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,⦠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 < α < š
0, we have a different truth predicate T
α
that intuitively expresses truth at the level α: we have \(Tr_{1}, \mathellipsis , Tr_{\omega }, \mathellipsis , Tr_{\omega ^{\omega }},\mathellipsis , Tr_{\omega ^{\omega ^{\omega }}}, \mathellipsis ,\) where for αā β < š
0, we have T
r
α
ā T
r
β
.Footnote 21 For all ordinals 1 ⤠α ⤠š
0, we define the language \(\mathcal {L}_{<\alpha }\) as the language \(\mathcal {L}\) of PA extended with all the truth predicates T
β
for 0 < β < α:
$$\mathcal{L}_{<\alpha}=_{def}\mathcal{L}\cup\{Tr_{\beta}~|~0<\beta<\alpha\}.$$
Then we set:
$$\mathcal{L}_{\alpha}=_{def} \mathcal{L}_{<\alpha+1},$$
for all ordinals 0 ⤠α < 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 ⤠α ⤠š
0, the language \(\mathcal {L}_{<\alpha }\) talks about the truths at the levels strictly below α and \(\mathcal {L}_{\alpha }\) talks about the truths at all levels up to and including α. When we are operating on the ordinal level α, the language \(\mathcal {L}_{<\alpha }\) is our intended object-language, 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 ⤠α ⤠š
0, the language \(\mathcal {L}_{<\alpha }^{\lhd }\) is \(\mathcal {L}_{<\alpha }\) extended with our binary ground predicate \(\lhd \):
$$\mathcal{L}_{<\alpha}^{\lhd}=_{def}\mathcal{L}_{<\alpha}\cup\{\lhd\}.$$
And we set:
$$\mathcal{L}_{\alpha}^{\lhd}=_{def}\mathcal{L}_{\alpha}\cup\{\lhd\},$$
for all ordinals 0 ⤠α < š
0. When we are operating on the ordinal level α, weāll use \(\mathcal {L}_{\alpha }^{\lhd }\) as our meta-language for the object-language 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 < α < š
0, the theory P
A
T
<α
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
<α
is the result of extending P
A
T
<α
with all the missing instances of the induction scheme over \(\mathcal {L}_{\alpha }^{\lhd }\). For 0 ⤠α < š
0, the theory P
A
T
α
, then, is P
A
T
<α+1 and similarly P
A
G
α
is P
A
G
<α+1. Thus, P
A
T
0 is PAT and P
A
G
0 is PAG. In P
A
T
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 \(\varphi \in \mathcal {L}_{<\alpha }\).Footnote 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 #
T
r
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 S
e
n
t
<α
.
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 < α < š
0. Thus, we wish to talk about ground between truths on all the ordinal levels 0 < β < α. 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 < β < α:
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 β in the hierarchy, then it is true at level β 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 β, then it is true at level β. To illustrate, if we let α = 2, we get the following axiom pair:
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 α. Thus, for all ordinals 0 < β < α, 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 α.
