Disquotation and Infinite Conjunctions

One of the main logical functions of the truth predicate is to enable us to express so-called ‘infinite conjunctions’. Several authors claim that the truth predicate can serve this function only if it is fully disquotational (transparent), which leads to triviality in classical logic. As a consequence, many have concluded that classical logic should be rejected. The purpose of this paper is threefold. First, we consider two accounts available in the literature of what it means to express infinite conjunctions with a truth predicate and argue that they fail to support the necessity of transparency for that purpose. Second, we show that, with the aid of some regimentation, many expressive functions of the truth predicate can actually be performed using truth principles that are consistent in classical logic. Finally, we suggest a reconceptualisation of deflationism, according to which the principles that govern the use of the truth predicate in natural language are largely irrelevant for the question of what formal theory of truth we should adopt. Many philosophers think that the paradoxes pose a special problem for deflationists; we will argue, on the contrary, that deflationists are in a much better position to deal with the paradoxes than their opponents. & Lavinia Picollo Lavinia.Picollo@lrz.uni-muenchen.de Thomas Schindler thomas.schindler1980@gmail.com 1 Munich Center for Mathematical Philosophy, LMU Munich, Germany 2 Clare College, University of Cambridge, Cambridge, UK 123 Erkenn DOI 10.1007/s10670-017-9919-x

Due to the paradoxes, a truth predicate satisfying a transparency principle is not possible in a classical context, on pain of triviality.
Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions Due to the paradoxes, a truth predicate satisfying a transparency principle is not possible in a classical context, on pain of triviality.
Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions We cast doubt on the necessity or sufficiency of transpacency principles to grant truth these expressive powers.
1. We argue that the two most promising accounts of what it means for sentences like (1) to express all the Ps place unreasonable requirements on truth theories.
2. We show that the reasonable bits of both accounts can be met in classical logic, adopting a consistent subprinciple of transparency. We conclude that so far the expression of infinite conjunctions carries no need to abandon classical logic, nor to sacrifice this expressive power if one wishes to remain classical. The infinite conjunction response In formal terms, that (1) implies and is implied by all the Ps means that the following inferences hold: Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The infinite conjunction response In formal terms, that (1) implies and is implied by all the Ps means that the following inferences hold: Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The infinite conjunction response In formal terms, that (1) implies and is implied by all the Ps means that the following inferences hold: Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The infinite conjunction response in trouble It's only reasonable to demand that -Elim holds in a formal theory of truth, not -Intro.
In classical logic the left-to-right direction of the T-schema, this is, or, equivalently, T-Elim, suffices to guarantee the inference -Elim.
E.g. in classical logic T-Out is consistent with . Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The infinite conjunction response in trouble It's only reasonable to demand that -Elim holds in a formal theory of truth, not -Intro.
In classical logic the left-to-right direction of the T-schema, this is, or, equivalently, T-Elim, suffices to guarantee the inference -Elim.
E.g. in classical logic T-Out is consistent with . Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The infinite conjunction response in trouble It's only reasonable to demand that -Elim holds in a formal theory of truth, not -Intro.
In classical logic the left-to-right direction of the T-schema, this is, or, equivalently, T-Elim, suffices to guarantee the inference -Elim.
E.g. in classical logic T-Out is consistent with . Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response Observation Let be any classical theory extending that contains T-Out. Then -Elim holds in .
Proof : Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The finite axiomatisation response in trouble In classical logic (typed) T-Out or T-Elim suffice to guarantee finite axiomatizations in the typed case.
In the untyped case it's only reasonable to demand the right-to-left direction of the biconditional because, taking C to be ∀x(Px → Tr(x)), the opposite direction entails -Intro.
For the left-to-right direction (untyped) T-Out or T-Elim are sufficient.
Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The finite axiomatisation response in trouble In classical logic (typed) T-Out or T-Elim suffice to guarantee finite axiomatizations in the typed case.
In the untyped case it's only reasonable to demand the right-to-left direction of the biconditional because, taking C to be ∀x(Px → Tr(x)), the opposite direction entails -Intro.
For the left-to-right direction (untyped) T-Out or T-Elim are sufficient.
Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions The infinite conjunction response The finite axiomatisation response The finite axiomatisation response in trouble In classical logic (typed) T-Out or T-Elim suffice to guarantee finite axiomatizations in the typed case.
In the untyped case it's only reasonable to demand the right-to-left direction of the biconditional because, taking C to be ∀x(Px → Tr(x)), the opposite direction entails -Intro.
For the left-to-right direction (untyped) T-Out or T-Elim are sufficient.
of the language, T-In or T-Intro allow for the following inference: So ∀x(Px → Tr(x)) is the 'finite conjunction' of the Ps. But finite conjunctions are already possible in the language without the truth predicate.
of the language, T-In or T-Intro allow for the following inference: So ∀x(Px → Tr(x)) is the 'finite conjunction' of the Ps. But finite conjunctions are already possible in the language without the truth predicate.
Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions

Embedding generalizations in conditionals
Consider the following definition of knowledge: Here the truth predicate allows to finitely express all the instances If we know that B and that C('B'), we would like to conclude from (2) that K ('B'). That requires T-In or T-Intro.
However, there is no need to generalise on the instances of (3) by (2). We may well do so by a generalisation of the form where Px is true exactly of all instances of (3).
In classical logic T-Out or T-Elim allow us to infer K ('B') from (4) and C('B') ∧ B.

Introduction Expressing infinite conjunctions
What is full transparency good for?
The epistemological role of truth Drawing conclusions

Embedding generalizations in conditionals
Consider the following definition of knowledge: Here the truth predicate allows to finitely express all the instances If we know that B and that C('B'), we would like to conclude from (2) that K ('B'). That requires T-In or T-Intro.
However, there is no need to generalise on the instances of (3) by (2). We may well do so by a generalisation of the form where Px is true exactly of all instances of (3).
In classical logic T-Out or T-Elim allow us to infer K ('B') from (4) and C('B') ∧ B.

