The Semantics of Predicate Logic

Question 1

L2-structure

Answer 1

An L2-structure is an ordered pair ⟨D, I⟩ where D is some non-empty set and I is a function from the set of all constants, sentence letters, and predicate letters such that

1. the value of every constant is an element of D
2. the value of every sentence letter is a truth-value T or F.
3. the value of every n-ary predicate letter is an n-ary relation.

Question 2

Variable assignment

Answer 2

A variable assignment over an :2-structure A assigns an element of the domain DA of A to each variable.

Question 3

Satisfaction

Answer 3

Assume A is an L2-structure, α is a variable assignment over A, φ and ψ are formulae of L2, and v is a variable. For a sentence letter φ either |φ|αA = T or |φ|αA = F obtains. Formulae other than sentence letters receive the following semantic values:

(i) |Φt1 …tn|αA = T iff ⟨|t1|αA, … , |tn|αA⟩ ∈ |Φ|αA, where Φ is an n-ary predicate letter for n 􏰀 1 and each of t1, . . . , tn is either a variable or a constant
(ii) |¬φ|αA = T iff |φ|αA = F
(iii) |φ∧ψ|αA =T iff|φ|αA =T and|ψ|αA =T
(iv) |φ∨ψ|αA =T iff|φ|αA =T or|ψ|αA =T
(v) |φ→ψ|αA =T iff|φ|αA =F or|ψ|αA =T
(vi) |φ↔ψ|αA =T iff|φ|αA =|ψ|αA
(vii) |∀vφ|αA = T iff |φ|βA = T for all variable assignments β over A differing from α in v at most
(viii) |∃vφ|αA = T iff |φ|βA = T for at least one variable assignment β over A differing from α in v
at most.

Question 4

Truth

Answer 4

A sentence φ is true in an L2-structure A iff |φ|αA = T for all variable assignments α over
A.

Question 5

Logical truth, etc. (L2 version)

Answer 5

(i) A sentence φ of L2 is logically true iff φ is true in all L2-structures.
(ii) A sentence φ of L2 is a contradiction iff φ is not true in any L2-structures.
(iii) Sentences φ and ψ of L2 are logically equivalent iff both are true in exactly the same L2- structures.
(iv) A set Γ of L2-setences is semantically consistent iff there is an L2-structure A in which all sentences in Γ are true. A set of L2-sentences is semantically inconsistent iff it is not semantically consistent.

Question 6

Validity (L2 version)

Answer 6

Let Γ be a set of sentences of L2 and φ a sentence of L2. The argument with all sentences in Γ as premisses and φ as conclusion is valid iff there is no L2 structure in which all sentences in Γ are true and φ is false. This is abbreviated as Γ |= φ.