we extend the axioms of ordinary arithmetic to cover ordinal arithmetic up to š _{0}. For simplicity, weāll use the same terminology for ordinal arithmetic and ordinary arithmetic. Thus, for example, we can now write \(\ulcorner \alpha \urcorner \times \ulcorner \beta \urcorner \) in \(\mathcal {L}\) to denote the product of ordinals \(\alpha ,\beta \in On_{<\epsilon _{0}}\). Moreover, we get: \(\vdash _{PA}\ulcorner \alpha \urcorner \times \ulcorner \beta \urcorner =\ulcorner \gamma \urcorner \) iff Ī± Ć Ī² = Ī³, for all ordinals \(\alpha ,\beta ,\gamma \in On_{<\epsilon _{0}}\). PA can represent the set of codes of ordinals below š _{0} and weāll use \(\underset {.}{On_{<\epsilon _{0}}}\) as a predicate for this. In particular, we get for all natural numbers \(n\in \mathbb {N}\): \(\vdash _{PA} \underset {.}{On_{<\epsilon _{0}}}(\overline {n})\) iff \(n\in \#On_{<\epsilon _{0}}=\{\#\alpha ~~\alpha \in On_{<\epsilon _{0}}\}.\) Finally, PA can represent the standard wellordering < of the ordinals below š _{0} and weāll use the relation symbol \(\underset {.}{<}\) to represent this ordering. So we get that for all ordinals \(\alpha ,\beta \in On_{<\epsilon _{0}}\): \(\vdash _{PA}\ulcorner \alpha \urcorner \underset {.}{<}\ulcorner \beta \urcorner \) iff Ī± < Ī². With these preliminaries in place,^{Footnote 25} weāll define a slightly nonstandard notion of complexity for the formulas in \(\mathcal {L}_{<\epsilon _{0}}\):
Definition 4 (ā Ļcomplexityā)
For all ordinals 1 ā¤ Ī± ā¤ š _{0}, we define the function \(\phantom {\varphi }_{\omega }:\mathcal {L}_{<\alpha }\to On_{<\epsilon _{0}}\) that assigns to every formula \(\varphi \in \mathcal {L}_{\alpha }\) its Ļcomplexity Ļ_{ Ļ } recursively by saying that:

(i)
\(\varphi _{\omega }=\left \{\begin {array}{ll} \omega \times \alpha & \text {if}~\varphi =Tr_{\alpha }(t)\\ 0 & \text {if}~\varphi ~\text {is another atomic formula} \end {array}\right .\)

(ii)
Ā¬Ļ_{ Ļ } = Ļ_{ Ļ } + 1;

(iii)
Ļ ā Ļ_{ Ļ } = l u b(Ļ_{ Ļ },Ļ_{ Ļ }) + 1, for ā = ā§,āØ;^{Footnote 26} and

(iv)
Q x Ļ_{ Ļ } = Ļ_{ Ļ } + 1, for Q = ā,ā.
Note that Ļcomplexity agrees with ordinary complexity on the formulas of \(\mathcal {L}_{<1}\). Note furthermore that the function xā¦Ļ Ć x is strictly monotonically increasing on the ordinals below š _{0}:
Lemma 4
For all \(\alpha ,\beta \in On_{<\epsilon _{0}}\) , if Ī± < Ī²,then Ļ Ć Ī± < Ļ Ć Ī² .
Note that as a consequence, we get that for all ordinals 0 < Ī± < š _{0}, if \(\varphi \in \mathcal {L}_{<\alpha },\) then \(\varphi _{\omega }<Tr_{\alpha }(\ulcorner \varphi \urcorner )_{\omega }\). In other words, Ļcomplexity has a sort of ātracking property:ā it can ātrackā the levels of Tarskiās hierarchy. Moreover, we can represent Ļcomplexity in PA. More specifically, the function \(c_{\omega }:\#\mathcal {L}\to \mathbb {N}\) that maps the code # Ļ of a formula \(\varphi \in \mathcal {L}\) to its Ļcomplexity Ļ_{ Ļ } is recursive and thus representable in PA. We represent c _{ Ļ } by the unary function symbol \(\underset {.}{c_{\omega }}\). Thus, we get for all \(\varphi \in \mathcal {L}_{<\epsilon _{0}}\) and all \(\alpha \in On_{<\epsilon _{0}}\): \(\vdash _{PA}\underset {.}{c_{\omega }}(\ulcorner \varphi \urcorner )=\ulcorner \alpha \urcorner \) iff Ļ_{ Ļ } = Ī±. Using this representation, we can show that Peano arithmetic proves that Ļcomplexity has the ātrackingpropertyā in the following sense:
Lemma 5
For all ordinals 0 < Ī² ā¤ Ī± < _{0} :
^{Footnote 27}
Using Ļcomplexity, weāll obtain the main result of this section:
Theorem 1
For all 0 ā¤ Ī± < š _{0},the theory P G A _{ Ī± } is a prooftheoretically conservative extension of the theory P R T _{ Ī± } .
Proof
First, we define the translation function \(\tau :\mathcal {L}_{\alpha }^{\lhd }\to \mathcal {L}_{\alpha }\) by saying that:

\(\tau (\varphi )=\left \{\begin {array}{ll} Tr_{\alpha }(s)\land Tr_{\alpha }(t)\land \underset {.}{c_{\omega }}(s)\underset {.}{<}\underset {.}{c_{\omega }}(t) & \text {if}~\varphi =s\lhd t\\\varphi & \text {if}~\varphi ~\text {is another atomic formula}\end {array}\right .\)

Ļ(Ā¬Ļ) = Ā¬Ļ(Ļ);

Ļ(Ļ ā Ļ) = Ļ(Ļ) ā Ļ(Ļ), for ā = ā§,āØ; and

Ļ(Q x Ļ) = Q x(Ļ(Ļ)), for Q = ā,ā.
Then, we note that (a) for all \(\varphi \in \mathcal {L}_{\alpha }\), Ļ(Ļ) = Ļ. Next, we check that (b) for all \(\varphi \in \mathcal {L}_{\alpha }^{\lhd },\) if \(\vdash _{PGA_{\alpha }}\varphi ,\) then \(\vdash _{PRT_{\alpha }}\tau (\varphi )\). The typed truth axioms of P G A _{ Ī± } are also axioms of P R T _{ Ī± }, so we only need to check the typed ground axioms and the typed upward and downward directed axioms. Here we only go through a few cases to illustrate the idea:

In the case of the axiom \(G_{4}^{\alpha },\) we get:
$$\tau(G_{4}^{\alpha})=\forall x\forall y((Tr_{\alpha}(x)\land Tr_{\alpha}(y)\land \underset{.}{c_{\omega}}(x)\underset{.}{<}\underset{.}{c_{\omega}}(y))\to Sent_{<\alpha}(x)\land Sent_{<\alpha}(y))$$This is provable (almost) immediately from the typed truth axiom T\(_{3}^{\alpha }\) of P R T _{ Ī± }:
$$\forall x(Tr_{\alpha}(x)\to Sent_{<\alpha}(x)).$$ 
Finally, consider the axioms (APU\(_{T}^{\beta }\)):
$$\forall x(Tr_{\beta}(x)\to x\lhd\underset{.}{Tr_{\beta}}(\dot{x})),$$where Ī² < Ī±. We get:
$$\tau(APU_{T}^{\beta})=\forall x(Tr_{\beta}(x)\to Tr_{\alpha}(x)\land Tr_{\alpha}(\underset{.}{Tr_{\beta}}(\dot{x}))\land \underset{.}{c_{\omega}}(x)\underset{.}{<} \underset{.}{c_{\omega}}(\underset{.}{Tr_{\beta}}(\dot{x}))).$$Now let x be a fresh variable for āIntro and assume T r _{ Ī² }(x) for a āIntro. Using the axiom \(T_{3}^{\beta }\) of P R T _{ Ī± }, we can infer that S e n t _{<Ī² }(x). Moreover, since Ī² < Ī± by assumption, we can infer that \(Tr_{\alpha }(\underset {.}{Tr_{\beta }}(\dot {x}))\) and T r _{ Ī² }(x) using the axiom \(RP_{12}^{\alpha }\) of P R T _{ Ī± }. Finally, by Lemma 5, we get \(Sent_{<\beta }(x) \to \underset {.}{c_{\omega }}(x)\underset {.}{<}\underset {.}{Tr_{\beta }}(\dot {x})\). Since we know already that S e n t _{<Ī² }(x), we get the final piece \(\underset {.}{c_{\omega }}(x)\underset {.}{<}\underset {.}{c_{\omega }}\underset {.}{Tr_{\beta }}(\dot {x})\). Putting all of this together, by āIntro, we have
$$Tr_{\beta}(x)\to Tr_{\alpha}(x)\land Tr_{\alpha}(\underset{.}{Tr_{\beta}}(\dot{x}))\land \underset{.}{c_{\omega}}(x)\underset{.}{<} \underset{.}{c_{\omega}}(\underset{.}{Tr_{\beta}}(\dot{x})),$$and since x was a fresh variable, by āIntro, we get the desired theorem.
Putting (a) and (b) together, the claim follows. ā”
The theorem has the following immediate consequence:^{Footnote 28}
Corollary 1
For all 0 ā¤ Ī± < š _{0},the theory P G A _{ Ī± } is consistent.
The proof of Theorem 1 essentially works because of the ātracking propertyā of Ļcomplexity. The idea of the proof is the same as in the proof of the corresponding result in the first part of this paper, but the translation we used there would not work here. Sentences of the form \(Tr_{\beta }(\ulcorner \varphi \urcorner )\) involving the truth predicate all have a classical complexity of zero, while the sentence Ļ may have arbitrary complexity. Thus, we would not be able to derive the translations of the (typed versions of the) Aristotelian principles under the translation from the previous paper. The trick is to use Ļcomplexity in the translationāthis is what allowed us to prove the result. The technique of the proof works for all ordinals Ī± < š _{0}, since PA can represent the wellordering of these ordinals, which is required for the proof. The theory \(PGA_{<\epsilon _{0}}\) is the first theory where our proof doesnāt work anymore, because in this theory we donāt have a āhighestā truth predicate as required for the definition of Ļ. But we can extend our result to this theory using a simple compactness argument:
Corollary 2
The theory \(PGA_{<\epsilon _{0}}\) is a prooftheoretically conservative extension of the theory \(PRT_{<\epsilon _{0}}\) .
Proof
Assume that there is a sentence \(\varphi \in \mathcal {L}_{<\epsilon _{0}}\) such that \(\vdash _{PGA_{<\epsilon _{0}}}\varphi ,\) but \(\not \vdash _{PRT_{<\epsilon _{0}}}\varphi \). Since proofs are finite objects, there can only be finitely many occurrences of different truth predicates \(Tr_{\beta _{1}},\mathellipsis , Tr_{\beta _{n}},\) for 0 < Ī² _{1} < ā¦ < Ī² _{ n } < š _{0}, in the proof. But then the proof of Ļ, is also a proof in \(PGA_{\beta _{n}}\) and \(\varphi \in \mathcal {L}_{\beta _{n}}\). Now by Theorem 1, \(PGA_{\beta _{n}}\) is conservative over \(PRT_{\beta _{n}}\). This means that \(\vdash _{PRT_{\beta _{n}}}\varphi \) and thus also \(\vdash _{PRT_{<\epsilon _{0}}}\varphi \). Contradiction! Thus, there is no such Ļ and the claim holds. ā”
We get immediately:
Corollary 3
The theory \(PGA_{<\epsilon _{0}}\) is consistent.
The theory \(PGA_{<\epsilon _{0}}\) is a natural stopping point for the methods weāve developed in this paper.^{Footnote 29}
We have shown the consistency of our theories P G A _{<Ī± }, where 1 ā¤ Ī± ā¤ š _{0}, by proof theoretic means. But for reasons of perspicuity, it would also be good to have an idea what models for these theories look like. In the rest of this section, we will show how to extend the construction from the previous paper to obtain models for P G A _{<Ī± }, where 1 ā¤ Ī± ā¤ š _{0}.
As in the case of P T, there is a standard model of P R T _{<Ī± }, for 1 ā¤ Ī± ā¤ š _{0}. A model for the language \(\mathcal {L}_{<\alpha }\) is a structure of the form (N,(S _{ Ī² })_{ Ī²<Ī± }), where for Ī² < Ī±, the set S _{ Ī² } interprets the truth predicate \(Tr_{\beta }\in \mathcal {L}_{<\alpha }\). For 1 ā¤ Ī± ā¤ š _{0}, we define the sets (S _{ Ī² })_{ Ī²<Ī± } by the following (transfinite) recursion:

\(\mathbf {S}_{1}=\{\#\varphi ~~\varphi \in \mathcal {L}_{<1}, \mathbf {N}\vDash \varphi \};\)

\(\mathbf {S}_{\alpha +1}=\mathbf {S}_{\alpha }\cup \{\#\varphi ~~\varphi \in \mathcal {L}_{<\alpha }, (\mathbf {N},(\mathbf {S}_{\beta })_{\beta <\alpha })\vDash \varphi \}\)

\(\mathbf {S}_{\alpha }=\bigcup _{\beta <\alpha }\mathbf {S}_{\beta },\) if Ī± is a limit ordinal.
Then we get, for all 1 ā¤ Ī± ā¤ š _{0}, that \((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha })\vDash PRT_{<\alpha }\). For 1 ā¤ Ī± ā¤ š _{0}, the model (N,(S _{ Ī² })_{ Ī²<Ī± }) is the standard model of P R T _{<Ī± }āit models Tarskiās hierarchy of truths.
We now extend our definition of groundingtrees from the previous paper to groundingtrees over the standard model of P R T _{<Ī± }:
Definition 5
Let 1 ā¤ Ī± ā¤ Ļµ_{0} and let (N,(S _{ Ī² })_{ Ī²<Ī± }) be the standard model of P R T _{<Ī± }. We define the groundingtrees over (N,(S _{ Ī² })_{ Ī²<Ī± }) by the following clauses for all formulas \(\varphi \in \mathcal {L}_{<\alpha }\):

(i)
\(\#\varphi \in \bigcup _{\beta <\alpha }\mathbf {S}_{\beta }\), then # Ļ is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with # Ļ as its root;

(ii)
if is a groundingtree \(\mathcal {T}\) over (N,(S _{ Ī² })_{ Ī²<Ī± }) with # Ļ as its root, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬Ā¬Ļ as its root;

(iii)
if is a groundingtree \(\mathcal {T}\) over (N, (S _{ Ī² })_{ Ī²<Ī± }) with # Ļ as its root, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #(Ļ āØ Ļ) as its root;

(iv)
if is a groundingtree \(\mathcal {T}\) over (N, (S _{ Ī² })_{ Ī²<Ī± }) with # Ļ as its root, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #(Ļ āØ Ļ) as its root;

(v)
if are groundingtrees \(\mathcal {T}_{1},\mathcal {T}_{2}\) over (N, (S _{ Ī² })_{ Ī²<Ī± }) with # Ļ,# Ļ as their roots respectively, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #(Ļ ā§ Ļ) as its root;

(vi)
if is a groundingtree \(\mathcal {T}\) over (N, (S _{ Ī² })_{ Ī²<Ī± }) with # Ļ(t) as its root, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #āx Ļ(x) as its root;

(vii)
if , ā¦ are groundingtrees \(\mathcal {T}_{1},\mathcal {T}_{2}, \mathellipsis \) over (N,(S _{ Ī² })_{ Ī²<Ī± }) with # Ļ(t _{1}),# Ļ(t _{2}),ā¦ as their roots respectively, where t _{1},t _{2},ā¦ are all and only the terms of \(\mathcal {L}_{PA}\), then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #āx Ļ(x) as its root;

(viii)
if is a groundingtree \(\mathcal {T}\) over (N, (S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬Ļ as its root, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬(Ļ ā§ Ļ) as its root;

(ix)
if is a groundingtree \(\mathcal {T}\) over (N, (S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬Ļ as its root, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬(Ļ ā§ Ļ) as its root;

(x)
if are groundingtrees \(\mathcal {T}_{1},\mathcal {T}_{2}\) over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬Ļ,#Ā¬Ļ as their roots respectively, then is a groundingtree over (N, (S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬(Ļ āØ Ļ) as its root;

(xi)
if is a groundingtree \(\mathcal {T}\) over (N, (S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬Ļ(t) as its root, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬āx Ļ(x) as its root;

(xii)
if , ā¦ are groundingtrees \(\mathcal {T}_{1},\mathcal {T}_{2}, \mathellipsis \) over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬Ļ(t _{1}),#Ā¬Ļ(t _{2}),ā¦ as their roots respectively, where t _{1},t _{2},ā¦ are all and only the terms of \(\mathcal {L}_{PA}\), then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #āx Ļ(x) as its root;

(xiii)
if is a groundingtree \(\mathcal {T}\) over (N,(S _{ Ī² })_{ Ī²<Ī± }) with # Ļ as its root and # Ļ ā S _{ Ī² }, for Ī² < Ī±, then is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with \(\#Tr_{\beta }(\ulcorner \varphi \urcorner )\) as its root;

(xiv)
if is a groundingtree \(\mathcal {T}\) over (N,(S _{ Ī² })_{ Ī²<Ī± }) with #Ā¬Ļ as its root and # Ļ āS _{ Ī² }, for Ī² < Ī±, then is a groundingtree over (N, (S _{ Ī² })_{ Ī²<Ī± }) with \(\#\neg Tr_{\beta }(\ulcorner \varphi \urcorner )\) as its root;

(xv)
nothing else is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }).
Now, in contrast to groundingtrees over (N,S), groundingtrees over (N,(S _{ Ī² })_{ Ī²<Ī± }) can have an infinite height:
Definition 6
We define the height \(h(\mathcal {T})\) of a grounding tree over (N,(S _{ Ī² })_{ Ī²<Ī± }) by saying that:

(i)
all groundingtrees over (N,(S _{ Ī² })_{ Ī²<Ī± }) of the form # Ļ āS have height one;