In the case of T 3, we wish to make sure that a truth predicate T
r
β
, for an ordinal 0 < β < α, only applies to sentences on levels below βāin compliance with Tarskiās distinction. Thus, we postulate for all 0 < β < α:
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:
Thus, on the level α = 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 meta-language. Finally, we have to modify our upwards and downwards directed grounding axioms to apply on all levels below α. 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 < β < α:
-
(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 ⤠š
0, the predicational theory P
G
<α
of
ground up to α, consists of the axioms of P
A
G
<α
plus the following axioms for all 0 < β < α:
Typed Ground Axioms: Typed Truth Axioms:
G1
\(\forall x\neg (x\lhd x)\) T\(_{1}^{\beta } \forall s\forall t(Tr_{\beta }(s\underset {.}{=}t)\leftrightarrow s^{\circ }=t^{\circ })\)
G2
\(\forall x\forall y\forall z(x\lhd y\land y\lhd z\to x\lhd z)\) 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:
D1 \(\forall x(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\neg }x)\to x\lhd \underset {.}{\neg }\underset {.}{\neg }x)\)
-
D2 \(\forall x\forall y (Tr_{\beta }(x\underset {.}{\lor }y)\to (Tr_{\beta }(x)\to x\lhd x\underset {.}{\lor }y )\land (Tr_{\beta }(y)\to y\lhd x\underset {.}{\lor }y))\)
-
D3 \(\forall x\forall y (Tr_{\beta }(x\underset {.}{\land }y)\to (x\lhd x\underset {.}{\land } y)\land (y\lhd x\underset {.}{\land } y))\)
-
D4 \(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg } (x\underset {.}{\land } y))\to (Tr_{\beta }(\underset {.}{\neg }x)\to \underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (Tr_{\beta }(\underset {.}{\neg }y)\to \underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)
-
D5 \(\forall x\forall y (Tr_{\beta }(\underset {.}{\neg } (x\underset {.}{\lor } y))\to (\underset {.}{\neg } x\lhd \underset {.}{\neg }(x\underset {.}{\lor } y))\land (\underset {.}{\neg } y\lhd \underset {.}{\neg }(x\underset {.}{\lor } y)))\)
-
D6 \(\forall x (Tr_{\beta }(\underset {.}{\exists } v x(v))\to \exists t(x(t/v)\lhd \underset {.}{\exists } v x))\)
-
D7
\(\forall x \forall v(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\exists }v x)\to \forall t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\exists }v x))\)
-
D8 \(\forall x \forall v(Tr_{\beta }(\underset {.}{\forall }v x\to \forall t(x(t/v)\lhd \underset {.}{\forall }v x))\)
-
D9 \(\forall x \forall v(Tr_{\beta }(\underset {.}{\neg }\underset {.}{\forall } v x)\to \exists t(\underset {.}{\neg }x(t/v)\lhd \underset {.}{\neg }\underset {.}{\forall } v x))\)
For 0 ⤠α < š
0, we define P
G
α
as P
G
<α+1.
To illustrate what P
G
α
looks like for different αās, 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 truth-predicate 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 < α < š
0, P
G
α
contains P
G
1, we can say that P
G
α
essentially is (in the precise sense sketched above) an extension of PG. For α bigger than one, P
G
α
essentially consists of α-many copies of P
G, one for every \(\mathcal {L}_{\beta }\) and truth predicate T
r
β
, where 1 < β < α. 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 < β ⤠α, 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\(_{1}^{2}\):
$$Tr_{2}(\ulcorner Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner)\to \ulcorner Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner\lhd \ulcorner \neg\neg Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner.$$
Indeed, using G\(_{3b}^{2}\) we can infer from this that:
$$\ulcorner Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner\lhd \ulcorner \neg\neg Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner\to Tr_{2}(\ulcorner \neg\neg Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner),$$
which together with the previous formula gives us:
$$Tr_{2}(\ulcorner Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner)\to Tr_{2}(\ulcorner \neg\neg Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner).$$
The other direction:
$$Tr_{2}(\ulcorner \neg\neg Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner)\to Tr_{2}(\ulcorner Tr_{1}(\ulcorner\overline{0}=\overline{0}\urcorner)\urcorner)$$
can be shown analogously using D\(_{1}^{2}\) and G\(_{3a}^{2}\). Generalizing this idea, we get more substantial truth-theoretic theorems in P
G
2, such as:
$$\forall x(Tr_{2}(\ulcorner Tr_{1}(\dot{x})\urcorner)\leftrightarrow Tr_{2}(\ulcorner \neg\neg Tr_{1}(\dot{x})\urcorner)),$$
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 α > 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 α is pretty straight-forward. We get the following axioms for every γ < α:
-
(APU\(_{T}^{\gamma }\)) \(\forall x(Tr_{\gamma }(x)\to x\lhd \underset {.}{Tr_{\gamma }}(\dot {x}))\)
-
(APU\(_{F}^{\gamma }\)) \(\forall x(Tr_{\gamma }(\underset {.}{\neg }x)\to \underset {.}{\neg }x\lhd \underset {.}{\neg }\underset {.}{Tr_{\gamma }}(\dot {x}))\)
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, straight-forwardly, we get for all γ < β ⤠α:
-
(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 ⤠α < š
0, the theory P
G
A
α
of partial ground with the Aristotelian principles up to α consists of the axioms of P
G
α
plus the following axioms for all γ < β ⤠α:
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
<α
is defined as \(\bigcup _{\beta <\alpha }PGA_{\alpha },\) for all 0 < α ⤠š
0.
