Outline Truth, Truth, Subderivations Subderivations and the Liar and the Liar 1. Why Should I Care about the Liar Sentence? Why Should I Why Should I Care about the Care about the Liar Sentence? Liar Sentence? Uses of the Uses of the Truth, Subderivations and the Liar 2. Uses of the Truth Concept - (i) Disquotation. Truth Concept - Truth Concept - (i) Disquotation. (i) Disquotation. Uses of the Uses of the Truth Concept - Truth Concept - (ii) Challenging (ii) Challenging 3. Uses of the Truth Concept - (ii) Challenging Assumptions. Assumptions. Assumptions. Formalization of Formalization of the Main the Main Intuition. Intuition. 4. Formalization of the Main Intuition. Main Technical Main Technical Results. Results. 5. Main Technical Results. 1 / 41 2 / 41 The Liar Sentence. Truth, Truth, Subderivations Subderivations and the Liar and the Liar • Let L be the sentence: Why Should I Why Should I This sentence is false. Care about the Care about the Liar Sentence? Liar Sentence? Uses of the Uses of the • This sentence causes trouble. Truth Concept - Truth Concept - (i) Disquotation. (i) Disquotation. Uses of the Uses of the If it is true, then it is false. So it can’t be true. Thus, it is false. Why Should I Care about the Liar Sentence? Truth Concept - Truth Concept - (ii) Challenging (ii) Challenging Assumptions. Assumptions. If it is false, then it is true. So it can’t be false. Thus, it is true. Formalization of Formalization of the Main the Main Thus, L is true and false. Intuition. Intuition. Main Technical Main Technical Results. Results. • But no sentence can be true and false. So we have a problem. • Is there an obvious way out? And if not, who cares? Is this a puzzle that should keep us up at night? 3 / 41 4 / 41
The Collapse of Logic. Truth, Truth, • Using the principles of logic alone, we have argued that L is both Subderivations Subderivations and the Liar and the Liar true and false; i.e., L & ¬ L . • (Aristotle) In classical logic, anything follows from a contradiction. Why Should I Why Should I Care about the Care about the Liar Sentence? Liar Sentence? • So using only the principles of logic, we can prove anything . • To see why - L is true. So Uses of the Uses of the Truth Concept - Truth Concept - • But the whole point of logic was to provide tools for distinguishing (i) Disquotation. (i) Disquotation. L or Santa Claus exists things that we are entitled to assert or infer from things we aren’t Uses of the Uses of the Truth Concept - Truth Concept - entitled to assert or infer. is true. (ii) Challenging (ii) Challenging Assumptions. Assumptions. • So unless something we have said is wrong, the whole project of • But L is also false. And from Formalization of Formalization of the Main the Main logic collapses. Intuition. Intuition. L or Santa Claus exists Main Technical Main Technical Results. Results. and L is false, it follows that Santa Claus exists. 5 / 41 6 / 41 A Quick Solution? (i) A Quick Solution? (i) Truth, Truth, Subderivations Subderivations and the Liar and the Liar • Di ffi culty: So we are rejecting the claim Why Should I Why Should I L or ¬ L . Care about the Care about the Liar Sentence? Liar Sentence? Uses of the Uses of the • Presumably this means that we are endorsing Truth Concept - • Our reasoning presupposes that the Liar sentence is true or false: Truth Concept - (i) Disquotation. (i) Disquotation. If it is true, then it is false. So it can’t be true. Thus, it is false. Uses of the Uses of the ¬ ( L or ¬ L ). Truth Concept - Truth Concept - (ii) Challenging (ii) Challenging • But perhaps some sentences (such as L ) are neither true nor false. Assumptions. Assumptions. • But in classical logic, this is equivalent to Formalization of Formalization of the Main the Main ¬ L & ¬¬ L . Intuition. Intuition. Main Technical Main Technical Results. Results. • And so we still have a contradiction. • So if we want to take this approach, we have to revise classical logic quite deeply. This requires care. 7 / 41 8 / 41