(ii)
if \(\mathcal {T}\) is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) that is constructed from groundingtrees \(\mathcal {T}_{1}, \mathcal {T}_{2}, \mathellipsis \) over (N,(S _{ Ī² })_{ Ī²<Ī± }), then the height of \(\mathcal {T}\) is one plus the least upper bound of the heights of \(\mathcal {T}_{1}, \mathcal {T}_{2}, \mathellipsis \):
$$h(\mathcal{T})=lub\{h(\mathcal{T}_{1}), h(\mathcal{T}_{2}), \mathellipsis\}+1,$$where lub is the operation of taking the least upper bound.
We call a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± })degenerate iff it is of height one.
To see that there are groundingtrees of infinite height, let \(DN_{\overline {0}=\overline {0}}(x)\) represent the property of being an instance of \(\overline {0}=\overline {0}\) preceded by an even number of negations. Then it is easily checked that for all Ļ such that \(DN_{\overline {0}=\overline {0}}(\#\varphi ),\) there is a groundingtree of the form which has height \(\frac {n}{2}+3,\) where n is the number of negations in Ļ. A consequence of this is that the least upper bound of the heights of \(\mathcal {T}_{1}, \mathcal {T}_{2}, \mathellipsis \) in the groundingtree is at least Ļ and thus the height of this tree is at least Ļ + 1.^{Footnote 30}
Now, an important consequence of this observation is that we canāt use ordinary induction on the height of trees to prove claims about all groundingtrees. We need to use transfinite induction on the height of the groundingtrees. This doesnāt add any further complications, but to be explicit letās state the form of the principle that weāre going to use. Consider a property of groundingtrees. Then, if we can show that any degenerate groundingtree has the property and we can show that if we can show that assuming that all trees of a height smaller than a given tree have the property, then the tree itself has the property, it follows that all groundingtrees have the property. Note that in this form of the principle, the induction step also includes limit cases, where we consider a tree of the height of a limit ordinal and need to show that the tree has the property in question, given that all trees of a lower height have the property.
Analogously to the case of groundingtrees over (N,S), we can now show that groundingtrees over (N,(S _{ Ī² })_{ Ī²<Ī± }) are: (i) rooted graphs over \(\bigcup _{\beta <\alpha }\mathbf {S}_{\beta }\); (ii) indeed rooted trees over \(\bigcup _{\beta <\alpha }\mathbf {S}_{\beta }\), i.e. they donāt contain any cycles; and finally, (iii) transitive.
Lemma 6
Let 1 ā¤ Ī± ā¤ š _{0}, (N,(S _{ Ī² })_{ Ī²<Ī± })be the standard model of P R T _{<Ī± },and let \(\mathcal {T}\) be a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }). Then for all formulas \(\varphi \in \mathcal {L}_{<\alpha },\) if # Ļ is a vertex in \(\mathcal {T},\) then \(\#\varphi \in \bigcup _{\beta <\alpha }\mathbf {S}_{\beta }\) .
Proof
The new cases for clauses (xiii) and (xiv) follow by the fact that (N,(S _{ Ī² })_{ Ī²<Ī± }) is a model of P R T _{<Ī± }. ā”
Remember the notion of a code of a formula occurring below another code in a groundingtree over (N,S). We now adapt this notion to groundingtrees over (N,(S _{ Ī² })_{ Ī²<Ī± }) by recursively saying that, for all 1 ā¤ Ī± ā¤ š _{0}, no code of any formula occurs below the code of any other formula in a degenerate groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }), and if \(\mathcal {T}\) is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) that was constructed from groundingtrees \(\mathcal {T}_{1}, \mathcal {T}_{2},\mathellipsis \) over (N,(S _{ Ī² })_{ Ī²<Ī± }) according to the rules (iiāxvi) of Definition 5, then all occurrences of all formulas in \(\mathcal {T}_{1}, \mathcal {T}_{2},\mathellipsis \) occur below the root of \(\mathcal {T}\) in \(\mathcal {T}\).
Then we can show:
Lemma 7
Let 1 ā¤ Ī± ā¤ š _{0} and let (N,(S _{ Ī² })_{ Ī²<Ī± })be the standard model of P R T _{<Ī± } . If \(\mathcal {T}\) is a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± })with # Ļ as its root, for some formula \(\varphi \in \mathcal {L}_{<\alpha }\) . Then, all formulas \(\psi \in \mathcal {L}_{<\alpha }\) whose code # Ļ occurs below # Ļ in \(\mathcal {T}\) have a lower Ļ complexity than Ļ.
Lemma 8
Let 1 ā¤ Ī± ā¤ š _{0}, (N,(S _{ Ī² })_{ Ī²<Ī± })be the standard model of P R T _{<Ī± },and let \(\mathcal {T}\) be a groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }). Then between any two nodes # Ļ and # Ļ in \(\mathcal {T},\) for formulas \(\varphi ,\psi \in \mathcal {L}_{<\alpha },\) there is exactly one path.
Lemma 9
Let 1 ā¤ Ī± ā¤ š _{0} and let (N,(S _{ Ī² })_{ Ī²<Ī± })be the standard model of P R T _{<Ī± } . If there is a groundingtree \(\mathcal {T}_{1}\) over (N,(S _{ Ī² })_{ Ī²<Ī± })with # Ļ as its root and # Ļ _{1}, # Ļ _{2},ā¦as its leaves and there is groundingtree \(\mathcal {T}_{2}\) over (N,(S _{ Ī² })_{ Ī²<Ī± })with # Ļ, # Ļ _{1}, # Ļ _{2},ā¦as its leaves and # š as its root, then there is a groundingtree \(\mathcal {T}_{3}\) over (N,(S _{ Ī² })_{ Ī²<Ī± })with # Ļ _{1}, # Ļ _{2},ā¦,# Ļ _{1}, # Ļ _{2},ā¦as its leaves and # š as its root.
Finally, we define the standard model of P G A _{<Ī± } by saying that:
Definition 7
Let 1 ā¤ Ī± ā¤ š _{0} and let (N,(S _{ Ī² })_{ Ī²<Ī± }) be the standard model of P R T _{<Ī± }. We define the relation \(\mathbf {R}\subseteq \mathbb {N}^{2}\) by saying that for all \(n,m\in \mathbb {N}\), R(m,n) iff there is a nondegenerate groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }) with n as a leaf and m as its root.
Putting Lemmas 6, 8, and 9 together, we obtain:
Theorem 2
(N,(S _{ Ī² })_{ Ī²<Ī± },R)is a model of P G A _{<Ī± },for 1 ā¤ Ī± ā¤ š _{0} , i.e. \((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha }, \mathbf {R})\vDash PGA_{<\alpha }\)
Proof
By Lemmas 6, 8, and 9, groundingtrees over (N,(S _{ Ī² })_{ Ī²<Ī± }) behave appropriately and satisfy the basic ground axioms. Since (N,(S _{ Ī² })_{ Ī²<Ī± }) is a model of P R T _{<Ī± }, the typed truth axioms are satisfied. The only new cases are the axioms for the typed Aristotelian principles. Here we only show that \(APU_{T}^{\gamma },\) for Ī³ < Ī± holds:

\((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha }, \mathbf {R})\vDash \forall x (Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}\dot {x})\) for all Ī³ < Ī±.
Let Ļ be a variable assignment over (N,(S _{ Ī² })_{ Ī²<Ī± },R) and \({\sigma ^{\prime }}\) some xvariant of Ļ. Assume that \((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha }, \mathbf {R})\vDash _{\sigma ^{\prime }} Tr_{\gamma }(x)\). This means that \({\sigma ^{\prime }}(x)\in \mathbf {S}_{\gamma }\). Since \(\mathbf {S}_{\gamma }=\{\#\varphi ~~\varphi \in \mathcal {L}_{<\gamma },(\mathbf {N},(\mathbf {S}_{\delta })_{\delta <\gamma })\vDash \varphi \},\) we know that \({\sigma ^{\prime }}(x)=\#\varphi ,\) for some formula \(\varphi \in \mathcal {L}_{<\gamma }\). Now, # Ļ is a degenerate groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }). But then, by clause (xiii) of Definition 5, is a nondegenerate groundingtree over (N,(S _{ Ī² })_{ Ī²<Ī± }). Moreover, the root of this tree is \(\#Tr_{\gamma }(\ulcorner \varphi \urcorner )\) and its only leaf is # Ļ. Now consider \({\sigma ^{\prime }}(\underset {.}{Tr_{\gamma }}\dot {x})\). Since we know that \({\sigma ^{\prime }}(x)=\#\varphi \) and \(\underset {.}{Tr_{\gamma }}\) expresses the function that maps codes of formulas to the code of T r _{ Ī³ } applied to the formula, we know that \({\sigma ^{\prime }}(\underset {.}{Tr_{\gamma }}\dot {\ulcorner \varphi \urcorner })=\#Tr_{\gamma }(\ulcorner \varphi \urcorner )\). Thus, \(\mathbf {R}({\sigma ^{\prime }}(x), {\sigma ^{\prime }}(\underset {.}{Tr_{\gamma }}\dot {x}))\) meaning \(\vDash _{\sigma ^{\prime }} x\lhd \underset {.}{Tr_{\gamma }}\dot {x}\). And since Ļ was arbitrary, we get \((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha }, \mathbf {R})\vDash \forall x (Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}\dot {x}),\) as desired.
We can show analogously that the other axioms hold. ā”
Paradoxes of SelfReferential Ground
In the setting of PG (as well as P G A _{<Ī± } for 0 < Ī± < š _{0}), we can accommodate new nary predicates R by stipulating:
If we furthermore add a theory for the new predicate R to our background theory, the resulting theory will be a theory of partial ground over the hierarchy of truths of that extended background theory. For example, we could formulate theories of partial ground over the truths of analysis, of mereology, or even of settheory.^{Footnote 31} The resulting theory will then, of course, also prove all the instances of the Tscheme \(Tr(\ulcorner \varphi \urcorner )\leftrightarrow \varphi \) over sentences of the new language \(\mathcal {L}\cup \{R\}\). But this approach has limits. Of course, we canāt let R be our unary truth predicate T r itself. It is wellknown that if we affirm:
our theory will fall prey to the liar paradox and its ilk. It might be somewhat surprising to learn, however, that we also canāt affirm:
in the context of predicational theories of ground. This will be the main result of this section.^{Footnote 32} Weāll call the resulting problem the new puzzle of ground.
Letās move to a setting where we can affirm applications of the ground predicate to sentences involving the ground predicate. For this purpose, we have to make a couple of adjustments to our theory setup. In the following, let \(\mathcal {L}_{\lhd }\) be the language \(\mathcal {L}\cup \{\lhd \}\). Now first, we have to assume that we work in the context of a proper coding for \(\mathcal {L}_{\lhd }\). In particular, we now assume that we have a name \(\ulcorner \varphi \urcorner \) for every sentence \(\varphi \in \mathcal {L}_{\lhd }\). We assume that we have a function symbol \(\underset {.}{\lhd }\) such that:
for all terms s and t. And we let \(Sent_{\lhd }\) abbreviate the formula that allows us to represent the set (of codes of) sentences of \(\mathcal {L}_{\lhd }\). In particular, we assume that for all \(n\in \mathbb {N}\):
Second, we need to adjust the axiom T _{3} to:
which allows sentences involving the ground predicate to occur in the context of the truth predicate (and thus in the context of the ground predicate). We arrive at a modified predicational theory of ground:
Definition 8
The predicational theory PUG of untyped ground consists of the axioms of PG without the axiom T _{3}, plus the axiom \(T_3^{\lhd }\) and the axioms:
We will now show that PUG is inconsistent. To see this, first note that we can show in the same way as in the case of PG that PUG proves the uniform Tscheme for sentences involving the ground predicate:
Lemma 10
For all sentences \(\varphi \in \mathcal {L}_{\lhd },\)
Proof
By induction on the positive complexity of formulas. The new axioms \(T_{\lhd }^{1/2}\) take care of the new basecase. ā”
Note that this lemma doesnāt entail yet that PUG is inconsistent: the truth predicate T r is not in the language \(\mathcal {L}_{\lhd }\) and thus Lemma 10 doesnāt entail that weāre applying the truth predicate to sentences involving the same truth predicate. To see that PUG is inconsistent, we need to do some more work. Since we work in the context of a coding for \(\mathcal {L}_{\lhd },\) we can get the diagonal lemma for the language:
Lemma 11
For all formulas \(\varphi \in \mathcal {L}_{\lhd }\) with exactly one free variable, there is a formula \(\delta \in \mathcal {L}_{\lhd }\) such that
With these two lemmas in place, we can show our main result:
Theorem 3
PUG is inconsistent.
Proof
Let Ļ(x) be the formula \(\neg (x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\). By the diagonal lemma for \(\mathcal {L}_{\lhd }\), we know that there is a formula \(\delta \in \mathcal {L}_{\lhd }\) such that:
It is easily checked that we can now both prove Ī“ and Ā¬Ī“. ā”
We are left with yet another puzzle of ground: by letting the truth predicate (and thus the ground predicate) apply to truths involving the ground predicate, we made our intuitively plausible theory of ground inconsistent. But intuitively, we want to be able to talk about the truth of sentences involving the ground predicate. So what went wrong?
First, note that the new puzzle is different from Fineās puzzle of ground. Fineās puzzle consists in the fact that different intuitively plausible principles for partial ground and truth entail that the truth of some sentences partially ground themselvesāin contradiction to the irreflexivity of partial ground. Our puzzle, in contrast, consists in the fact that letting the truth predicate apply to the ground predicate makes our previously consistent principles of ground inconsistent.
Moreover, note that the use of double negation (and of the corresponding upward directed ground axiom) in the proof of Theorem 3 is dispensable. We could equally well have applied the diagonal lemma to the formula \(\neg (x\lhd x\underset {.}{\lor }x)\) or \(\neg (x\lhd x\underset {.}{\land }x)\) or ā¦ and we would still have gotten the same inconsistency result (using the corresponding upward directed ground axioms for these connectives). The point is that our paradox is not a paradox of double negation, or disjunction, or conjunction, or the likeāit has another source.