To see how P
G
A
α
works for different 0 ⤠α < 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:
$$Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\to \ulcorner \overline{0}=\overline{0}\urcorner\lhd \ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner,$$
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:
$$Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner).$$
So putting the two together, we get:
$$\ulcorner \overline{0}=\overline{0}\urcorner\lhd \ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner.$$
Now, using the instance:
$$\ulcorner \overline{0}=\overline{0}\urcorner\lhd \ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner\!\to\! (Sent_{<2}(\ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner)\to Tr_{2}(\ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner))$$
of the axiom G\(_{3b}^{2}\), and since:
$$Sent_{<2}(\ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner)$$
is derivable in PA, we can infer:
$$Tr_{2}(\ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner).$$
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 \(AP{U_{T}^{1}}\):
$$\forall x(Tr_{1}(x)\to x\lhd\underset{.}{Tr_{1}}(\dot{x}))$$
and \(APD_{T}^{2,1}\):
$$\forall x(Tr_{2}(\underset{.}{Tr_{1}}(\dot{x}))\to x\lhd\underset{.}{Tr_{1}}(\dot{x}))$$
together, we can actually prove:
$$\forall x(Tr_{2}(\underset{.}{Tr_{1}}(\dot{x}))\leftrightarrow Tr_{1}(x))$$
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 truth-theoretic 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 < α < š
0, P
G
A
α
proves the theory P
R
T
α
of positive ramified truth up to α. Indeed, we can show that P
G
A
α
is a proof-theoretically conservative extension of P
R
T
α
.
Conservativity and Models
The theory P
T
<α
of positive ramified truth up to an ordinal level 1 ⤠α ⤠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
<α
:
Definition 3 (āPositive Ramified Truthā)
For all ordinals 1 ⤠α ⤠0, the theory P
R
T
<α
of positive ramified truth up to α consists of the axioms of P
A
T
<α
plus the following axioms for all γ < β < α:
-
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
β
(x) ā S
e
n
t
<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 P
R
T
α
, for 0 ⤠α < š
0, is defined as P
R
T
<α+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
<α
, for 1 < α ⤠š
0, however, is a much stronger theory of truth than PTāit formalizes the Tarskian hierarchy up to the level α.Footnote 24 For example, P
G
T
2 contains the axioms:
$$Tr_{2}(\ulcorner Tr_{1}(\ulcorner \overline{0}=\overline{0}\urcorner)\urcorner), $$
and
$$\forall x(Tr_{2}(\underset{.}{Tr_{1}}(\dot{x}))\leftrightarrow Tr_{1}(x)), $$
just like P
G
A
2. Indeed, we get:
Proposition 1
For all ordinals 1 ⤠α ⤠š
0,the theory P
G
A
<α
proves the theory P
R
T
<α
:
\(PGA_{<\alpha }\vdash PRT_{<\alpha }\)
.
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 β < α. The interesting cases are the new axioms \(RP_{10-13}^{\beta },\) for β < α, which can be shown from the typed Aristotelian principles \(APU^{\gamma }_{T/F}\) and \(APD_{T/F}^{\beta ,\gamma },\) for γ < β ⤠α, the typed truth axiom \(T_{3}^{\beta },\) and the typed ground axioms \(G_{3a/b}^{\beta },\) for β < α, 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 γ < β < α
Let x be a fresh variable for ā-Intro. 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 ā-Intro. 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
<γ
(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
γ
(x). Thus, we get \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}(x))\to Tr_{\gamma }(x)\) by ā-Intro.
(\(\Leftarrow \)): Assume (ā”) T
r
γ
(x) for another ā-Intro. Using PA and \(G_{4}^{\gamma },\) we get S
e
n
t
<γ
(x). From this and PA, we can derive for all γ < β < α 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 ā-Intro.
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 ā-Intro as desired.
