Axiomatic Theories of Partial Ground II
Abstract
This is part two of a two-part 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 truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to 𝜖_{0} and thus is consistent. We argue that this theory provides a natural solution to Fine’s “puzzle of ground” about the interaction of truth and ground. Finally, we show that if we apply the truth-predicate to sentences involving our ground-predicate, we run into paradoxes similar to the semantic paradoxes: we get ground-theoretical paradoxes of self-reference.
Keywords
Metaphysical grounding Axiomatic theories of truth Predicational theories of ground Positive truth1 Introduction
This is part two of a two-part paper in which we develop axiomatic theories of partial ground.^{1} Partial ground, remember, is the relation of one truth holding (either wholly or partially) in virtue of another, cf. [5, 7].^{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.^{3} Consequently, the relation of partial ground induces a hierarchy of grounds on the truths, in which the partial grounds of a truth rank “strictly 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.^{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.
(Aristotelian Principle): If φ is a true sentence, then the truth of \(Tr(\ulcorner \varphi \urcorner )\) holds either wholly or partially in virtue of the truth of φ.^{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 themselves—in direct contradiction to the irreflexivity of partial ground. To give a quick example, take the sentence ∃xTr(x), which says that there is at least one true sentence. This sentence is itself (provably) true, and so we have that ∃xTr(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 ∃xTr(x). But by the transitivity of partial ground this immediately gives us that ∃xTr(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. 108–10] calls a “predicativist” solution to the puzzle of ground. What makes our solution predicativist is our appeal to Tarski’s hierarchy of object-language, meta-language, meta-meta-language, and so on to rule out the problematic cases of apparently self-grounding sentences. We show that by observing Tarski’s distinction between object and meta-language, 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 truth-predicate to sentences involving the same truth-predicate are ruled out.^{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.^{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 T-scheme \(Tr(\ulcorner \varphi \urcorner )\leftrightarrow \varphi \) over formulas φ involving the ground-predicate; for if we were do so, we’d get what we call ground-theoretic paradoxes of self-reference. We will show that these paradoxes are closely related to the well-known semantic paradoxes of self-reference, such as the infamous liar paradox, and furthermore that the paradoxes are genuinely different from Fine’s puzzle of ground. Fine’s 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 ground-theoretic paradoxes of self-reference, in contrast, concern an entirely different aspect of partial ground: the problem is, as we’ll argue, that the ground predicate behaves too much like a truth-predicate.^{8}
Here is the plan for the paper: Section 1.1 contains a précis of the first part of the paper. Then, in Section 2, we’ll 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 Section 3, we’ll 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 Section 4, we will show that if we’re not careful, partial ground, just like truth, can give rise to paradoxes of self-reference. In Section 5, we conclude with some general observations and a map of possible responses to the ground-theoretic paradoxes of self-reference.
1.1 Précis of “Axiomatic Theories of Partial Ground I. The Base Theory”
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, let’s 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 (Section 2).
In the following sections (Sections 3 and 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 Tr is the unary truth-predicate, and the language \(\mathcal {L}_{Tr}^{\lhd }\) is \(\mathcal {L}_{Tr}\cup \{\lhd \}\), where \(\lhd \) is the binary (partial) ground-predicate. We formulate our theory in \(\mathcal {L}_{Tr}^{\lhd }\).
Now our predicational theory of ground PG has all the axioms of PAG, plus the following axioms:^{10}
G_{1}\(\forall x\neg (x\lhd x)\) T_{1}∀s∀t(Tr(s =̣ \(t)\leftrightarrow s^{\circ }=t^{\circ })\)
G_{2}\(\forall x\forall y\forall z(x\lhd y\land y\lhd z\to x\lhd z)\) T\(_{2} \forall s\forall t(Tr(s\underset {.}{\neq }t)\leftrightarrow s^{\circ }\neq t^{\circ })\)
G_{3}\(\forall x\forall y(x\lhd y\to Tr(x)\land Tr(y))\) T_{3}∀x(Tr(x) → Sent(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 proof-theoretically 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).
2 The Aristotelian Principle and Typed Truth
In the context of partial ground, this motivates the following principle: If φ is true, then it is natural to say that φ is true in virtue of what it says being the case, and if φ is false, then φ is false in virtue of what it says not being the case. We will thus call the corresponding informal ground-theoretic principles the Aristotelian principles about truth and falsehood respectively.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 1051b6–9)
(AP _{T}) \(\forall x(Tr(x)\to x\lhd \underset {.}{Tr}(\dot {x})),\) and
(AP _{F}) \(\forall x(Tr(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr}(\dot {x})),\)
Unfortunately, as [6] shows, the Aristotelian principles are ground-theoretically inconsistent:^{12}
Lemma 1 (Puzzle of Ground)
AP_{T}andAP_{F}are inconsistent over PGT.
Proof
- 1.
\(\overline {0}=\overline {0}\) (Arithmetic)
- 2.
\(Tr(\ulcorner \overline {0}=\overline {0}\urcorner )\) (T-scheme 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.
⊥ (10.,11., ⊥-Intro)
We are left with a ground-theoretic puzzle about truth.^{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 first-order logic, and the Aristotelian principle for truth is plausible from considerations about truth. (The principles required to show that AP _{F} is ground-theoretically 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 predicate—a move familiar from typed theories of truth.
Lemma 2
The existence of the liar sentence λ, then, follows by a simple application of the diagonal lemma to the formula \(\neg Tr(x)\in \mathcal {L}_{Tr}\). It is well-known that the existence of a liar sentence is inconsistent with the T-scheme over the language \(\mathcal {L}_{Tr}\).