We argue that the paradox is a paradox of selfreference in the context of partial ground. In support of this claim, first note that all the different sentences that we could use for the proof of Theorem 3 have in common that they (provably) āsay of themselvesā that they violate certain principles of partial ground, and as a consequence they are inconsistent over our theory of untyped ground. Thus, the new puzzle of ground bears a strong resemblance to the classic semantic paradoxes. Indeed, we can make this analogy even more explicit in the following proposition. Letās define the predicate T r0= by the following explicit definition:
And letās define the predicate \(Tr_0^{\lhd }\) by the following explicit definition:
So, intuitively T r0= and \(Tr_0^{\lhd }\) are truth predicates for the literals of \(\mathcal {L}_{\lhd }\). We can then show the following proposition:
Proposition 2
PUG proves that the formula \(Tr_0^=(x)\lor Tr_0^{\lhd }(x)\lor \exists y(y\lhd x)\) satisfies the Tscheme for the formulas of \(\mathcal {L}_{\lhd },\) i.e. for all \(\varphi \in \mathcal {L}_{\lhd }\) :
Proof
By induction on the positive complexity of Ļ for both directions of the biconditional. The base cases for Ļ being s = t or sā t, for terms s and t, are covered by the basic truth axioms T _{1} and T _{2} and the definition of T r0=. Similarly, the base cases for Ļ being \(s\lhd t\) or \(s\ntriangleleft t,\) for terms s and t, are covered by the axioms \(T_{\lhd }^1\) and \(T_{\lhd }^2\) and the definition of \(Tr_0^{\lhd }\). The remaining cases are can be dealt with using Lemma 10. ā”
In other words, in PUG we can define a truth predicate for \(\mathcal {L}_{\lhd }\) that satisfies the Tscheme for \(\mathcal {L}_{\lhd }\). It follows by Tarskiās theorem of the undefinability of truth that PUG is inconsistent [20]. Our Theorem 3 is only a special case of this more general fact, as it were. This is the precise sense in which the ground predicate ābehaves too much like a truth predicateāāin other words: the new puzzle of ground is at heart a paradox of selfreference.
Conclusion
In this paper, we have provided a typed solution to the puzzle of ground. But there are a couple of worries that should be addressed: First, we might be worried that we used āa sledgehammer to crack a walnut:ā our use of typing up to š _{0} could be seen as a slight overkill to solve such a simple problem as the puzzle of ground. We would like to say two things in response to this worry: First, the use of typing up to š _{0} is not really necessary to formulate the solution. We could already have stopped at level two, where in the typed setting grounding between arithmetic truths and truths involving a truth predicate occurs for the first time: it follows from our results that the theory P G A _{2} is consistent and proves the following simple version of the typed Aristotelian principle for truth:
But there is no real reason to stop at level two. We have shown that the methods of the paper work up to the level š _{0}. Moreover, the result was an intuitive and mathematically nontrivial theory of partial ground and typed truth. Thus, the results of this paper support our plea for collaboration between groundtheorists and truththeorists.
Second, there is the worry that by āgoing typedā we have āthrown out the baby with the bathwater:ā There are several unproblematic selfreferential sentences that fall out of the scope of our typed theory. For example, by a simple application of the diagonal lemma to the formula T r(x), we get the truthteller sentence Ļ such that:
This sentence can consistently be added to a theory that proves the instance of the Tscheme for it. It might be argued that this shows that we should include it in our best theory of partial ground and truth. PA and the Tscheme donāt decide this sentence: we can both consistently affirm it and its negation (although, of course, not at the same time). Thus, it might be argued, a good theory of partial ground and truth should at least prove the following principle:
This sentence, however, is in none of the languages \(\mathcal {L}_{<\alpha }^{\lhd }\), for 0 < Ī± < š _{0}, and so in these languages we cannot say anything about its partial grounds. To make things worse, it could be argued that the principles of partial ground and truth should even apply to the liar sentence Ī» such that \(\vdash _{PA}\lambda \leftrightarrow \neg Tr(\ulcorner \lambda \urcorner )\). In particular, it might be argued that both
\(\lambda \to \ulcorner \lambda \urcorner \lhd \ulcorner \neg Tr(\ulcorner \lambda \urcorner )\urcorner \) and \(\neg \lambda \to \ulcorner \neg \lambda \urcorner \lhd \ulcorner Tr(\ulcorner \lambda \urcorner )\urcorner \) should be the provable. The argument would be that assuming that if Ī» is true, then since Ī» āsays of itselfā that it is not true, the truth of Ī» grounds the truth of \(\neg Tr(\ulcorner \lambda \urcorner )\) and analogously for Ā¬Ī». In the context of the factivity of ground, this would lead to a nonclassical theory of truth. If we have:
then we would immediately get \(Tr(\ulcorner \lambda \urcorner )\) and \(Tr(\ulcorner \neg \lambda \urcorner )\) from the above principles. But there are nonclassical theories of truth on the market, where this would indeed be plausibleāso whatās the problem?
So far, we have axiomatized a classical theory of partial ground and truth, and we have achieved this by typing our truth predicate. We could also try to axiomatize nonclassical theories of partial ground and truth, for example in the spirit of the KripkeFeferman theory of truth KF [8, p. 195ā227]. In such a setting we could consistently affirm the existence of the liar sentence and the truthteller sentence, and even apply the principles of partial ground to themāwhich is, of course, ruled out on our approach. We would get a theory of partial ground and untyped truth. But such a project would require the development of new methods and this goes beyond the scope of this paper. Most importantly, in such a setting, we would have to find a new solution to the puzzle of ground, since we relied on the typing of the truth predicate for our solution. In the terminology of [6, p. 110], our solution is a predicativist solution to the puzzle of ground: On our approach, the grounds of a truth of the form āx T r _{ Ī± }(x) never includes truths of involving a truth predicate T r _{ Ī± }, where Ī± is some ordinal with 0 < Ī± < š _{0}āin particular, the grounds of the truth of āx T r _{ Ī± }(x) donāt include the truth of āx T r _{ Ī± }(x) itself. We have shown that by respecting the typing restrictions of Tarskiās hierarchy, we can obtain a consistent theory of partial ground that proves (typed versions of) the Aristotelian principles. There is, of course, room for impredicativist solutions in the terminology of [6, p. 110], but developing such solutions is beyond the scope of this paper. We leave this for future research.