-
\(\vdash _{PGA_{<\alpha }}\forall x(Sent_{<\gamma }(x)\to (Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\leftrightarrow Tr_{\beta }(x)))\) for γ < β < α.
Let x be a fresh variable for ā-Intro. Assume S
e
n
t
<γ
(x) for a ā-Intro. 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 ā-Intro. 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
<β
(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
β
(x). Thus, we have \(Tr_{\beta }(\underset {.}{Tr_{\gamma }}\dot {x})\to Tr_{\beta }(x)\) by ā-Intro.(\(\Leftarrow \)): Assume T
r
β
(x) for yet another ā-Intro. From this and \(APU_{T}^{\beta },\) we get \(x\lhd Tr_{\beta }(x)\). Since we have assumed S
e
n
t
<γ
(x), we get T
r
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 ā-Intro.
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 ā-Intro. 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 β < α < š
0, the theory P
G
A
<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 < 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 ⤠α < š
0, the theory P
G
A
α
is a proof-theoretically conservative extension of the theory P
R
T
α
. But first, we need to introduce some more technical preliminaries: It is well-known that we can extend the technique of Gƶdel 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 α is the numeral of the code #
α of α. 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 α à β = γ, 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 well-ordering < 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 α < β. With these preliminaries in place,Footnote 25 weāll define a slightly non-standard notion of complexity for the formulas in \(\mathcal {L}_{<\epsilon _{0}}\):
Definition 4 (ā Ļ-complexityā)
For all ordinals 1 ⤠α ⤠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 Ļ-complexity |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)
|¬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 Ļ-complexity 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 α < β,then Ļ Ć Ī± < Ļ Ć Ī²
.
Note that as a consequence, we get that for all ordinals 0 < α < š
0, if \(\varphi \in \mathcal {L}_{<\alpha },\) then \(|\varphi |_{\omega }<|Tr_{\alpha }(\ulcorner \varphi \urcorner )|_{\omega }\). In other words, Ļ-complexity has a sort of ātracking property:ā it can ātrackā the levels of Tarskiās hierarchy. Moreover, we can represent Ļ-complexity 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 Ļ-complexity |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
= α. Using this representation, we can show that Peano arithmetic proves that Ļ-complexity has the ātracking-propertyā in the following sense:
Lemma 5
For all ordinals 0 < β ⤠α < 0
:
$$\vdash_{PAG_{\alpha}}\forall x(Sent_{<\beta}(x)\to \underset{.}{c_{\omega}}(x)\underset{.}{<}\underset{.}{c_{\omega}}(\underset{.}{Tr_{\beta}}(\dot{x}))). $$
Footnote 27
Using Ļ-complexity, weāll obtain the main result of this section:
Theorem 1
For all 0 ⤠α < š
0,the theory P
G
A
α
is a proof-theoretically conservative extension of the theory P
R
T
α
.
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(¬Ļ) = ¬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
α
are also axioms of P
R
T
α
, 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
α
:
$$\forall x(Tr_{\alpha}(x)\to Sent_{<\alpha}(x)).$$
-
Finally, consider the axioms (APU\(_{T}^{\beta }\)):
$$\forall x(Tr_{\beta}(x)\to x\lhd\underset{.}{Tr_{\beta}}(\dot{x})),$$
where β < α. 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 ā-Intro and assume T
r
β
(x) for a ā-Intro. Using the axiom \(T_{3}^{\beta }\) of P
R
T
α
, we can infer that S
e
n
t
<β
(x). Moreover, since β < α by assumption, we can infer that \(Tr_{\alpha }(\underset {.}{Tr_{\beta }}(\dot {x}))\) and T
r
β
(x) using the axiom \(RP_{12}^{\alpha }\) of P
R
T
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 S
e
n
t
<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 ā-Intro, 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 ā-Intro, 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 ⤠α < š
0,the theory P
G
A
α
is consistent.
The proof of Theorem 1 essentially works because of the ātracking propertyā of Ļ-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 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 Ļ-complexity in the translationāthis is what allowed us to prove the result. The technique of the proof works for all ordinals α < š
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 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 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
< 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
<α
, where 1 ⤠α ⤠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
<α
, where 1 ⤠α ⤠š
0.
