Interpretations and Validity¶
4 Interpretations and Validity
4.1 Interpretations of L : Given a sentence Φ in L assign a denotation to each non-logical constant occurring in Φ
4.1.1 Specify Universe of discourse : non-empty domain D
4.1.2 Assign to each individual constant an element of D
4.1.3 Assign to each n-ary predicate an n-ary relation among element of D
4.1.4 Assign to each sentential letter one of truth values T or F
4.2 Truth : Φ is true under I
4.2.1 Let I be any interpretation and Φ be any quantifier-free sentence of L
4.2.1.1 If Φ is sentential letter, then Φ is true under I iff I assigns T to Φ
4.2.1.2 If Φ is atomic and not a sentential , then Φ is true under I iff the objects I assigns to the individual constants of Φ are related by the relation that I assigns to the predicate of Φ
4.2.1.3 If Φ = - Ψ, then Φ is true under I iff Ψ is not true under I
4.2.1.4 If Φ = (Ψ V χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I or χ is true under I or both
4.2.1.5 If Φ = (Ψ & χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I and χ is true under I
4.2.1.6 If Φ = (Ψ \(\implies) χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is not true under I or χ is true under I or both
4.2.1.7 If Φ = (Ψ \(\iff\) χ) for sentences Ψ, χ, then Φ is true under I iff either Ψ and χ are both true or both not true under I.
4.2.2 Φ is false under I iff Φ is not true under I
4.2.3 Every quantifier-free sentence is an atomic sentence or is a molecular compound of shorter sentences
4.2.4 Let I and I’ be interpretations of L, and β be an individual constant; the I is β-variant of I’ iff I and I’ are the same or differ only in what they assign to β
4.2.5 Let Φ be any sentence of L, α a variable, and β the first individual constant not occurring in Φ.
4.2.5.1 If Φ is sentential letter, then Φ is true under I iff I assigns T to Φ
4.2.5.2 If Φ is atomic and not a sentential , then Φ is true under I iff the objects I assigns to the individual constants of Φ are related by the relation that I assigns to the predicate of Φ
4.2.5.3 If Φ = - Ψ, then Φ is true under I iff Ψ is not true under I
4.2.5.4 If Φ = (Ψ V χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I or χ is true under I or both
4.2.5.5 If Φ = (Ψ & χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I and χ is true under I
4.2.5.6 If Φ = (Ψ \(\implies) χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is not true under I or χ is true under I or both
4.2.5.7 If Φ = (Ψ \(\iff\) χ) for sentences Ψ, χ, then Φ is true under I iff either Ψ and χ are both true or both not true under I.
4.2.5.8 If Φ = (α)Ψ, then Φ is true under I iff Ψ α/β is true under every β-variant of I
4.2.5.8.1 α/β : replace all free occurrences of α by occurrences of some individual constant β
4.2.5.9 If Φ = (∃α)Ψ, then Φ is true under I iff Ψ α/β is true under at least one β-variant of I
4.2.6 Φ is true/false under I = I assigns the truth-value T/F to Φ
4.2.7 Complete Interpretations : for each element of the domain of I, there is an individual constant to which I assigns that element as denotation
4.2.7.1 Φ is true under all complete interpretations iff Φ is true under all interpretations
4.3 Validity, Consequence, Consistency
4.3.1 Valid (a sentence Φ) : Φ is true under every interpretation
4.3.1.1 Not valid : a nonempty set D and an assignment of appropriate entities to the non-logical constants of L which makes Φ false exists
4.3.2 Consequence of a set of sentences Γ ( a sentence Φ) : there is no interpretation under which all the sentences of Γ are true and Φ is false
4.3.3 Consistent ( a set of sentences Γ ) : there is an interpretation under which all the sentences of Γ are true
4.3.4 Deduction theorem : For any sentence Φ, Φ is a consequence of the sentences Γ together with a sentence Ψ iff (Ψ \(\implies) Φ) is a consequence of Γ alone.