A Quick Solution? (ii) A Quick Solution? (ii) Truth, Truth, Subderivations Subderivations and the Liar and the Liar • Response 2: One can explicitly construct mathematical analogs of Why Should I Why Should I the Liar. Care about the Care about the • Is the Liar Sentence really a grammatical sentence? If not, we don’t Liar Sentence? Liar Sentence? • Take the first-order language of arithmetic ( { 0 , 0 , + , ⇥ } ) and add a have to worry about it. Uses of the Uses of the Truth Concept - Truth Concept - one-place predicate T ( · ). (i) Disquotation. (i) Disquotation. • Response 1: There are versions of the Liar that are not manifestly Uses of the Uses of the • Then one can construct a sentence L in this language such that self-referential. For example: Truth Concept - Truth Concept - (ii) Challenging (ii) Challenging Assumptions. Assumptions. The topmost sentence written on the blackboard in my o ffi ce is false. PA ` L $ ¬ T ( p L q ) Formalization of Formalization of the Main the Main Intuition. Intuition. • Does whether a sentence is grammatical really depend on where it • On what grounds could we say that L is ungrammatical? Main Technical Main Technical is or isn’t written? Results. Results. • Given the di ffi culty of these challenges, we follow the modern tradition that takes grammatical Liar sentences to really exist. 9 / 41 10 / 41 Truth, Truth, Subderivations Subderivations and the Liar and the Liar Why Should I Why Should I Care about the Care about the Liar Sentence? Liar Sentence? Uses of the Uses of the • It follows that some part of logic (as it is traditionally understood) Truth Concept - Truth Concept - (i) Disquotation. (i) Disquotation. or the way in which we understand the concept of truth must be Uses of the Uses of the revised. Uses of the Truth Concept - (i) Disquotation. Truth Concept - Truth Concept - (ii) Challenging (ii) Challenging Assumptions. Assumptions. • Here, there is no consensus. Formalization of Formalization of the Main the Main Intuition. Intuition. Main Technical Main Technical Results. Results. 11 / 41 12 / 41
Uses of the Truth Concept - (i) Disquotation. Uses of the Truth Concept - (i) Disquotation. Truth, Truth, Subderivations Subderivations and the Liar • What is the utility of the concept of truth? and the Liar • In order for the concept of truth to be usuble in this way, we need • (Tarski, Quine) We use the concept of truth to endorse sentences / to endorse something like the T-schema Why Should I Why Should I Care about the Care about the endorse descriptions of sentences. Liar Sentence? Liar Sentence? for any proposition p , T ( p p q ) $ p . Uses of the Uses of the • One use of the truth concept - ’disquotational’ Truth Concept - Truth Concept - (i) Disquotation. (i) Disquotation. Uses of the Uses of the ’Grass is green’ is true. • This means we must endorse Truth Concept - Truth Concept - (ii) Challenging (ii) Challenging Assumptions. Assumptions. • Another use - to endorse a sentence under a description: T ( p L q ) $ L . Formalization of Formalization of the Main the Main Intuition. Intuition. ’What John said was probably true.’ Main Technical Main Technical which gets us into trouble. Results. Results. (Note: maybe the speaker doesn’t even know what John said) • I would like to argue that the concept of truth is used in other ways, and that realizing this o ff ers us another way out. ( 9 s ) (John-just-said( s ) & Probably(True( s ))) . 13 / 41 14 / 41 Uses of the Truth Concept - (ii) Challenging Assumptions. Truth, Truth, Subderivations Subderivations and the Liar and the Liar • I want to point out a di ff erent use of the truth concept in Why Should I Why Should I challenging assumptions. Care about the Care about the Liar Sentence? Liar Sentence? Uses of the Uses of the Imagine a prosecutor arguing: Truth Concept - Truth Concept - (i) Disquotation. (i) Disquotation. Suppose John was in the park at 5pm. Uses of the Truth Concept - (ii) Challenging Uses of the Uses of the Truth Concept - Truth Concept - Then he would have had access to a weapon, (ii) Challenging (ii) Challenging Assumptions. Assumptions. Assumptions. He would have had ample opportunity to kill the victim, Formalization of Formalization of ... the Main the Main Intuition. Intuition. He should be convicted. Main Technical Main Technical Results. Results. John’s lawyer retorts: But suppose it’s not true that John was in the park at 5pm ... 15 / 41 16 / 41
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