Propositional Logic

Syntactically Well-Formed Formulas

Propositions: The letters 'A', 'B', 'C', ... 'Z' are syntactically well-formed formulas. They stand for propositions.
Negation: If 'A' is a syntactically well-formed formula, then 'not (A)' is a syntactically well-formed formula too. The bracket is required.
Conjunction: If 'A' and 'B' are syntactically well-formed formulas, then '(A and B)' is a syntactically well-formed formula too.
Disjunction (inclusive or): If 'A' and 'B' are syntactically well-formed formulas, then '(A or B)' is a syntactically well-formed formula too.
Exclusive or: If 'A' and 'B' are syntactically well-formed formulas, then '(A xor B)' is a syntactically well-formed formula too.
Implication: If 'A' and 'B' are syntactically well-formed formulas, then '(A -> B)' is a syntactically well-formed formula too.
Equivalence: If 'A' and 'B' are syntactically well-formed formulas, then '(A <-> B)' is a syntactically well-formed formula too.

Examples of Syntactically Well-Formed Formulas

'((A and B) or not (C))' is a syntactically well-formed formula.
'((A or B) and not ((C and D)))' is a syntactically well-formed formula. Two brackets are required around 'C and D'. One bracket is required for the conjunction of C and D. Another bracket is required for the negation of the conjunction.

Truth Valuations of Syntactically Well-Formed Formulas

The Truth Table for the Negation

Anot (A)
truefalse
falsetrue

The Truth Table for the Conjunction

AB(A and B)
truetruetrue
truefalsefalse
falsetruefalse
falsefalsefalse

The Truth Table for the Disjunction

AB(A or B)
truetruetrue
truefalsetrue
falsetruetrue
falsefalsefalse

The Truth Table for the Exclusive 'Or'

AB(A xor B)
truetruefalse
truefalsetrue
falsetruetrue
falsefalsefalse

The Truth Table for the Implication

AB(A -> B)
truetruetrue
truefalsefalse
falsetruetrue
falsefalsetrue

The Truth Table for the Equivalence

AB(A <-> B)
truetruetrue
truefalsefalse
falsetruefalse
falsefalsetrue

Rules to Draw Logically Inferences Pertaining to Negation, Conjunction, and Disjunction

Unpacking a Double Negation: If 'not ( not (A) )' is a true formula, then 'A' is a true formula too.
Unpacking a Conjunction: If '( A and B )' is a true formula, then both 'A' and 'B' are true formulas too.
Unpacking the Negation of a Disjunction: If 'not ( A or B )' is a true formula, then 'not ( A )' and 'not ( B )' are true formulas too.
Unpacking a Disjunction: If '( A or B )' is a true formula, then at least one of the formulas 'A' and 'B' is a true formula too.
Unpacking the Negation of a Conjunction: If 'not ( A and B )' is a true formula, then at least one of the formulas 'not ( A )' and 'not ( B )' is a true formula too.

Rules to Draw Logically Inferences Pertaining to Negation, Implication, and Equivalence

Unpacking an Implication: If '( A -> B )' is a true formula, then at least one of the formulas 'not ( A )' and 'B' is a true formula too.
Unpacking the Negation of an Implication: If 'not ( A -> B )' is a true formula, then 'A' and 'not ( B )' are true formulas too.
Unpacking an Equivalence: If '( A <-> B )' is a true formula, then either 'A' and 'B' are true formulas or 'not ( A )' and 'not ( B )' are true formulas.
Unpacking the Negation of an Equivalence: If 'not ( A <-> B )' is a true formula, then either 'A' and 'not ( B )' are true formulas or 'not ( A )' and 'B' are true formulas.

 
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