How should we respond to the paradox of selfreferential ground? Three natural ways in which we could try to block the inconsistency theorem suggest themselves: First, we could try to rule out selfreferential sentences of ground like the one used in the proof of the inconsistency theorem. Second, we could try to restrict the principles of partial ground used in the proof of the inconsistency theorem. And third, we could try to formulate a nonstandard logic of ground that does not sanction the logical principles used in the proof of the inconsistency theorem. The analogy between our inconsistency theorem (Theorem 3) and the Tarskiās theorem of the undefinability of truth (Proposition 2) suggests a neat terminology for these approaches. Analogously to theories of truth [8], we get: typed theories of partial ground, which avoid paradox by putting typerestrictions on the relation of partial ground, effectively ruling out selfreferential sentences like the one in the proof; untyped theories of partial ground, which avoid paradox by restricting the principles of partial ground; and finally nonclassical theories of partial ground, which avoid paradox (or: triviality) by abandoning classical logic in favor of alternative logics.
The results of this paper all point in the direction of a typed theory of partial ground. Indeed our theories P G A _{<Ī± } are typed theories of partial ground: the axiom \(G_4^{\alpha }\) is effectively a typing axiom. Moreover, we have shown that the theories P G A _{<Ī± }, for 1 ā¤ Ī± ā¤ š _{0}, are consistent (Corollary 1). Thus, there is good evidence that a typed solution to the new puzzle of ground works. A natural direction to take from here would be to extend Tarskiās truththeoretic hierarchy to a truth and groundtheoretic hierarchy. We would type the ground predicate to get a family of predicates: \(\lhd _1, \lhd _2, \mathellipsis \). We would end up with a doublytyped theory: typed with respect to truth and typed with respect to ground. The results of this paper suggest that this theory will turn out consistent and this provides further support for the claim that we should use typing in the context of theories of ground. But carrying out the details of this proposal is beyond the scope of this paper. Also this we leave for future research.^{Footnote 33}
Notes
For part one, see [12] in this journal.
This is, in any case, the standard view of partial ground. Some authors have challenged the view, though: [11] challenges the claim that partial ground is irreflexive and [19] challenges the claim that partial ground is transitive. See [16] and [18] for a defense of the standard view against these challenges.
For a precis of the first part of the paper, see SectionĀ 1.1 below.
Here we use T r as our truthpredicate and the GĆ¶delcorners \(\ulcorner \phantom {\varphi }\urcorner \) as our quotation device. For more on these syntactic matters, see SectionĀ 1.1 below.
For a detailed exposition of typed theories of truth, see [8, p. 49ā286].
Stephan KrĆ¤mer [14] and Thomas Donaldson [3] tentatively sketch predicativist solutions without developing the formal details. Fine [6], Correia [2], and Litland [15] propose nonpredicativist solutions to the puzzle of ground. While Correia gives up the irreflexivity of ground, Litland and Fine do not.
This generalizes some previously results obtained by the author in [13].
In formulating these axioms, we make use of the function symbol ^{ā} to represent the valuation function val in the standard model, i.e. s ^{ā} = t means that the denotation of s is t. Officially, however, this is merely an abbreviation for the more complicated defining formula of the valuation function (which exists by standard representation results). Officially, we cannot have symbol representing the valuation function in our language, since then weād run the risk of inconsistency. For more on this, compare [8, p. 32].
To illustrate how the quantified axioms work, consider a formula Ļ such that \(Tr(\ulcorner \varphi \urcorner )\). Then the principle AP _{ T } says that \(\ulcorner \varphi \urcorner \lhd \underset {.}{Tr}\dot {\ulcorner \varphi \urcorner }\). But the latter is \(\ulcorner \varphi \urcorner \lhd \ulcorner Tr(\ulcorner \varphi \urcorner )\urcorner ,\) since \(\dot {\ulcorner \varphi \urcorner }=\ulcorner \ulcorner \varphi \urcorner \urcorner \) and \(\underset {.}{Tr}(\ulcorner \ulcorner \varphi \urcorner \urcorner )=\ulcorner Tr(\ulcorner \varphi \urcorner )\urcorner \). So AP _{ T } allows us to prove \(Tr(\ulcorner \varphi \urcorner )\to \ulcorner \varphi \urcorner \lhd \ulcorner Tr(\ulcorner \varphi \urcorner )\urcorner ,\) for every sentence Ļ. The quantified principle AP _{ F } works analogously.
Fine, in his original paper [6], discusses a range of groundtheoretic puzzles that arise in a similar way from principles similar to the Aristotelian principle. Here weāll focus on the puzzle about truth and partial ground, because the problem arises most naturally in the present context.
For a proof, see for example [1, p. 220ā224].
Here and in the following, we will use a weak concept of selfreference, where a sentence that is (provably) equivalent to a sentence that involves its own GĆ¶del numeral is said to be selfreferential. In this sense, the diagonal lemma as stated above gives us a sentence that is selfreferential. There is a stronger sense of selfreference, in which a sentence is only selfreferential if it is identical to a sentence that involves itās own GĆ¶del numeral. There are stronger versions of the diagonal lemma that would give us selfreference in this sense as well, but for simplicityās sake, weāll stick to the weak concept of selfreference throughout the paper. For a more comprehensive discussion of these issues, see, e.g., [9].
However, as [21] shows there are paradoxes without selfreference. Moreover, there are selfreferential sentences that are not paradoxical. But still, the intuitive view is that in the case of the liar and similar paradoxes, selfreference plays an essential role.
This language is then, of course, an extension of the language of PA: it extends the purely arithmetic vocabulary with names for the sentences of \(\mathcal {L}\) and a truth predicate for those sentences.
This also applies to paradoxes without selfreference, such as Yabloās paradox: Yablo similarly formulates his paradox in a semantically closed language.
There is also a kind of semantic selfreference involved in the case of puzzle. The existential quantifier in āx T r(x) semantically ranges over all sentences of \(\mathcal {L}_{Tr}\), including āx T r(x) itself. Thus the truth of āx T r(x) is partially witnessed by the truth of āx T r(x). But here we do not wish to push this point any further.
