5 The Semantics of Predicate Logic Flashcards
What is a domain of discourse
a non-empty set of objects
Define an L2 structure
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
What is the semantic value of a constant in an L2 expression
object
What is the semantic value of a sentence letter in an L2 expression
truth-value (T or F)
What is the semantic value of a unary predicate letter in an L2 expression
set, unary relation
What is the semantic value of a binary predicate letter in an L2 expression
binary relation (set of ordered pairs)
What is the semantic value of a ternary predicate letter in an L2 expression
3-place relation (set of ordered triples)
Define truth in L2
A sentence ∅ is true in an L2-structure iff ∣∅∣𝛼Α=T for all variable assignments 𝛼 over Α
Define logical truth in L2 (4 parts)
(i). A sentence ∅ of L2 is logically true iff ∅ is true in all L2 structures.
(ii). A sentence ∅ of L2 is a contradiction iff ∅ is not true in any L2 structures.
(iii). Sentences ∅ and Ψ of L2 are logically equivalent iff both are true in exactly the same L2 structures.
(iv). A set ⨡ of L2 sentences is semantically consistent iff there is an L2 structure Α in which all sentences in ⨡ are true. A set of L2 sentences are semantically inconsistent iff it is not semantically consistent.
Define validity in L2
Let ⨡ be a set of sentences of L2 and ∅ a sentence of L2. The argument with all sentences in ⨡ as premisses and ∅ as conclusion is valid iff there is no L2 structure in which the premisses are true and the conclusion is false.
