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The rule states that P implies Q is logically equivalent to not-or and that either form can replace the other in logical proofs. In other words, if P {\displaystyle P} is true, then Q {\displaystyle Q} must also be true, while if Q {\displaystyle Q} is not true, then P {\displaystyle P} cannot be true either; additionally, when P {\displaystyle ...
In the abstract (ideal) case the simplest oscillating formula is a NOT fed back to itself: ~(~(p=q)) = q. Analysis of an abstract (ideal) propositional formula in a truth-table reveals an inconsistency for both p=1 and p=0 cases: When p=1, q=0, this cannot be because p=q; ditto for when p=0 and q=1.
material conditional (material implication) implies, if P then Q, it is not the case that P and not Q propositional logic, Boolean algebra, Heyting algebra: is false when A is true and B is false but true otherwise.
A Propositional Function, or a predicate, in a variable x is an open formula p(x) involving x that becomes a proposition when one gives x a definite value from the set of values it can take. According to Clarence Lewis, "A proposition is any expression which is either true or false; a propositional function is an expression, containing one or ...
The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, [1] in which formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false. [19]
For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values is 2×2, or four. So the result is four possible outputs of C and R. If one were to use base 3, the size would increase to 3×3, or nine possible outputs.
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A string is considered true if it contains at least one non-whitespace character (thus, for example, the #if function interprets the strings "0" and "FALSE" as true values, not false). Any string containing only whitespace or no characters at all will be treated as false (thus #if interprets " " and "", as well as undefined parameters, as false ...