For more on axiomatizations of Tarskiās hierarchy, see [8, p. 125ā29].
We assume that the reader is familiar with the basic theory of ordinals. For the relevant definitions, see [10, p. 17ā26], for example. The ordinal š _{0} is the limit of the sequence \(1,\omega , \omega ^{\omega }, \omega ^{\omega ^{\omega }},\mathellipsis \); in other words, š _{0} is the first ordinal that satisfies the equation Ļ ^{x} = x. This ordinal š _{0} is still countable, i.e. it has the same cardinality as the set of the natural numbers. But it provides a natural stopping point for our infinitary hierarchy, since (i) we can code the ordinals below š _{0} and (ii) PA represents the wellordering of the ordinals below š _{0}. We will not go into the details here, as the infinitary nature of our hierarchy is not particularly important to our philosophical point. Nevertheless, for reasons of generality, we will extend our hierarchy to this level, since š _{0} is the limit up to where we can apply the methods of this paper.
Here it is important that weāre restricting ourselves to countable ordinals, because otherwise we would ārun out of codesā at some point.
For the proof note that if we assume that T r _{1}(t) for a term t, it follows by axiom T\(_{1}^{1}\) that S e n t _{<1}(t) and thus we can prove in PA that \(Sent_{<2}(\underset {.}{Tr_{1}}(t))\). Similarly, if we assume \(Tr_{2}(\underset {.}{Tr_{1}}\dot {t}),\) we can prove that S e n t _{<1}(t) by T\(^{2}_{3}\) and PA.
We can formulate a slightly stronger version of P R T _{<Ī± } by replacing the schematic axioms \(RP_{12}^{\beta }\) and \(RP_{13}^{\beta }\) with the quantified axioms: \(\forall x\forall \gamma \underset {.}{<}\ulcorner \beta \urcorner (Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(x)))\) and \(\forall x\forall \gamma \underset {.}{<}\ulcorner \beta \urcorner (Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{\neg }\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(\underset {.}{\neg }x)))\), where āĪ³ quantifies over codes of ordinals, \(\ulcorner \beta \urcorner \) is a term for a code of the ordinal Ī², and \(\underset {.}{<}\) represents the wellordering on ordinals. To properly formulate these axioms, we require a coding of the ordinals up to š _{0}, a representation of the natural wellordering of these ordinals, and a justification for quantifying into ordinal indexes in S e n t _{<Ī³ } and \(\underset {.}{Tr_{\gamma }}\). We will discuss such a coding below, but for reasons of perspicuity, we will stick with the slightly weaker schematic version of P R T _{<Ī± }. Also, the (schematic) theory P R T _{<Ī± } is equivalent to the (schematic) theory R T _{<Ī± } of ramified truth up to Ī±, which is the typed version of CT. As in the case of PT and C T, we will take the metatheorems of R T _{<Ī± } and apply them immediately to P R T _{<Ī± }.
Now we could define quantification over ordinals by saying that āĪ³ Ļ means \(\forall x(\underset {.}{On_{<\epsilon _{0}}}(x)\to \varphi )\) and work with the more general versions of the axioms mentioned before. But for reasons of perspicuity, we refrain from doing so.
Here lub isthe operation of taking the least upper bound with respect to the standard partial ordering ā¼on the ordinals.
We don't give the detailed proof here, but it essentially proceeds by using induction on ordinals below Ļµ_{0} in PA.
We could also use the theorem to determine the proof theoretic strength of P G A _{ Ī± }, but there is a small āhiccup:ā the version of P R T _{ Ī± } that we discussed here is not exactly the one that is usually discussed in the literature. As mentioned in Footnote 24, P R T _{ Ī± } is usually formulated using quantification over ordinals, which we avoided here for reasons of perspicuity. The version of P R T _{ Ī± } with axioms quantifying over ordinals proves the same arithmetical theorems as the theory R A _{ Ī± } of ramified analysis up to (and including) Ī±. For a proof of this result, see [4]. We suspect that the proof theoretic strength of our version of P R T _{ Ī± } is very close to this, although weāre not going to prove anything to this effect.
The situation is quite similar to the corresponding theories of truth. For a discussion of the natural stopping point see [8, p. 32229].
In fact, by a slightly more complicated argument we can show that the height of this tree is exactly Ļ + 1.
The truths of new atomic sentences will, of course, not have any provable grounds in the theory.
Here \(s\ntriangleleft t,\) for terms s and t, is an abbreviation of \(\neg (s\lhd t),\) analogous to the case of sā t. Similarly, the notation \(s\underset {.}{\ntriangleleft }t\) is an abbreviation for the complex function term \(\underset {.}{\neg }(s\underset {.}{\lhd }t)\) for terms s and t.
A particularly interesting point that Iāll leave for future research has been brought to my attention by an anonymous referee: next to the kind of selfreference we obtain, for example, by means of the diagonal lemma, there is also whatās sometimes called empirical selfreference: selfreference that depends on contingent facts about the world. Consider the sentence āthe last example sentence in the paper Axiomatic Theories of Partial Ground II. Partial Ground and Hierarchies of Typed Truth does not partially ground its own double negation,ā for example. Properly formalized, using, for example, a description operator, this sentence will create a groundtheoretic paradox of selfreference in the same way as the sentence Ī“ in the proof of TheoremĀ 3ābut only because it is in fact the last such example sentence, otherwise it would simply be false. To deal with such forms of selfreference in a typed theory, we would presumably need to introduce some kind of context dependent typing. But alas, working out the details goes beyond the scope of this paper.
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Acknowledgements
Iād like to thank Hannes Leitgeb, Benjamin Schnieder, Albert Anglberger, Thomas Schindler, Lavinia Picollo, Johannes Stern, Martin Fischer, Tim Button, the participants of the workshop āLogical and Metaphysical Perspectives on Groundingā at the GAP.9 in OsnabrĆ¼ck, O. Foisch, and two anonymous referees for their helpful comments and suggestions. Part of this research was supported by the Alexander von Humboldt Foundation.
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Korbmacher, J. Axiomatic Theories of Partial Ground II. J Philos Logic 47, 193ā226 (2018). https://doi.org/10.1007/s109920179444z
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DOI: https://doi.org/10.1007/s109920179444z
Keywords
 Metaphysical grounding
 Axiomatic theories of truth
 Predicational theories of ground
 Positive truth