Truth-Tables:		

	~ (NOT)		∧ (AND)		∨ (OR)	
	P	~P		P	Q	P∧Q		P	Q	P∨Q	
	T	F		T	T	T		T	T	T	
	F	T		T	F	F		T	F	T	
				F	T	F		F	T	T	
				F	F	F		F	F	F	
											
⊃ (ONLY IF)		≡ (EQUIV)					
P	Q	P⊃Q		P	Q	P≡Q					
T	T	T		T	T	T					
T	F	F		T	F	F					
F	T	T		F	T	F					
F	F	T		F	F	T					

Formalisation: Translating into logical notation, revealing logical or propositional structure, e.g.

It’s raining.							P
It’s not raining.						~P
There are clouds in the sky.				Q
There aren’t clouds in the sky.				~Q
If it’s raining, then there are clouds in the sky.		P⊃Q
It’s raining only if there are clouds in the sky.		P⊃Q
Chomsky is a linguist.					R
It’s raining and Chomsky is a linguist.			P∧R
Either it’s raining or Chomsky is not a linguist.		P ∨ ~R
It’s raining if and only if there are clouds in the sky.	P≡Q 
The syntax of PC (Propositional Calculus)

Primitive vocabulary
	Propositional variables (PV):	P, Q, R, …
	Truth-functional connectives:	~ (not, it is not the case that, it is false that)
∧ (and), ∨ (or)
⊃ (if … then …, only if, implies)
≡ (if and only if)

Formation rules:
	Every propositional variable is a wff (well-formed formula)
	If α if a wff, so is ~α
	If α and β are wffs, then so are: (α∧β), (α∨β), (α⊃β) and (α≡β)
	Nothing else is a wff.

Semantics of PC
Interpretation: An interpretation I is an assignment of truth-values to atomic wffs, such that 
	for any wff α, I(~α)=1 iff I(α)=0;
	for any wffs α and β, I(α∧β)=1 iff I(α)=1 and I(β)=1;
	for any wffs α and β, I(α∨β)=1 iff I(α)=1 or I(β)=1;
	for any wffs α and β, I(α⊃β)=1 iff I(α)=0 or I(β)=1;
	for any wffs α and β, I(α≡β)=1 iff I(α)=I(β).
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Tautology: true on all interpretations, e.g. ⊩(P∨~P).
Contradiction: False on all interpretations, e.g. (P∧~P).
Contingent: True on at least one interpretation and false on at least one interpretation, e.g. (P⊃Q).
Logical equivalence: Any two wffs α and β are said to be logically equivalent if and only if they have the same truth-value on all interpretations, e.g. (P⊃Q) and (~P∨Q).
Valid argument: There is no interpretation on which the premises are true but not the conclusion, e.g. (P⊃Q), P ⊩ Q. If an argument is valid then (α⊃β) is a tautology, where α is the conjunction of its premises and β its conclusion.
Consistency: A set of wffs is consistent iff there is at least one interpretation on which they are all true; inconsistent otherwise.