As in the case of P
T, there is a standard model of P
R
T
<α
, for 1 ⤠α ⤠š
0. A model for the language \(\mathcal {L}_{<\alpha }\) is a structure of the form (N,(S
β
)
β<α
), where for β < α, the set S
β
interprets the truth predicate \(Tr_{\beta }\in \mathcal {L}_{<\alpha }\). For 1 ⤠α ⤠š
0, we define the sets (S
β
)
β<α
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 α is a limit ordinal.
Then we get, for all 1 ⤠α ⤠š
0, that \((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha })\vDash PRT_{<\alpha }\). For 1 ⤠α ⤠š
0, the model (N,(S
β
)
β<α
) is the standard model of P
R
T
<α
āit models Tarskiās hierarchy of truths.
We now extend our definition of grounding-trees from the previous paper to grounding-trees over the standard model of
P
R
T
<α
:
Definition 5
Let 1 ⤠α ⤠ϵ0 and let (N,(S
β
)
β<α
) be the standard model of P
R
T
<α
. We define the grounding-trees over (N,(S
β
)
β<α
) 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 grounding-tree over (N,(S
β
)
β<α
) with #
Ļ as its root;
-
(ii)
if
is a grounding-tree \(\mathcal {T}\) over (N,(S
β
)
β<α
) with #
Ļ as its root, then
is a grounding-tree over (N,(S
β
)
β<α
) with #Ā¬Ā¬Ļ as its root;
-
(iii)
if
is a grounding-tree \(\mathcal {T}\) over (N, (S
β
)
β<α
) with #
Ļ as its root, then
is a grounding-tree over (N,(S
β
)
β<α
) with #(Ļ āØ Ļ) as its root;
-
(iv)
if
is a grounding-tree \(\mathcal {T}\) over (N, (S
β
)
β<α
) with #
Ļ as its root, then
is a grounding-tree over (N,(S
β
)
β<α
) with #(Ļ āØ Ļ) as its root;
-
(v)
if
are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}\) over (N, (S
β
)
β<α
) with #
Ļ,#
Ļ as their roots respectively, then
is a grounding-tree over (N,(S
β
)
β<α
) with #(Ļ ā§ Ļ) as its root;
-
(vi)
if
is a grounding-tree \(\mathcal {T}\) over (N, (S
β
)
β<α
) with #
Ļ(t) as its root, then
is a grounding-tree over (N,(S
β
)
β<α
) with #āx
Ļ(x) as its root;
-
(vii)
if
, ⦠are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}, \mathellipsis \) over (N,(S
β
)
β<α
) with #
Ļ(t
1),#
Ļ(t
2),⦠as their roots respectively, where t
1,t
2,⦠are all and only the terms of \(\mathcal {L}_{PA}\), then
is a grounding-tree over (N,(S
β
)
β<α
) with #āx
Ļ(x) as its root;
-
(viii)
if
is a grounding-tree \(\mathcal {T}\) over (N, (S
β
)
β<α
) with #Ā¬Ļ as its root, then
is a grounding-tree over (N,(S
β
)
β<α
) with #¬(Ļ ā§ Ļ) as its root;
-
(ix)
if
is a grounding-tree \(\mathcal {T}\) over (N, (S
β
)
β<α
) with #Ā¬Ļ as its root, then
is a grounding-tree over (N,(S
β
)
β<α
) with #¬(Ļ ā§ Ļ) as its root;
-
(x)
if
are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}\) over (N,(S
β
)
β<α
) with #¬Ļ,#Ā¬Ļ as their roots respectively, then
is a grounding-tree over (N, (S
β
)
β<α
) with #¬(Ļ āØ Ļ) as its root;
-
(xi)
if
is a grounding-tree \(\mathcal {T}\) over (N, (S
β
)
β<α
) with #¬Ļ(t) as its root, then
is a grounding-tree over (N,(S
β
)
β<α
) with #¬āx
Ļ(x) as its root;
-
(xii)
if
, ⦠are grounding-trees \(\mathcal {T}_{1},\mathcal {T}_{2}, \mathellipsis \) over (N,(S
β
)
β<α
) with #¬Ļ(t
1),#¬Ļ(t
2),⦠as their roots respectively, where t
1,t
2,⦠are all and only the terms of \(\mathcal {L}_{PA}\), then
is a grounding-tree over (N,(S
β
)
β<α
) with #āx
Ļ(x) as its root;
-
(xiii)
if
is a grounding-tree \(\mathcal {T}\) over (N,(S
β
)
β<α
) with #
Ļ as its root and #
Ļ ā S
β
, for β < α, then
is a grounding-tree over (N,(S
β
)
β<α
) with \(\#Tr_{\beta }(\ulcorner \varphi \urcorner )\) as its root;
