The Logic Manual, Chapter 2: Syntax and Semantics of Propositional Logic

Recursive definition of a sentence of L1:

(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.

Bracketing Convention 1

The outer brackets may be omitted from a sentence that is not part of another sentence

Bracketing Convention 2

The inner set of brackets may be omitted from a sentence of the form ((ϕ∧ψ)∧χ). An analogous convention applies to ∨

Bracketing Convention 3

Assume ϕ, ψ, and χ are sentences of L1, ● is either ∧ or ∨, and ○ is either → or ↔. 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.

A sentence of L1 is logically true if and only if

It is true in all L1-structures

A sentence of L1 is a contradiction if and only if

It is not true in any L1-structure

Two sentences of L1 are logically equivalent if and only if

They are true in exactly the same L1-structures

A sentence of L1 is logically true (truth-table) if and only if

There are only T’s in the main column of its truth-table

A sentence of L1 is a contradiction (truth-table) if and only if

There are only F’s in the main column of its truth-table

Two sentences of L1 are logically equivalent (truth table) if and only if

They agree on the truth-values in their main columns

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 if and only if

There is no L1-structure in which all sentences in Γ are true and ϕ is false

An L1-structure is a counterexample to the argument with Γ as the set of premisses and ϕ as conclusion if and only if

∣ψ∣A = T for all ψ ∈ Γand ∣ϕ∣A =F.

A set Γ of L1-sentences is semantically consistent if and only if

There is an L1-structure A such that ∣ϕ∣A =T for all sentences ϕ of Γ.

A set Γ of L1-sentences is semantically inconsistent if and only if

Γ is not consistent.