A common intuitive response to the liar paradox is that it somehow arises from the self-reference involved.^{15} On this informal view, the problem is that the liar sentence “says something of itself,” namely that it is not true.^{16} Thus, so the intuitive response, we should put restrictions on our language that prevent self-reference. Tarski makes this response precise by introducing the distinction between object-language and meta-language. 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 PA—our object-language for arithmetic. But if we wish talk about the truth of sentences in \(\mathcal {L}\), we have to do so in the language \(\mathcal {L}_{Tr}\)—our meta-language for the truths of arithmetic.^{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- meta-language, which has a distinct truth-predicate 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 meta-language, as it were, and thus we get self-referential paradoxes. Tarski shows that if we obey the distinction between object-language and meta-language, we can formulate a consistent theory of truth: In an appropriate meta-language, which is not semantically closed, we can consistently affirm the T-scheme for the sentences of the object-language 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 Gödel-numbering for the terms of \(\mathcal {L}_{Tr}\) within \(\mathcal {L}_{Tr}\) and at the same time affirm the T-scheme over the sentences of \(\mathcal {L}_{Tr},\) i.e. if we use \(\mathcal {L}_{Tr}\) as its own meta-language, 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 meta-language. Intuitively, the picture is that semantic truths, such as truths about the truths of arithmetic, are on a “higher level” than non-semantic 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 “higher level” than the truths about the truths of arithmetic and so on. What emerges is Tarski’s hierarchy of truths. Following Tarski, if we work in a semantically closed language, we mix the levels of the hierarchy of truths—and ultimately this is the source of the semantic paradoxes.^{18}
Our original, unmodified predicational theory of ground PG respects Tarski’s distinction between object- and meta-language: 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 meta-language for our object-language \(\mathcal {L}\)—in compliance with Tarski’s distinction. When we move to the modified theory PGT, however, we no longer conform with Tarski’s distinction. Since PGT is formulated in \(\mathcal {L}_{Tr}^{\lhd }\) in the context of a Gödel 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 PGT, we have used \(\mathcal {L}_{Tr}^{\lhd }\) as its own meta-language—we 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 truth-theoretic inconsistency, i.e. an inconsistency with plausible principles for truth, while in the case of the puzzle, we get a ground-theoretic 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 self-reference is involved: while the liar sentence λ intuitively says something of itself, the principles of partial ground entail that truth of ∃xTr(x) partially grounds itself. Thus, we can say that the self-reference in the case of the liar is semantic, while the self-reference in the case of the puzzle of ground is ground-theoretic.^{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 Tr, we have a family Tr_{1},Tr_{2},… of typed truth predicates. These truth predicates intuitively express truth on the first, second, … level of Tarski’s 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.
3 Axiomatic Theories of Partial Ground and Typed Truth
Typed theories of truth aim to axiomatize Tarski’s hierarchy of truths.^{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 \(\mathcal {L}_{0}=_{def}\mathcal {L}\)—the language of PA. The truth predicate for sentences of arithmetic is, then, Tr_{1} and the language \(\mathcal {L}_{1}\) extends \(\mathcal {L}_{0}\) with Tr_{1}. The truth predicate for sentences of \(\mathcal {L}_{1}\), in turn, is Tr_{2} and the language \(\mathcal {L}_{2}\) extends \(\mathcal {L}_{1}\) with Tr_{2}. And so on. Thus, typed theories of truth are formulated using a hierarchical family of truth predicates Tr_{1},Tr_{2},… that intuitively correspond to truth on the different levels of Tarski’s hierarchy.
3.1 Language and Background Theory
For an ordinal 0 < α < 𝜖_{0}, the theory PAT_{<α} is the result of extending PA with all the instances of the induction scheme over the language \(\mathcal {L}_{<\alpha }\) and the theory PAG_{<α} is the result of extending PAT_{<α} with all the missing instances of the induction scheme over \(\mathcal {L}_{\alpha }^{\lhd }\). For 0 ≤ α < 𝜖_{0}, the theory PAT_{α}, then, is PAT_{<α+1} and similarly PAG_{α} is PAG_{<α+1}. Thus, PAT_{0} is PAT and PAG_{0} is PAG. In PAT_{α}, 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 α, we assume that in \(\mathcal {L}_{\alpha }^{\lhd },\) via some appropriate Gödel coding, we have names \(\ulcorner \varphi \urcorner \) for all formulas \(\varphi \in \mathcal {L}_{<\alpha }\).^{22} Moreover, we assume that for every 0 < β < α, we have a function symbol \(\underset {.}{Tr_{\beta }}\) that represents the function which maps the code #t of a term t to the code #Tr_{β}(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 Sent_{<α}.
3.2 Axioms for Partial Ground and Typed Truth
(G\(_{3a}^{\beta }\)) \(\forall x\forall y (x\lhd y\to (Sent_{<\beta }(x)\to Tr_{\beta }(x)))\)
(G\(_{3b}^{\beta }\)) \(\forall x\forall y (x\lhd y\to (Sent_{<\beta }(y)\to Tr_{\beta }(y)))\)
(G\(_{3a}^{1}\)) \(\forall x\forall y(x\lhd y\to (Sent_{<1}(x)\to Tr_{1}(x)))\)
(G\(_{3b}^{1}\)) \(\forall x\forall y(x\lhd y\to (Sent_{<1}(y)\to Tr_{1}(y)))\)
(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\(_{3}^{\beta }\)) ∀x(Tr_{β}(x) → Sent_{<β}(x))
(G\(_{4}^{\alpha }\)) \(\forall x\forall y(x\lhd y\to Sent_{<\alpha }(x)\land Sent_{<\alpha }(y))\)
(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 < β < α, if a sentence is true on that level, then the sentence grounds its double-negation 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 ≤ α ≤ 𝜖_{0}, the predicational theoryPG_{<α}ofground up toα, consists of the axioms of PAG_{<α} plus the following axioms for all 0 < β < α:
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\(_{4}^{\alpha }\)
\(\forall x\forall y(x\lhd y\to Sent_{<\alpha }(x)\land Sent_{<\alpha }(y))\)
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 ≤ α < 𝜖_{0}, we define PG_{α} as PG_{<α+1}.
To illustrate what PG_{α} looks like for different α’s, let’s consider at a few examples. First, note that PG_{0} is PAG. Next, note PG_{1} is a functional analog of our original theory PG, where the truth-predicate has been “renamed” Tr_{1}. In particular, we get that PG_{1} proves the theory PT of positive truth.