-
(xiv)
if
is a grounding-tree \(\mathcal {T}\) over (N,(S
β
)
β<α
) with #Ā¬Ļ as its root and #
Ļ āS
β
, for β < α, then
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
We define the height \(h(\mathcal {T})\) of a grounding tree over (N,(S
β
)
β<α
) by saying that:
-
(i)
all grounding-trees over (N,(S
β
)
β<α
) of the form #
Ļ āS 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 \(\mathcal {T}_{1}, \mathcal {T}_{2}, \mathellipsis \):
$$h(\mathcal{T})=lub\{h(\mathcal{T}_{1}), h(\mathcal{T}_{2}), \mathellipsis\}+1,$$
where lub is the operation of taking the least upper bound.
We call a grounding-tree over (N,(S
β
)
β<α
)degenerate iff it is of height one.
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 Ļ such that \(DN_{\overline {0}=\overline {0}}(\#\varphi ),\) there is a grounding-tree 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 grounding-tree
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 grounding-trees. We need to use transfinite induction on the height of the grounding-trees. 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 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 donāt contain any cycles; and finally, (iii) transitive.
Lemma 6
Let 1 ⤠α ⤠š
0, (N,(S
β
)
β<α
)be the standard model of P
R
T
<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 #
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
β
)
β<α
) is a model of P
R
T
<α
. ā”
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 ⤠š
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 (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 ⤠α ⤠š
0
and let (N,(S
β
)
β<α
)be the standard model of P
R
T
<α
.
If
\(\mathcal {T}\)
is a grounding-tree over (N,(S
β
)
β<α
)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 ⤠α ⤠š
0, (N,(S
β
)
β<α
)be the standard model of P
R
T
<α
,and let
\(\mathcal {T}\)
be a grounding-tree over (N,(S
β
)
β<α
).
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 ⤠α ⤠š
0
and let (N,(S
β
)
β<α
)be the standard model of P
R
T
<α
.
If there is a grounding-tree
\(\mathcal {T}_{1}\)
over (N,(S
β
)
β<α
)with #
Ļ
as its root and #
Ļ
1, #
Ļ
2,ā¦as its leaves and there is grounding-tree
\(\mathcal {T}_{2}\)
over (N,(S
β
)
β<α
)with #
Ļ, #
Ļ
1, #
Ļ
2,ā¦as its leaves and #
š
as its root, then there is a grounding-tree
\(\mathcal {T}_{3}\)
over (N,(S
β
)
β<α
)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
<α
by saying that:
Definition 7
Let 1 ⤠α ⤠š
0 and let (N,(S
β
)
β<α
) be the standard model of P
R
T
<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 of P
G
A
<α
,for 1 ⤠α ⤠š
0
,
i.e.
\((\mathbf {N}, (\mathbf {S}_{\beta })_{\beta <\alpha }, \mathbf {R})\vDash PGA_{<\alpha }\)
Proof
By Lemmas 6, 8, and 9, grounding-trees over (N,(S
β
)
β<α
) behave appropriately and satisfy the basic ground axioms. Since (N,(S
β
)
β<α
) is a model of P
R
T
<α
, 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 γ < α 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 γ < α.
Let Ļ be a variable assignment over (N,(S
β
)
β<α
,R) and \({\sigma ^{\prime }}\) some x-variant 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 grounding-tree over (N,(S
β
)
β<α
). But then, by clause (xiii) of Definition 5,
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 #
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
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 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”