Chapter 5: The Semantics of Predicate Logic

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9 Terms

1
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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

• the value of every constant is an element of D

• the value of every sentence letter is a truth-value T or F

• the value of every n-ary predicate letter is an n-ary relation.

2
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A variable assignment over an 𝐿₂ structure 𝓐…

assigns an element of the domain Dᴀ of 𝓐 to each variable.

3
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Satisfaction

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

  • |φt₁ . . . tₙ|ᵅᴀ = T if and only if ⟨|t₁|ᵅᴀ,…,|tₙ|ᵅᴀ⟩ ∈ |φ|ᵅᴀ, where φ is an n-ary predicate letter (n must be 1 or higher), and each of t₁ . . . tₙ is either a variable or a constant.

  • |¬φ|ᵅᴀ = T if and only if |φ|ᵅᴀ = F.

  • |φ ∧ ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = T and |ψ|ᵅᴀ = T.

  • |φ ∨ ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = T or |ψ|ᵅᴀ = T.

  • |φ → ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = F or |ψ|ᵅᴀ = T.

  • |φ ↔ ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = |ψ|ᵅᴀ.

  • |∃vφ|ᵅᴀ = T if and only if |φ|ᵅᴀ = T for at least one variable assignment β over 𝓐 differing from ɑ in v at most.

  • |∀vφ|ᵅᴀ = T if and only if |φ|ᴀᵝ = T for all variable assignments β over 𝓐 differing from ɑ in v at most.

<ul><li><p>|φt₁ . . . tₙ|ᵅᴀ = T if and only if ⟨|t₁|ᵅᴀ,…,|tₙ|ᵅᴀ⟩ <strong>∈ |</strong>φ|ᵅᴀ, where φ is an n-ary predicate letter (n must be 1 or higher), and each of t₁ . . . tₙ is either a variable or a constant.</p></li><li><p>|¬φ|ᵅᴀ = T if and only if |φ|ᵅᴀ = F.</p></li><li><p>|φ ∧ ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = T and |ψ|ᵅᴀ = T.</p></li><li><p>|φ ∨ ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = T or |ψ|ᵅᴀ = T.</p></li><li><p>|φ → ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = F or |ψ|ᵅᴀ = T.</p></li><li><p>|φ <span data-name="left_right_arrow" data-type="emoji">↔</span> ψ|ᵅᴀ = T if and only if |φ|ᵅᴀ = |ψ|ᵅᴀ.</p></li><li><p>|∃vφ|ᵅᴀ = T if and only if |φ|ᵅᴀ = T for at least one variable assignment β over 𝓐 differing from ɑ in v at most.</p></li><li><p>|∀vφ|ᵅᴀ = T if and only if |φ|ᴀᵝ = T for all variable assignments β over 𝓐 differing from ɑ in v at most.</p></li></ul><p></p>
4
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A sentence φ is true in an 𝐿₂-structure 𝓐 iff…

|φ|ᵅᴀ = T for all variable assignments ɑ over 𝓐.

5
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A sentence φ of 𝐿₂ is logically true iff…

φ is true in all 𝐿₂-structures.

6
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A sentence φ of 𝐿₂ is a contradiction iff…

φ is not true in any 𝐿₂-structure.

7
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A sentence φ and a sentence ψ are logically equivalent iff…

both are true in exactly the same 𝐿₂-structures.

8
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A set Γ of 𝐿₂-sentences is semantically consistent iff…

there is an 𝐿₂-structure 𝓐 in which all sentences in Γ are true. As in propositional logic, a set of 𝐿₂-sentences is semantically inconsistent if and only if it is not semantically consistent.

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Validity in 𝐿₂

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 if and only if there is no 𝐿₂-structure in which all sentences in Γ are true and φ is false.