Logical concepts – Rewriting formulas

We can rewrite formulas, with the aim of better understanding the meaning of the logical ‘bones’ (=form) of the statement, and to produce provable formulas, or proofs.

If the rewritten formula has the same meaning as its original formula, both formulas are equivalent (≡). If instead we place a biconditional (↔) between the rewritten and its original formula, and we calculate under this operand, then we get a tautology.

1. Same meaning

Rewriting formulas with operands ˅, ˄, ~
If we use the operands ˅, ˄, ~, which together form a functionally complete set, then we can rewrite these formulas using the following rules (for statements with 2 variables and 1 operand):

Double negative if A is true, then not not-A is true and its converse: if not not-A is true, then A is true. The rule allows us to introduce or eliminate a negation from a logical proof  ~~AA |10|  negation
De Morgan’s laws The negation of a conjunction = the disjunction of the negations ~(A˄B) ≡ (~A˅~B) |0111|  neg.and
The negation of a disjunction = the conjunction of the negations (~~B) ≡ ~(A˅B) |0001|  context
commutativity changing the order or sequence of the operands within an expression of disjunctionp˅q q˅p |1110|  disjunction
of conjunctionp˄q q˄p |1000|  conjunction
Complete commutative law of equivalence  of biconditional(pq) ≡ (qp) |1001|  biconditional

Rewriting the implication
Expressions that are often used to rewrite the implication.

expression

form

description of the operation

outcome1)

diagram

implication

pq

first statement implies truth of second

|1011|

 implication
contrapositive or transposition

~q→~p

reversal and negation of both statements. (switch the antecedent with the consequent and also negate both)

|1011|

equivalence

~p˅q

p implies q is defined to mean that either p is false or q is true.

|1011|

 2. Different meaning
Expressions opposing/contrary to the implication

expression

form

description of the operation

outcome1)

diagram

inverse

~p→~q

negation of both statements

|1101|

 inverse,converse
converse

qp

reversal of both statements

|1101|

Negating the implication

expression

form

description of the operation

outcome1)

diagram

negation

p˄~q

contradicts the implication

|0100|

 neg.implication

1) For spatial efficiency, instead of vertically, I noted the formula’s outcome in the truth table columns horizontally and between ||.