(APU\(_{T}^{\gamma }\)) \(\forall x(Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)
(APU\(_{F}^{\gamma }\)) \(\forall x(Tr_{\gamma }(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
(APD\(_{T}^{\beta ,\gamma }\)) \(\forall x(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)
(APD\(_{F}^{\beta ,\gamma }\)) \(\forall x(Tr_{\beta }(\underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
Definition 2
For every ordinal 0 ≤ α < 𝜖_{0}, the theory PGA_{α} of partial ground with the Aristotelian principles up to α consists of the axioms of PG_{α} plus the following axioms for all γ < β ≤ α:
(APU\(_{T}^{\gamma }\)) \(\forall x(Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)
(APU\(_{F}^{\gamma }\)) \(\forall x(Tr_{\gamma }(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
(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 PGA_{<α} is defined as \(\bigcup _{\beta <\alpha }PGA_{\alpha },\) for all 0 < α ≤ 𝜖_{0}.
3.3 Conservativity and Models
The theory PT_{<α} of positive ramified truth up to an ordinal level 1 ≤ α ≤ 𝜖_{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 PG_{<α}:
Definition 3 (‘Positive Ramified Truth’)
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\(_{3}^{\beta }\)
∀x(Tr_{β}(x) → Sent_{<β}(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 PRT_{α}, for 0 ≤ α < 𝜖_{0}, is defined as PRT_{<α+1}.
Proposition 1
For all ordinals 1 ≤ α ≤ 𝜖_{0},the theoryPGA_{<α}proves the theoryPRT_{<α}:\(PGA_{<\alpha }\vdash PRT_{<\alpha }\).
Proof
\(\vdash _{PGA_{<\alpha }}\forall x(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\gamma }(x))\) for γ < β < α
Let x be a fresh variable for ∀-Intro. We now prove both directions of the biconditional \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\gamma }(x)\).(\(\Rightarrow \)): Assume (∗) \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(x))\) for a →-Intro. By \(T_{3}^{\beta },\) we can derive \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}(x))\). Using PA and \(T_{3}^{\gamma },\) we can derive (∗∗) Sent_{<γ}(x) from this. Moreover, using (∗) and \(APD_{T}^{\beta ,\gamma }\):we can derive \(x\lhd \underset {.}{Tr_{\gamma }}(\dot {x})\). Using (∗∗) and \(G_{4}^{\gamma }\):$$\forall x(Tr_{\beta}(\underset{.}{Tr_{\gamma}}(\dot{x}))\to x\lhd\underset{.}{Tr_{\gamma}}(\dot{x})),$$we can in turn derive: Tr_{γ}(x). Thus, we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(x))\to Tr_{\gamma }(x)\) by →-Intro.$$\forall x\forall y(x\lhd y\to (Sent_{<\gamma}(x)\to Tr_{\gamma}(x))),$$(\(\Leftarrow \)): Assume (‡) Tr_{γ}(x) for another →-Intro. Using PA and \(G_{4}^{\gamma },\) we get Sent_{<γ}(x). From this and PA, we can derive for all γ < β < α that (‡‡) \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\). Moreover, using (‡) and \(APU_{T}^{\gamma }\):we get \(x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\). From this, using (‡‡) 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 →-Intro.$$\forall x(Tr_{\gamma}(x)\to x\lhd\underset{.}{Tr_{\gamma}}(\dot{x})),$$Putting both “\(\Rightarrow \)” and “\(\Leftarrow \)” 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:by ∀-Intro as desired.$$\forall x (Tr_{\beta}(\underset{.}{Tr_{\gamma}}\dot{x})\leftrightarrow Tr_{\gamma}(x))$$
\(\vdash _{PGA_{<\alpha }}\forall x(Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(x)))\) for γ < β < α.
Let x be a fresh variable for ∀-Intro. Assume Sent_{<γ}(x) for a →-Intro. We now prove both directions of the biconditional \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(x)\).(\(\Rightarrow \)): Assume (†) \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\) for →-Intro. From this, T\(_{3}^{\beta },\) and PA, we can derive \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\) and thus also (††) Sent_{<β}(x). As before, we get \(x\lhd \underset {.}{Tr_{\gamma }}\dot {x}\) using \(APD_{T}^{\beta ,\gamma }\). Using (††) and \(G_{3a}^{\beta },\) we can derive Tr_{β}(x). Thus, we have \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\to Tr_{\beta }(x)\) by →-Intro.(\(\Leftarrow \)): Assume Tr_{β}(x) for yet another →-Intro. From this and \(APU_{T}^{\beta },\) we get \(x\lhd Tr_{\beta }(x)\). Since we have assumed Sent_{<γ}(x), we get Tr_{γ}(x) from this and \(G_{3a}^{\gamma }\). From this and \(APU_{T}^{\gamma },\) we get (§) \(x\lhd \underset {.}{Tr_{\gamma }}(\dot {x})\). But now since γ < β, we can show in PA that \(Sent_{<\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\). But from this and (§), we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\). So, we have \(Tr_{\beta }(x)\to Tr_{\beta }(\underset {.}{Tr_{\gamma }}(\dot {x}))\) by →-Intro.
