Syntax and Semantics of Propositional Logic

Question 1

Sentence letters

Answer 1

P, Q, R, P1, Q1, R1, P2, Q2, R2 and so on are sentence letters.

Question 2

Sentence 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

Bracketing convention

Answer 3

1 The outer brackets may be omitted from a sentence that is not part of another sentence.
2 The inner set of brackets may be omitted from a sentence of the form ((φ ∧ ψ) ∧ χ) and
analogously for ∨.
3 Suppose⋄∈{∧, ∨}and◦∈{→, ↔}. Then if (φ◦(ψ⋄χ))or((φ⋄ψ)◦χ) occurs as part of
the sentence that is to be abbreviated, the inner set of brackets may be omitted.

Question 4

L1-structure

Answer 4

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

Question 5

Truth in an L1-structure

Answer 5

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 6

Logical truth, contradiction, equivalence (L1 version)

Answer 6

(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 7

Validity (L1 version)

Answer 7

Let Γ be a set of sentences of L1 and φ a sentence of L1. The argument with all sentences in Γ as premisses and φ as conclusion is valid iff there is no L1-structure in which all sentences in Γ are true and φ is false.

Question 8

Counterexamples

Answer 8

An L1-structure is a counterexample to the argument with Γ as the set of premisses and φ as the conclusion iff for all γ ∈ Γ we have |γ|A =T but |φ|A =F.

Question 9

Semantic consistency

Answer 9

A set Γ of L1-sentences is semantically consistent iff there is an L1- structure A such that for all sentence γ ∈ Γ we have |γ|A = T. A set Γ of L1-sentences is semantically inconsistent iff Γ is not semantically consistent.