2 Syntax and Semantics of Propositional Logic Flashcards
Define a sentence of L1
(i). All sentence letters are sentences of L1
(ii). If ∅ and Ψ are sentences of L1, then ¬∅, (∅＾Ψ), (∅ or Ψ), (∅→Ψ) and (∅↔︎Ψ) are sentences.
(iii). Nothing else is a sentence of L1.
What is the first bracketing convention?
The outer brackets may be omitted from a sentence that is not part of another sentence.
What is the second bracketing convention?
The inner set of brackets may be omitted from a sentence of the form ((∅＾Ψ)＾X) and analogously for ‘or’.
Define what an L1 structure is
An L1 structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.
Define a logical truth in L1
A sentence ∅ of L1 is logically true iff ∅ is true in all L1 structures.
Define a contradiction in L1
A sentence ∅ of L1 is a contradiction iff ∅ is not true in any L1 structures.
Define logical equivalence in L1
A sentence ∅ and a sentence Ψ of L1 are logically equivalent iff ∅ and Ψ are true in exactly the same L1 structures.
Define logical validity in L1
Let ⨡ be a set of sentences of L1 and ∅ a sentence of L1. The argument with all sentences in ⨡ as premisses and ∅ as a conclusion is valid iff there is no L1 structure in which all sentences in ⨡ are true and ∅ is false.
Define a counterexample in L1
An L1-structure is a counterexample to the argument ⨡ as the set of premisses and ∅ as the conclusion iff for all γ as an element of ⨡ we have the truth value of γ as T but the conclusion is false.
Define semantic consistency in L1
A set ⨡ of L1 sentences is semantically consistent iff there is an L1-structure such that for all sentence γ in ⨡ we have an L1 structure such that γ has a true truth-value.
Define semantic inconsistency in L1
A set ⨡of L1 sentences is semantically inconsistent iff ⨡ is not semantically consistent.
