The semantics of the connectives can be given by truth tables. It determines the semantics for complex formulae.
What is a logic?
A logic consists of:
A formal system for expressing knowledge about a domain consisting of
Syntax: Sentences(well formed formulae)
Semantics: Meaning
A proof theory: rules of inference for deducing sentences from a knowledge base
Provability
λ ⊢ ρ: we can construct a proof for ρ from λ using axioms and rules of inference
If λ is empty (i.e., 0⊢ρ) and ρ is a single formula, then we say that ρ is a theorem of the logic
Entailment
λ |= ρ: whenever the formula(s) λ are true, one of the formula(s) in ρ is true
In the case where ρ is a single formula, we can determine whether λ |= ρ by constructing a truth table for λ and ρ. If, in any row of the truth table where all the formulae in λ are true, ρ is also true, then λ |= ρ.
If λ is empty, we say that ρ is a tautology
Soundness and Completeness
λ |= a是语义蕴含， λ |- b是形式推演
An inference procedure (and hence a logic) is sound if and only if it preserves truth
In other words ⊢ is sound iff whenever λ ⊢ ρ, then λ |= ρ
Soundness 是说右侧推演的知识都是被λ蕴含的（推出来的知识都是正确的）
A logic is complete if and only if it is capable of proving all truths
In other words, whenever λ |= ρ, then λ ⊢ ρ
Completeness 是说，左侧蕴含出来的知识都可以推演出来
A logic is decidable if and only if we can write a mechanical procedure (computer program) which when asked λ ⊢ ρ it can eventually halt and answer “yes” or answer “no”