Now putting both “\(\Rightarrow \)” and “\(\Leftarrow \)” 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 →-Intro. Finally, since x was a fresh variable, we have:as desired.$$\forall x(Sent_{<\gamma}(x)\to (Tr_{\beta}(\underset{.}{Tr_{\gamma}}(\dot{x}))\leftrightarrow Tr_{\beta}(x))),$$
This has the immediate consequence that for all ordinals β < α < 𝜖_{0}, the theory PGA_{<α} proves the following typed version of the T-scheme for all languages \(\mathcal {L}_{<\beta }\):
Lemma 3
For all ordinals 0 < γ ≤ β < α < 𝜖_{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 ≤ α < 𝜖_{0}, the theory PGA_{α} is a proof-theoretically conservative extension of the theory PRT_{α}. But first, we need to introduce some more technical preliminaries: It is well-known that we can extend the technique of Gödel numbering to get terms for all ordinals below 𝜖_{0} [17, p. 17–42]. Let’s denote the set of all ordinals below 𝜖_{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 #σ of the other expressions σ 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 α is the numeral of the code #α of α. Moreover, we extend the axioms of ordinary arithmetic to cover ordinal arithmetic up to 𝜖_{0}. For simplicity, we’ll 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 α × β = γ, for all ordinals \(\alpha ,\beta ,\gamma \in On_{<\epsilon _{0}}\). PA can represent the set of codes of ordinals below 𝜖_{0} and we’ll 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 well-ordering < of the ordinals below 𝜖_{0} and we’ll 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 α < β. With these preliminaries in place,^{25} we’ll define a slightly non-standard notion of complexity for the formulas in \(\mathcal {L}_{<\epsilon _{0}}\):
Definition 4 (‘ ω-complexity’)
- (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)
|¬φ|_{ω} = |φ|_{ω} + 1;
- (iii)
|φ ∘ ψ|_{ω} = lub(|φ|_{ω},|ψ|_{ω}) + 1, for ∘ = ∧,∨;^{26} and
- (iv)
|Qxφ|_{ω} = |φ|_{ω} + 1, for Q = ∀,∃.
Note that ω-complexity agrees with ordinary complexity on the formulas of \(\mathcal {L}_{<1}\). Note furthermore that the function x↦ω × x is strictly monotonically increasing on the ordinals below 𝜖_{0}:
Lemma 4
For all\(\alpha ,\beta \in On_{<\epsilon _{0}}\),ifα < β,thenω × α < ω × β.
Note that as a consequence, we get that for all ordinals 0 < α < 𝜖_{0}, if \(\varphi \in \mathcal {L}_{<\alpha },\) then \(|\varphi |_{\omega }<|Tr_{\alpha }(\ulcorner \varphi \urcorner )|_{\omega }\). In other words, ω-complexity has a sort of “tracking property:” it can “track” the levels of Tarski’s hierarchy. Moreover, we can represent ω-complexity in PA. More specifically, the function \(c_{\omega }:\#\mathcal {L}\to \mathbb {N}\) that maps the code #φ of a formula \(\varphi \in \mathcal {L}\) to its ω-complexity |φ|_{ω} is recursive and thus representable in PA. We represent c_{ω} 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 |φ|_{ω} = α. Using this representation, we can show that Peano arithmetic proves that ω-complexity has the “tracking-property” in the following sense:
Lemma 5
Using ω-complexity, we’ll obtain the main result of this section:
Theorem 1
For all 0 ≤ α < 𝜖_{0},the theoryPGA_{α}is a proof-theoretically conservative extension of the theoryPRT_{α}.
Proof
\(\tau (\varphi )=\left \{\begin {array}{ll} Tr_{\alpha }(s)\land Tr_{\alpha }(t)\land \underset {.}{c_{\omega }}(s)\underset {.}{<}\underset {.}{c_{\omega }}(t) & \text {if}~\varphi =s\lhd t\\\varphi & \text {if}~\varphi ~\text {is another atomic formula}\end {array}\right .\)
τ(¬φ) = ¬τ(φ);
τ(φ ∘ ψ) = τ(φ) ∘ τ(ψ), for ∘ = ∧,∨; and
τ(Qxφ) = Qx(τ(φ)), for Q = ∀,∃.
- In the case of the axiom \(G_{4}^{\alpha },\) we get:This is provable (almost) immediately from the typed truth axiom T\(_{3}^{\alpha }\) of PRT_{α}:$$\tau(G_{4}^{\alpha})=\forall x\forall y((Tr_{\alpha}(x)\land Tr_{\alpha}(y)\land \underset{.}{c_{\omega}}(x)\underset{.}{<}\underset{.}{c_{\omega}}(y))\to Sent_{<\alpha}(x)\land Sent_{<\alpha}(y))$$$$\forall x(Tr_{\alpha}(x)\to Sent_{<\alpha}(x)).$$
- Finally, consider the axioms (APU\(_{T}^{\beta }\)):where β < α. We get:$$\forall x(Tr_{\beta}(x)\to x\lhd\underset{.}{Tr_{\beta}}(\dot{x})),$$Now let x be a fresh variable for ∀-Intro and assume Tr_{β}(x) for a →-Intro. Using the axiom \(T_{3}^{\beta }\) of PRT_{α}, we can infer that Sent_{<β}(x). Moreover, since β < α by assumption, we can infer that \(Tr_{\alpha }(\underset {.}{Tr_{\beta }}(\dot {x}))\) and Tr_{β}(x) using the axiom \(RP_{12}^{\alpha }\) of PRT_{α}. 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 Sent_{<β}(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 →-Intro, we have$$\tau(APU_{T}^{\beta})=\forall x(Tr_{\beta}(x)\to Tr_{\alpha}(x)\land Tr_{\alpha}(\underset{.}{Tr_{\beta}}(\dot{x}))\land \underset{.}{c_{\omega}}(x)\underset{.}{<} \underset{.}{c_{\omega}}(\underset{.}{Tr_{\beta}}(\dot{x}))).$$and since x was a fresh variable, by ∀-Intro, we get the desired theorem.$$Tr_{\beta}(x)\to Tr_{\alpha}(x)\land Tr_{\alpha}(\underset{.}{Tr_{\beta}}(\dot{x}))\land \underset{.}{c_{\omega}}(x)\underset{.}{<} \underset{.}{c_{\omega}}(\underset{.}{Tr_{\beta}}(\dot{x})),$$
Putting (a) and (b) together, the claim follows. □
The theorem has the following immediate consequence:^{28}
Corollary 1
For all 0 ≤ α < 𝜖_{0},the theoryPGA_{α}is consistent.
