Whats a Theory to Do? Classicality with the Purpose of Capturing - PowerPoint PPT Presentation

Opening remarks Introducing STTT Classicality Leitgeb’s criteria What’s a Theory to Do? Classicality with the Purpose of Capturing Maja Jaakson Institute for Logic, Language and Computation Universiteit van Amsterdam October 5, 2012

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Opening remarks

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Opening remarks “But the more di ffi cult the task is, the greater would be the merit of accomplishing what such excellent thinkers—to mention the most illustrious only—as Frege, Russell and Hilbert have tried in vain: namely, to avoid the logical paradoxes without infringing classical logic. ” – Kurt Grelling, “The Logical Paradoxes”

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Outline

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Outline • Strict-Tolerant Transparent Truth (STTT)

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Outline • Strict-Tolerant Transparent Truth (STTT) • If classical, then meets all of Hannes Leitgeb’s criteria (“What Theories of Truth Should Be Like (but Cannot Be)”)

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Outline • Strict-Tolerant Transparent Truth (STTT) • If classical, then meets all of Hannes Leitgeb’s criteria (“What Theories of Truth Should Be Like (but Cannot Be)”) • Cobreros et al. do not make a compelling case for the classicality of their logic

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Outline • Strict-Tolerant Transparent Truth (STTT) • If classical, then meets all of Hannes Leitgeb’s criteria (“What Theories of Truth Should Be Like (but Cannot Be)”) • Cobreros et al. do not make a compelling case for the classicality of their logic • That’s okay; STTT still meets Leitgeb’s “real” criteria.

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Outline • Strict-Tolerant Transparent Truth (STTT) • If classical, then meets all of Hannes Leitgeb’s criteria (“What Theories of Truth Should Be Like (but Cannot Be)”) • Cobreros et al. do not make a compelling case for the classicality of their logic • That’s okay; STTT still meets Leitgeb’s “real” criteria. • How to decide between largely classical theories of truth which o ff er di ff erent treatments of paradoxical arguments?

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Strict-Tolerant Transparent Truth

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Strict-Tolerant Transparent Truth STTT is a first-order logic with a transparent truth predicate T and a quotation device h i . Definition A truth predicate is transparent i ff , where ' is some sentence in the language, all occurrences of T h ' i and ' are intersubstitutable salva veritate in all extensional contexts. • Nice properties: validates all T-biconditionals, represents truth as a predicate which respects compositionality, no type restrictions. • The fact that STTT’s consequence relation is not transitive plays an important role in accounting for paradoxes.

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models • Start with a three-valued model (with values { 1, 1 2 , 0 } ) for a base language L , which does not contain a truth predicate T .

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models • Start with a three-valued model (with values { 1, 1 2 , 0 } ) for a base language L , which does not contain a truth predicate T . • Strong Valuation Schema:

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models • Start with a three-valued model (with values { 1, 1 2 , 0 } ) for a base language L , which does not contain a truth predicate T . • Strong Valuation Schema: ( 1 if ( g ( t 1 ) , . . . , g ( t n )) 2 P in M v M g ( Pt 1 , . . . , t n ) = 0 otherwise

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models • Start with a three-valued model (with values { 1, 1 2 , 0 } ) for a base language L , which does not contain a truth predicate T . • Strong Valuation Schema: ( 1 if ( g ( t 1 ) , . . . , g ( t n )) 2 P in M v M g ( Pt 1 , . . . , t n ) = 0 otherwise 8 1 if v ( ' ) = 0 > < v M g ( ¬ ' ) = 0 if v ( ' ) = 1 > 1 if v ( ' ) = 1 : 2 2

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models • Start with a three-valued model (with values { 1, 1 2 , 0 } ) for a base language L , which does not contain a truth predicate T . • Strong Valuation Schema: ( 1 if ( g ( t 1 ) , . . . , g ( t n )) 2 P in M v M g ( Pt 1 , . . . , t n ) = 0 otherwise 8 1 if v ( ' ) = 0 > < v M g ( ¬ ' ) = 0 if v ( ' ) = 1 > 1 if v ( ' ) = 1 : 2 2 v M g ( ' ^ ) = min { v ( ' ) , v ( ) }

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models • Start with a three-valued model (with values { 1, 1 2 , 0 } ) for a base language L , which does not contain a truth predicate T . • Strong Valuation Schema: ( 1 if ( g ( t 1 ) , . . . , g ( t n )) 2 P in M v M g ( Pt 1 , . . . , t n ) = 0 otherwise 8 1 if v ( ' ) = 0 > < v M g ( ¬ ' ) = 0 if v ( ' ) = 1 > 1 if v ( ' ) = 1 : 2 2 v M g ( ' ^ ) = min { v ( ' ) , v ( ) } v M g ( 8 x ' ) = min { v g [ x 7! a ] ( ' ) | for all a in M}

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models and L + • STTT’s full language, L + :

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Kripke-Kleene models and L + • STTT’s full language, L + : 1. h ' i names ' . 2. Valuations of ' and T h ' i agree on all models. 3. Reference to sentences of L + within L + made possible by arithmetizing L + ’s syntax using G¨ odel numbering and Peano arithmetic.

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Validity

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Validity Definition STTT consequence: Γ 2 STTT ∆ i ff there is a KK model whereby every member of Γ gets truth value 1 and every member of ∆ gets value 0.

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Other trivalent logics

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Other trivalent logics • LEM not a validity in K3TT. • MP not a validity in LPTT. • Neither LEM nor MP is a validity of S3TT.

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Other trivalent logics • LEM not a validity in K3TT. • MP not a validity in LPTT. • Neither LEM nor MP is a validity of S3TT. • STTT preserves all classically-valid arguments.

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Other trivalent logics • LEM not a validity in K3TT. • MP not a validity in LPTT. • Neither LEM nor MP is a validity of S3TT. • STTT preserves all classically-valid arguments. Ripley result 1: Γ ✏ CL ∆ i ff Γ ✏ ST ∆ Ripley result 2: If Γ ✏ CL ∆ , then Γ ? ✏ STTT ∆ ? for any uniform substitution ? on the full language L + .

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Non-transitive consequence relation • STTT’s notion of consequence is not transitive .

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Non-transitive consequence relation • STTT’s notion of consequence is not transitive . • We have a Liar sentence � , which says in L + that ¬ T h � i .

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Non-transitive consequence relation • STTT’s notion of consequence is not transitive . • We have a Liar sentence � , which says in L + that ¬ T h � i . • Assume that, for some sentences ' and , there is a KK model M on which v M ( ' ) = 1 and v M ( ) = 0. Then = STTT � , since no KK model makes v ( ' ) = 1 and ' | = STTT , since no KK model makes v ( � ) = 0; and � | v ( � ) = 1 and v ( ) = 0. But note that ' 2 STTT , because our M is a countermodel; for v M ( ' ) = 1 and v M ( ) = 0.

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Non-transitive consequence relation (Cont’d)

Opening remarks Introducing STTT Classicality Leitgeb’s criteria Non-transitive consequence relation (Cont’d) • The only counterexamples to generalized transitivity will be of the following form: the arguments from Γ , ' to ∆ and from Γ to ' , ∆ will be STTT-valid, but the argument from Γ to ∆ will fail because there is a KK model M where all � 2 Γ and � 2 ∆ are such that v M ( � ) = 1 and v M ( � ) = 0, but v M ( ' ) = 1 2 .