Introduction Expressing infinite conjunctions
What is full transparency good for?
The epistemological role of truth Drawing conclusions

Embedding generalizations in conditionals
Consider the following definition of knowledge: Here the truth predicate allows to finitely express all the instances If we know that B and that C('B'), we would like to conclude from (2) that K ('B'). That requires T-In or T-Intro.
However, there is no need to generalise on the instances of (3) by (2). We may well do so by a generalisation of the form where Px is true exactly of all instances of (3).
In classical logic T-Out or T-Elim allow us to infer K ('B') from (4) and C('B') ∧ B.

Introduction Expressing infinite conjunctions
What is full transparency good for?
The epistemological role of truth Drawing conclusions

Embedding generalizations in conditionals
Consider the following definition of knowledge: Here the truth predicate allows to finitely express all the instances If we know that B and that C('B'), we would like to conclude from (2) that K ('B'). That requires T-In or T-Intro.
However, there is no need to generalise on the instances of (3) by (2). We may well do so by a generalisation of the form where Px is true exactly of all instances of (3).
In classical logic T-Out or T-Elim allow us to infer K ('B') from (4) and C('B') ∧ B.

Introduction Expressing infinite conjunctions
What is full transparency good for?
The epistemological role of truth Drawing conclusions The epistemological role of truth Generalizations like (1) where P is satisfied only by a finite number of sentences can be useful not for logical but for epistemological reasons.
One might not know or remember the explicit articulation of one or (possibly finitely) many sentences while counting on a property P that applies only to them. Then, one can express the content of these sentences nonetheless, aided by the truth predicate: just utter (1).

Introduction Expressing infinite conjunctions
What is full transparency good for?
The epistemological role of truth Drawing conclusions The epistemological role of truth Generalizations like (1) where P is satisfied only by a finite number of sentences can be useful not for logical but for epistemological reasons.
One might not know or remember the explicit articulation of one or (possibly finitely) many sentences while counting on a property P that applies only to them. Then, one can express the content of these sentences nonetheless, aided by the truth predicate: just utter (1).

Introduction Expressing infinite conjunctions
What is full transparency good for?
The epistemological role of truth Drawing conclusions The epistemological role of truth Generalizations like (1) where P is satisfied only by a finite number of sentences can be useful not for logical but for epistemological reasons.
One might not know or remember the explicit articulation of one or (possibly finitely) many sentences while counting on a property P that applies only to them. Then, one can express the content of these sentences nonetheless, aided by the truth predicate: just utter (1). T-In or T-Intro allow us to introduce only finite generalizations, but these are in principle dispensable.
Many uses of T-In or T-Intro can be mimicked in classical T-Out-or T-Elim-theories.
So far there aren't enoguh reasons to abandon classical logic to have a truth predicate capable of expressing infinitely many sentences at once, nor to sacrifice part of this expressive power to remain classical.
In many cases, abandoning classical logic means that the truth predicate isn't capable of expressing infinite conjunctions, even if transparency principles hold in full. T-In or T-Intro allow us to introduce only finite generalizations, but these are in principle dispensable.
Many uses of T-In or T-Intro can be mimicked in classical T-Out-or T-Elim-theories.
So far there aren't enoguh reasons to abandon classical logic to have a truth predicate capable of expressing infinitely many sentences at once, nor to sacrifice part of this expressive power to remain classical.
In many cases, abandoning classical logic means that the truth predicate isn't capable of expressing infinite conjunctions, even if transparency principles hold in full. T-In or T-Intro allow us to introduce only finite generalizations, but these are in principle dispensable.
Many uses of T-In or T-Intro can be mimicked in classical T-Out-or T-Elim-theories.
So far there aren't enoguh reasons to abandon classical logic to have a truth predicate capable of expressing infinitely many sentences at once, nor to sacrifice part of this expressive power to remain classical.
In many cases, abandoning classical logic means that the truth predicate isn't capable of expressing infinite conjunctions, even if transparency principles hold in full. Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions

Remaining classical
In classical logic T-Out or T-Elim alone are enough to meet the reasonable requirements of both the 'infinite conjunction' and the 'finite axiomatisation' accounts of what it means for (1) to express all the Ps.
T-In or T-Intro allow us to introduce only finite generalizations, but these are in principle dispensable.
Many uses of T-In or T-Intro can be mimicked in classical T-Out-or T-Elim-theories.
So far there aren't enoguh reasons to abandon classical logic to have a truth predicate capable of expressing infinitely many sentences at once, nor to sacrifice part of this expressive power to remain classical.
In many cases, abandoning classical logic means that the truth predicate isn't capable of expressing infinite conjunctions, even if transparency principles hold in full. Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions

Remaining classical
In classical logic T-Out or T-Elim alone are enough to meet the reasonable requirements of both the 'infinite conjunction' and the 'finite axiomatisation' accounts of what it means for (1) to express all the Ps.
T-In or T-Intro allow us to introduce only finite generalizations, but these are in principle dispensable.
Many uses of T-In or T-Intro can be mimicked in classical T-Out-or T-Elim-theories.
So far there aren't enoguh reasons to abandon classical logic to have a truth predicate capable of expressing infinitely many sentences at once, nor to sacrifice part of this expressive power to remain classical.
In many cases, abandoning classical logic means that the truth predicate isn't capable of expressing infinite conjunctions, even if transparency principles hold in full. Introduction Expressing infinite conjunctions What is full transparency good for? Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions Introduction Expressing infinite conjunctions What is full transparency good for?
The epistemological role of truth Drawing conclusions