1 24.400 Proseminar in philosophy I Fall 2003 Tarski’s Definition of Truth Let L be a first-order language with identity, where the logical vocabulary is ‘∀’, ‘&’, ‘~’, together with an infinite supply of variables ‘x1’, ‘x2’,… The non-logical vocabulary of L comprises ‘Bernhard’, ‘Clare’, ‘Jason’, ‘Johanna’, ‘Selim’, and a two-place predicate ‘L’. The quantifiers of L are stipulated to range over MIT graduate students, ‘=’ is interpreted as expressing the identity relation, ‘Bernhard’, ‘Clare’,… as referring to Bernhard, Clare,… respectively, and ‘L’ as expressing the loving relation. A materially adequate and formally correct definition of ‘true sentence of L’ is as follows: Definition of the denotation of a term relative to a sequence (i) The ith variable of L denotes an object o relative to a sequence S iff o is the ith member of S. (So, relative to <Adina, Adam, Bernhard, Clare, Adam, Asta, …>, ‘x4’ denotes Clare.) (ii) A name ν of L denotes o relative to S iff ν=‘ Bernhard’ and o=Bernhard, or ν=‘Clare’ and o=Clare, or ν=‘Jason’ and o=Jason, or ν=‘Johanna’ and o=Johanna, or ν=‘Selim’ and o=Selim. Definition of the truth of a formula relative to a sequence (or of a sequence satisfying a formula) (i) An atomic formula ⎡(α = β)⎤ is true relative to a sequence S iff the object denoted by the term α relative to S is identical to the object denoted by the term β relative to S. ⎤[Alternatively: a sequence S satisfies an atomic formula ⎡(α = β) iff the object denoted by the term α relative to S is identical to the object denoted by the term β relative to S.] (ii) An atomic formula ⎡(Lαβ)⎤ is true relative to a sequence S iff the object denoted by the term α relative to S loves the object denoted by the term β relative to S. (iii) A formula ⎡(~φ)⎤ is true relative to S iff φ is not true relative to S.2 (iv) A formula ⎡(φ&ψ)⎤ is true relative to S iff φ is true relative to S and ψ is true relative to S. (v) A formula ⎡(∀νφ)⎤, where ν is the ith variable, is true relative to S iff φ is true relative to every sequence S* that differs from S at most in the ith place. Then, for all sentences x, x is a true sentence of L iff x is true relative to every sequence of graduate students (or: is satisfied by every sequence). So far we only have an “inductive definition” of ‘denotation relative to a sequence’ and ‘true relative to a sequence’. That is, we have fixed the application of these expressions without supplying other expressions to which they are equivalent. But this may be remedied as follows: α denotes o relative to S iff there is a set D such that <α, o, S> ∈ D, and the members of D are exactly the triples <γ, o#, S#> such that either (i) γ is the ith variable and o# is the ith member of S#; or (ii) γ is a name and either γ=‘Bernhard’ and o#=Bernhard, or γ=‘Clare’ and o#=Clare, or γ=‘Jason’ and o#=Jason, or γ=‘Johanna’ and o#=Johanna, or γ=‘Selim’ and o#=Selim. φ is true relative to S iff there is a set T such that <φ, S> ∈ T, and for all formulas x and sequences S#, the members of T are exactly the pairs <x, S#> such that either (i) x is ⎡(α = β)⎤ and the object denoted by the term α relative to S# is identical to the object denoted by the term β relative to S#; or (ii) x is ⎡(Lαβ)⎤ and the object denoted by the term α relative to S# loves the object denoted by the term β relative to S#; or (iii) x is ⎡(~φ)⎤ and <φ, S#> ∉ T; or (iv) x is ⎡(π&ψ)⎤ and <π, S#> ∈ T and <ψ, S#> ∈ T; or (v) x is ⎡(∀νψ)⎤, where n is the ith variable, and <ψ, S*> ∈ T, where S* differs from S at most in the ith place. Then the explicit definition of ‘true sentence of L’ is: For all sentences x, x is a true sentence of L iff there is a set TL such that x ∈ TL and for all sentences y of L, y ∈ TL iff there is a set T such that for all sequences S′ , <y, S′> ∈ T, and the members of T are exactly the pairs <x, S> such that either (i) x is ⎡(α = β)⎤ and there is a set D such that: for all terms γ, objects o#, and sequences S#, the members of D are exactly the triples <γ, o#, S#> such that either (a) γ is the ith variable and o# is the ith member of S#; or (b) γ is a name and either γ=‘Bernhard’ and o#=Bernhard, or γ=‘Clare’ and o#=Clare, or γ=‘Jason’ and o#=Jason, or γ=‘Johanna’ and o#=Johanna, or γ=‘Selim’ and o#=Selim. and there are objects o′, o′′ such that <α, o′, S> ∈ D and <β, o′′, S> ∈ D, and o′ is identical to o′′;3 or: (ii) x is ⎡(Lαβ)⎤ and there is a set D as above and there are objects o′, o′′ such that <α, o′, S> ∈ D and <β, o′′, S> ∈ D, and o′ loves o′′; or: (iii) x is ⎡(~φ)⎤ and <φ, S> ∉ T; or: (iv) x is ⎡(π&ψ)⎤ and <π, S> ∈ T and <ψ, S> ∈ T; or: (v) x is ⎡(∀νψ)⎤, where ν is the ith variable, and <ψ, S*> ∈ T, where S* differs from S at most in the ith