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The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q). It may also be ...
These are often denoted by uppercase letters such as P, Q and R. Examples: In P(x), P is a predicate symbol of valence 1. One possible interpretation is "x is a man". In Q(x,y), Q is a predicate symbol of valence 2. Possible interpretations include "x is greater than y" and "x is the father of y".
The propositional calculus [a] is a branch of logic. [1] It is also called propositional logic, [2] statement logic, [1] sentential calculus, [3] sentential logic, [4] [1] or sometimes zeroth-order logic. [b] [6] [7] [8] Sometimes, it is called first-order propositional logic [9] to contrast it with System F, but it should not be confused with ...
In propositional logic, modus ponens (/ ˈ m oʊ d ə s ˈ p oʊ n ɛ n z /; MP), also known as modus ponendo ponens (from Latin 'mode that by affirming affirms'), [1] implication elimination, or affirming the antecedent, [2] is a deductive argument form and rule of inference. [3] It can be summarized as "P implies Q. P is true. Therefore, Q ...
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
The material conditional (also known as material implication) is an operation commonly used in logic.When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false.
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.
Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.