The proof of Theorem 1 essentially works because of the “tracking property” of ω-complexity. 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 φ 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 ω-complexity in the translation—this is what allowed us to prove the result. The technique of the proof works for all ordinals α < 𝜖_{0}, since PA can represent the well-ordering of these ordinals, which is required for the proof. The theory \(PGA_{<\epsilon _{0}}\) is the first theory where our proof doesn’t work anymore, because in this theory we don’t have a “highest” truth predicate as required for the definition of τ. But we can extend our result to this theory using a simple compactness argument:
Corollary 2
The theory\(PGA_{<\epsilon _{0}}\)is a proof-theoretically 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 < β_{1} < … < β_{n} < 𝜖_{0}, in the proof. But then the proof of φ, 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 φ and the claim holds. □
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 we’ve developed in this paper.^{29}
We have shown the consistency of our theories PGA_{<α}, where 1 ≤ α ≤ 𝜖_{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 PGA_{<α}, where 1 ≤ α ≤ 𝜖_{0}.
\(\mathbf {S}_{1}=\{\#\varphi ~|~\varphi \in \mathcal {L}_{<1}, \mathbf {N}\vDash \varphi \};\)
\(\mathbf {S}_{\alpha +1}=\mathbf {S}_{\alpha }\cup \{\#\varphi ~|~\varphi \in \mathcal {L}_{<\alpha }, (\mathbf {N},(\mathbf {S}_{\beta })_{\beta <\alpha })\vDash \varphi \}\)
\(\mathbf {S}_{\alpha }=\bigcup _{\beta <\alpha }\mathbf {S}_{\beta },\) if α is a limit ordinal.
Then we get, for all 1 ≤ α ≤ 𝜖_{0}, that \((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha })\vDash PRT_{<\alpha }\). For 1 ≤ α ≤ 𝜖_{0}, the model (N,(S_{β})_{β<α}) is the standard model of PRT_{<α}—it models Tarski’s hierarchy of truths.
We now extend our definition of grounding-trees from the previous paper to grounding-trees over the standard model ofPRT_{<α}:
Definition 5
- (i)
\(\#\varphi \in \bigcup _{\beta <\alpha }\mathbf {S}_{\beta }\), then #φ is a grounding-tree over (N,(S_{β})_{β<α}) with #φ as its root;
- (ii)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N,(S_{β})_{β<α}) with #φ as its root, then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #¬¬φ as its root;
- (iii)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N, (S_{β})_{β<α}) with #φ as its root, then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #(φ ∨ ψ) as its root;
- (iv)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N, (S_{β})_{β<α}) with #ψ as its root, then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #(φ ∨ ψ) as its root;
- (v)
if Open image in new window are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}\) over (N, (S_{β})_{β<α}) with #φ,#ψ as their roots respectively, then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #(φ ∧ ψ) as its root;
- (vi)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N, (S_{β})_{β<α}) with #φ(t) as its root, then Open image in new windowis a grounding-tree over (N,(S_{β})_{β<α}) with #∃xφ(x) as its root;
- (vii)
if Open image in new window, … are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}, \mathellipsis \) over (N,(S_{β})_{β<α}) with #φ(t_{1}),#φ(t_{2}),… as their roots respectively, where t_{1},t_{2},… are all and only the terms of \(\mathcal {L}_{PA}\), then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #∀xφ(x) as its root;
- (viii)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N, (S_{β})_{β<α}) with #¬φ as its root, then Open image in new windowis a grounding-tree over (N,(S_{β})_{β<α}) with #¬(φ ∧ ψ) as its root;
- (ix)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N, (S_{β})_{β<α}) with #¬ψ as its root, then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #¬(φ ∧ ψ) as its root;
- (x)
if Open image in new window are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}\) over (N,(S_{β})_{β<α}) with #¬φ,#¬ψ as their roots respectively, then Open image in new window is a grounding-tree over (N, (S_{β})_{β<α}) with #¬(φ ∨ ψ) as its root;
- (xi)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N, (S_{β})_{β<α}) with #¬φ(t) as its root, then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #¬∀xφ(x) as its root;
- (xii)
if Open image in new window , … are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}, \mathellipsis \) over (N,(S_{β})_{β<α}) with #¬φ(t_{1}),#¬φ(t_{2}),… as their roots respectively, where t_{1},t_{2},… are all and only the terms of \(\mathcal {L}_{PA}\), then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with #∀xφ(x) as its root;
- (xiii)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N,(S_{β})_{β<α}) with #φ as its root and #φ ∈ S_{β}, for β < α, then Open image in new window is a grounding-tree over (N,(S_{β})_{β<α}) with \(\#Tr_{\beta }(\ulcorner \varphi \urcorner )\) as its root;
- (xiv)
if Open image in new window is a grounding-tree \(\mathcal {T}\) over (N,(S_{β})_{β<α}) with #¬φ as its root and #φ ∈S_{β}, for β < α, then Open image in new window is a grounding-tree over (N, (S_{β})_{β<α}) with \(\#\neg Tr_{\beta }(\ulcorner \varphi \urcorner )\) as its root;
- (xv)
nothing else is a grounding-tree over (N,(S_{β})_{β<α}).
Now, in contrast to grounding-trees over (N,S), grounding-trees over (N,(S_{β})_{β<α}) can have an infinite height:
Definition 6
- (i)
all grounding-trees over (N,(S_{β})_{β<α}) of the form #φ ∈S have height one;
- (ii)if \(\mathcal {T}\) is a grounding-tree over (N,(S_{β})_{β<α}) that is constructed from grounding-trees \(\mathcal {T}_{1}, \mathcal {T}_{2}, \mathellipsis \) over (N,(S_{β})_{β<α}), then the height of \(\mathcal {T}\) is one plus the least upper bound of the heights of \(\mathcal {T}_{1}, \mathcal {T}_{2}, \mathellipsis \):where lub is the operation of taking the least upper bound.$$h(\mathcal{T})=lub\{h(\mathcal{T}_{1}), h(\mathcal{T}_{2}), \mathellipsis\}+1,$$
To see that there are grounding-trees 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 φ such that \(DN_{\overline {0}=\overline {0}}(\#\varphi ),\) there is a grounding-tree of the form Open image in new windowwhich has height \(\frac {n}{2}+3,\) where n is the number of negations in φ. A consequence of this is that the least upper bound of the heights of \(\mathcal {T}_{1}, \mathcal {T}_{2}, \mathellipsis \) in the grounding-tree Open image in new window is at least ω and thus the height of this tree is at least ω + 1.^{30}
Now, an important consequence of this observation is that we can’t use ordinary induction on the height of trees to prove claims about all grounding-trees. We need to use transfinite induction on the height of the grounding-trees. This doesn’t add any further complications, but to be explicit let’s state the form of the principle that we’re going to use. Consider a property of grounding-trees. Then, if we can show that any degenerate grounding-tree 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 grounding-trees 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 grounding-trees over (N,S), we can now show that grounding-trees over (N,(S_{β})_{β<α}) 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 don’t contain any cycles; and finally, (iii) transitive.
