Syntactic Entailment (Propositional Logic)

A notion of proof consists of axioms and rules of deduction.

Axioms

In Propositional Logic we adopt three axioms schemes
as axioms. They are all Tautology (Propositional Logic)

A1

for all

A2

for all

A3

for

Deduction rule

We’ll have one rule of deduction called modus ponens:

MP

From and we can deduce

Proof

Given and , a proof of from is a finite sequence in s.t. and for each

1. Either is an axiom
2. or (premiss or hypothesis)
3. or is obtained by modus ponens from earlier lines:

Say proves or syntactically entails or
if there’s a proof of from .

Theorem (Propositional Logic)
Deduction Theorem (Propositional Logic)
Soundness Theorem (Propositional Logic)
Model Existence Lemma (Propositional Logic)
Adequacy Theorem (Propositional Logic)
Completeness Theorem