## 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.

Anot (A)
truefalse
falsetrue

AB(A and B)
truetruetrue
truefalsefalse
falsetruefalse
falsefalsefalse

AB(A or B)
truetruetrue
truefalsetrue
falsetruetrue
falsefalsefalse

AB(A xor B)
truetruefalse
truefalsetrue
falsetruetrue
falsefalsefalse

AB(A -> B)
truetruetrue
truefalsefalse
falsetruetrue
falsefalsetrue

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.