Lemma 6
Let 1 ≤ α ≤ 𝜖_{0}, (N,(S_{β})_{β<α})be the standard model ofPRT_{<α},and let\(\mathcal {T}\)be a grounding-tree over (N,(S_{β})_{β<α}).Then for all formulas\(\varphi \in \mathcal {L}_{<\alpha },\)if#φis 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_{β})_{β<α}) is a model of PRT_{<α}. □
Remember the notion of a code of a formula occurring below another code in a grounding-tree over (N,S). We now adapt this notion to grounding-trees over (N,(S_{β})_{β<α}) by recursively saying that, for all 1 ≤ α ≤ 𝜖_{0}, no code of any formula occurs below the code of any other formula in a degenerate grounding-tree over (N,(S_{β})_{β<α}), and if \(\mathcal {T}\) is a grounding-tree over (N,(S_{β})_{β<α}) that was constructed from grounding-trees \(\mathcal {T}_{1}, \mathcal {T}_{2},\mathellipsis \) over (N,(S_{β})_{β<α}) according to the rules (ii–xvi) 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 ≤ α ≤ 𝜖_{0}and let (N,(S_{β})_{β<α})be the standard model ofPRT_{<α}.If\(\mathcal {T}\)is a grounding-tree over (N,(S_{β})_{β<α})with#φas its root, for some formula\(\varphi \in \mathcal {L}_{<\alpha }\).Then, all formulas\(\psi \in \mathcal {L}_{<\alpha }\)whose code#ψoccurs below#φin\(\mathcal {T}\)have a lowerω-complexitythanφ.
Lemma 8
Let 1 ≤ α ≤ 𝜖_{0}, (N,(S_{β})_{β<α})be the standard model ofPRT_{<α},and let\(\mathcal {T}\)be a grounding-tree over (N,(S_{β})_{β<α}).Then between any two nodes#φand#ψin\(\mathcal {T},\)for formulas\(\varphi ,\psi \in \mathcal {L}_{<\alpha },\)there is exactly one path.
Lemma 9
Let 1 ≤ α ≤ 𝜖_{0}and let (N,(S_{β})_{β<α})be the standard model ofPRT_{<α}.If there is a grounding-tree\(\mathcal {T}_{1}\)over (N,(S_{β})_{β<α})with#ψas its root and#φ_{1}, #φ_{2},…as its leaves and there is grounding-tree\(\mathcal {T}_{2}\)over (N,(S_{β})_{β<α})with#ψ, #ψ_{1}, #ψ_{2},…as its leaves and#𝜃as its root, then there is a grounding-tree\(\mathcal {T}_{3}\)over (N,(S_{β})_{β<α})with#φ_{1}, #φ_{2},…,#ψ_{1}, #ψ_{2},…as its leaves and#𝜃as its root.
Finally, we define the standard model of PGA_{<α} by saying that:
Definition 7
Let 1 ≤ α ≤ 𝜖_{0} and let (N,(S_{β})_{β<α}) be the standard model of PRT_{<α}. 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 non-degenerate grounding-tree over (N,(S_{β})_{β<α}) with n as a leaf and m as its root.
Putting Lemmas 6, 8, and 9 together, we obtain:
Theorem 2
(N,(S_{β})_{β<α},R)is a model ofPGA_{<α},for 1 ≤ α ≤ 𝜖_{0},i.e.\((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha }, \mathbf {R})\vDash PGA_{<\alpha }\)
Proof
\((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha }, \mathbf {R})\vDash \forall x (Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}\dot {x})\) for all γ < α.
Let σ be a variable assignment over (N,(S_{β})_{β<α},R) and \({\sigma ^{\prime }}\) some x-variant of σ. 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, #φ is a degenerate grounding-tree over (N,(S_{β})_{β<α}). But then, by clause (xiii) of Definition 5, Open image in new window is a non-degenerate grounding-tree over (N,(S_{β})_{β<α}). Moreover, the root of this tree is \(\#Tr_{\gamma }(\ulcorner \varphi \urcorner )\) and its only leaf is #φ. 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 Tr_{γ} 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 σ 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. □
4 Paradoxes of Self-Referential Ground
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
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 T-scheme for sentences involving the ground predicate:
Lemma 10
Proof
By induction on the positive complexity of formulas. The new axioms \(T_{\lhd }^{1/2}\) take care of the new base-case. □
Note that this lemma doesn’t entail yet that PUG is inconsistent: the truth predicate Tr is not in the language \(\mathcal {L}_{\lhd }\) and thus Lemma 10 doesn’t entail that we’re 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
With these two lemmas in place, we can show our main result:
Theorem 3
PUG is inconsistent.
Proof
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 Fine’s puzzle of ground. Fine’s puzzle consists in the fact that different intuitively plausible principles for partial ground and truth entail that the truth of some sentences partially ground themselves—in 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 … 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 like—it has another source.
