Syntax and Semantics of Propositional Logic

Question 1

What are the sentence letters of L1?

Answer 1

P, Q, R, P1, Q1, R1, P2, Q2, R2, P3, Q3, R3, etc

Question 2

What are sentences of L1?

Answer 2

(i) All sentence letters are sentences of L1.
(ii) If ϕ and ψ are sentences of L1, then ¬ϕ, (ϕ ∧ ψ), (ϕ ∨ ψ), (ϕ → ψ) and (ϕ ↔ ψ) are sentences of L1.
(iii) Nothing else is a sentence of L1.

Question 3

What is an L1 structure?

Answer 3

An L1-structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1.

Question 4

How is truth defined in the L1 structure?

Answer 4

Truth in an L1-structure: Let A be some L1-structure. Then | . . . |A assigns either T or F to
every sentence of L1 in the following way.
(i) If φ is a sentence letter, |φ|A is the truth-value assigned to φ by the L1-structure A
(ii) |¬φ|A = T iff |φ|A = F
(iii) |φ ∧ ψ|A = T iff |φ|A = T and |ψ|A = T
(iv) |φ ∨ ψ|A = T iff |φ|A = T or |ψ|A = T
(v) |φ → ψ|A = T iff |φ|A = F or |ψ|A = T
(vi) |φ ↔ ψ|A = T iff |φ|A = |ψ|A

Question 5

How is logical truth, contradiction, logical equivalence defined in terms of L1 structures?

Answer 5

(i) A sentence φ of L1 is logically true iff φ is true in all L1-structures.
(ii) A sentence φ of L1 is a contradiction iff φ is not true in any L1-structures.
(iii) A sentence φ and a sentence ψ of L1 are logically equivalent iff φ and ψ are true in exactly the same L1-structures.

Question 6

How is logical truth, contradiction, logical equivalence defined in terms of truth tables?

Answer 6

(i) A sentence of L1 is logically true (or a tautology) if and only if there
are only T’s in the main column of its truth table.
(ii) A sentence is a contradiction if and only if there are only F’s in the main column of its truth table.
(iii) A sentence Φ and a sentence Ψ are logically equivalent if they agree on the truth-values in their main columns.