Proposition 2
Proof
By induction on the positive complexity of φ for both directions of the biconditional. The base cases for φ being s = t or s≠t, for terms s and t, are covered by the basic truth axioms T_{1} and T_{2} and the definition of Tr0=. Similarly, the base cases for φ 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. □
In other words, in PUG we can define a truth predicate for \(\mathcal {L}_{\lhd }\) that satisfies the T-scheme for \(\mathcal {L}_{\lhd }\). It follows by Tarski’s 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 “behaves too much like a truth predicate”—in other words: the new puzzle of ground is at heart a paradox of self-reference.
5 Conclusion
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 non-classical theories of partial ground and truth, for example in the spirit of the Kripke-Feferman theory of truth KF [8, p. 195–227]. In such a setting we could consistently affirm the existence of the liar sentence and the truth-teller sentence, and even apply the principles of partial ground to them—which 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 ∃xTr_{α}(x) never includes truths of involving a truth predicate Tr_{α}, where α is some ordinal with 0 < α < 𝜖_{0}—in particular, the grounds of the truth of ∃xTr_{α}(x) don’t include the truth of ∃xTr_{α}(x) itself. We have shown that by respecting the typing restrictions of Tarski’s 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 self-referential ground? Three natural ways in which we could try to block the inconsistency theorem suggest themselves: First, we could try to rule out self-referential 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 non-standard 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 Tarski’s 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 type-restrictions on the relation of partial ground, effectively ruling out self-referential sentences like the one in the proof; untyped theories of partial ground, which avoid paradox by restricting the principles of partial ground; and finally non-classical 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 PGA_{<α}are typed theories of partial ground: the axiom \(G_4^{\alpha }\) is effectively a typing axiom. Moreover, we have shown that the theories PGA_{<α}, for 1 ≤ α ≤ 𝜖_{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 Tarski’s truth-theoretic hierarchy to a truth- and ground-theoretic hierarchy. We would type the ground predicate to get a family of predicates: \(\lhd _1, \lhd _2, \mathellipsis \). We would end up with a doubly-typed 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.^{33}
Footnotes
- 1.
For part one, see [12] in this journal.
- 2.
- 3.
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.
- 4.
For a precis of the first part of the paper, see Section 1.1 below.
- 5.
Here we use Tr as our truth-predicate and the Gödel-corners \(\ulcorner \phantom {\varphi }\urcorner \) as our quotation device. For more on these syntactic matters, see Section 1.1 below.
- 6.
For a detailed exposition of typed theories of truth, see [8, p. 49–286].
- 7.
Stephan Krämer [14] and Thomas Donaldson [3] tentatively sketch predicativist solutions without developing the formal details. Fine [6], Correia [2], and Litland [15] propose non-predicativist solutions to the puzzle of ground. While Correia gives up the irreflexivity of ground, Litland and Fine do not.
- 8.
This generalizes some previously results obtained by the author in [13].
- 9.
- 10.
In formulating these axioms, we make use of the function symbol ^{∘} to represent the valuation function val in the standard model, i.e. s^{∘} = 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 we’d run the risk of inconsistency. For more on this, compare [8, p. 32].
- 11.
To illustrate how the quantified axioms work, consider a formula φ 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 φ. The quantified principle AP _{F} works analogously.
- 12.
- 13.
Fine, in his original paper [6], discusses a range of ground-theoretic puzzles that arise in a similar way from principles similar to the Aristotelian principle. Here we’ll focus on the puzzle about truth and partial ground, because the problem arises most naturally in the present context.
- 14.
For a proof, see for example [1, p. 220–224].
- 15.
Here and in the following, we will use a weak concept of self-reference, where a sentence that is (provably) equivalent to a sentence that involves its own Gödel numeral is said to be self-referential. In this sense, the diagonal lemma as stated above gives us a sentence that is self-referential. There is a stronger sense of self-reference, in which a sentence is only self-referential if it is identical to a sentence that involves it’s own Gödel numeral. There are stronger versions of the diagonal lemma that would give us self-reference in this sense as well, but for simplicity’s sake, we’ll stick to the weak concept of self-reference throughout the paper. For a more comprehensive discussion of these issues, see, e.g., [9].
- 16.
However, as [21] shows there are paradoxes without self-reference. Moreover, there are self-referential sentences that are not paradoxical. But still, the intuitive view is that in the case of the liar and similar paradoxes, self-reference plays an essential role.
- 17.
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.
- 18.
This also applies to paradoxes without self-reference, such as Yablo’s paradox: Yablo similarly formulates his paradox in a semantically closed language.
- 19.
There is also a kind of semantic self-reference involved in the case of puzzle. The existential quantifier in ∃xTr(x) semantically ranges over all sentences of \(\mathcal {L}_{Tr}\), including ∃xTr(x) itself. Thus the truth of ∃xTr(x) is partially witnessed by the truth of ∃xTr(x). But here we do not wish to push this point any further.
- 20.
For more on axiomatizations of Tarski’s hierarchy, see [8, p. 125–29].
- 21.
We assume that the reader is familiar with the basic theory of ordinals. For the relevant definitions, see [10, p. 17–26], for example. The ordinal 𝜖_{0} is the limit of the sequence \(1,\omega , \omega ^{\omega }, \omega ^{\omega ^{\omega }},\mathellipsis \); in other words, 𝜖_{0} is the first ordinal that satisfies the equation ω^{x} = x. This ordinal 𝜖_{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 𝜖_{0} and (ii) PA represents the well-ordering of the ordinals below 𝜖_{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 𝜖_{0} is the limit up to where we can apply the methods of this paper.
- 22.
Here it is important that we’re restricting ourselves to countable ordinals, because otherwise we would “run out of codes” at some point.
- 23.
For the proof note that if we assume that Tr_{1}(t) for a term t, it follows by axiom T\(_{1}^{1}\) that Sent_{<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 Sent_{<1}(t) by T\(^{2}_{3}\) and PA.
- 24.
We can formulate a slightly stronger version of PRT_{<α} 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 ∀γ quantifies over codes of ordinals, \(\ulcorner \beta \urcorner \) is a term for a code of the ordinal β, and \(\underset {.}{<}\) represents the well-ordering on ordinals. To properly formulate these axioms, we require a coding of the ordinals up to 𝜖_{0}, a representation of the natural well-ordering of these ordinals, and a justification for quantifying into ordinal indexes in Sent_{<γ} 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 PRT_{<α}. Also, the (schematic) theory PRT_{<α} is equivalent to the (schematic) theory RT_{<α} of ramified truth up to α, which is the typed version of CT. As in the case of PT and CT, we will take the meta-theorems of RT_{<α} and apply them immediately to PRT_{<α}.
- 25.
Now we could define quantification over ordinals by saying that ∀γφ 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.
- 26.
Here lub isthe operation of taking the least upper bound with respect to the standard partial ordering ≼on the ordinals.
- 27.
We don't give the detailed proof here, but it essentially proceeds by using induction on ordinals below ϵ_{0} in PA.
- 28.
We could also use the theorem to determine the proof theoretic strength of PGA_{α}, but there is a small “hiccup:” the version of PRT_{α} that we discussed here is not exactly the one that is usually discussed in the literature. As mentioned in Footnote 24, PRT_{α} is usually formulated using quantification over ordinals, which we avoided here for reasons of perspicuity. The version of PRT_{α} with axioms quantifying over ordinals proves the same arithmetical theorems as the theory RA_{α} of ramified analysis up to (and including) α. For a proof of this result, see [4]. We suspect that the proof theoretic strength of our version of PRT_{α} is very close to this, although we’re not going to prove anything to this effect.
- 29.
The situation is quite similar to the corresponding theories of truth. For a discussion of the natural stopping point see [8, p. 322-29].
- 30.
In fact, by a slightly more complicated argument we can show that the height of this tree is exactlyω + 1.
- 31.
The truths of new atomic sentences will, of course, not have any provable grounds in the theory.
- 32.
Here \(s\ntriangleleft t,\) for terms s and t, is an abbreviation of \(\neg (s\lhd t),\) analogous to the case of s≠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.
- 33.
A particularly interesting point that I’ll leave for future research has been brought to my attention by an anonymous referee: next to the kind of self-reference we obtain, for example, by means of the diagonal lemma, there is also what’s sometimes called empirical self-reference: self-reference that depends on contingent facts about the world. Consider the sentence “the 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,” for example. Properly formalized, using, for example, a description operator, this sentence will create a ground-theoretic paradox of self-reference in the same way as the sentence δ in the proof of Theorem 3—but only because it is in fact the last such example sentence, otherwise it would simply be false. To deal with such forms of self-reference 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.
Notes
Acknowledgements
I’d like to thank Hannes Leitgeb, Benjamin Schnieder, Albert Anglberger, Thomas Schindler, Lavinia Picollo, Johannes Stern, Martin Fischer, Tim Button, the participants of the workshop “Logical and Metaphysical Perspectives on Grounding” at the GAP.9 in Osnabrück, O. Foisch, and two anonymous referees for their helpful comments and suggestions. Part of this research was supported by the Alexander von Humboldt Foundation.
References
- 1.Boolos, G. et al. (2007). Computability and logic. Cambridge University Press.Google Scholar
- 2.Correia, F. (2013). Logical grounds, Review of Symbolic Logic 7.1 (pp. 31–59).Google Scholar
- 3.Donaldson, T. (2016). The (metaphysical) foundations of arithmetic?, Noûs. doi:10.1111/nous.12147.
- 4.Feferman, S. (1991). Reflecting on incompleteness. Journal of Symbolic Logic, 56.1, 1–49.CrossRefGoogle Scholar
- 5.Fine, K. (2012). Guide to Ground. In Correia, F., & Schnieder, B. (Eds.), Metaphysical Grounding: Understanding the Structure of Reality (pp. 37–80): Cambridge University Press.Google Scholar
- 6.Fine, K. (2010). Some puzzles of ground, Notre Dame Journal of Formal Logic 51.1 (pp. 97–118).Google Scholar
- 7.Fine, K. (2012). The pure logic of ground, Review of Symbolic Logic 5.1 (pp. 1–25).Google Scholar
- 8.Halbach, V. (2011). Axiomatic theories of truth. Cambridge University Press.Google Scholar
- 9.Halbach, V., & Visser, A. (2014). Self-reference in arithmetic I, Review of Symbolic Logic 7.4 (pp. 671–691).Google Scholar
- 10.Jech, T. (2002). Set theory. Springer.Google Scholar
- 11.Jenkins, C.S. (2011). Is metaphysical dependence irreflexive?, The Monist 94.2 (pp. 267–276).Google Scholar
- 12.Korbmacher, J. (forthcoming). Axiomatic Theories of Partial Ground I, Journal of Philosophical Logic (pp. 1–31).Google Scholar
- 13.Korbmacher, J. (2015). Yet another puzzle of ground, Kriterion 29.2 (pp. 1–10).Google Scholar
- 14.Krämer, S. (2013). A simpler puzzle of ground, Thought: A Journal of Philosophy 2.2 (pp. 85–89).Google Scholar
- 15.Litland, J.E. (2015). Grounding, Explanation, and the limit of internality, Philosophical review 124.4.Google Scholar
- 16.Litland, J.E. (2013). On some counterexamples to the transitivity of grounding, Essays in Philosophy 14.1 (pp. 19–32).Google Scholar
- 17.Pohlers, W. (2009). Proof Theory. The first step into impredicativity. Springer.Google Scholar
- 18.Raven, M.J. (2013). Is ground a strict partial order?, American Philosophical Quarterly 50.2 (pp. 191–99).Google Scholar
- 19.Schaffer, J. (2012). Grounding, transitivity, and contrastivity. In Correia, F., & Schnieder, B. (Eds.), Metaphysical Grounding: Understanding the Structure of Reality (pp. 122–138): Cambridge University Press.Google Scholar
- 20.Tarski, A. (1944). The semantic conception of truth: and the foundations of semantics, Philosophy and Phenomenological Research 4.3 (pp. 341–76).Google Scholar
- 21.Yablo, S. (1993). Paradox Without Self-Reference, Analysis 53.4 (pp. 251–52).Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.