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Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it.. For example, the two-place truth function that always returns false is not definable from → and arbitrary propositional variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are ...
The formulations here use implication and negation {,} as functionally complete set of basic connectives. Every logic system requires at least one non-nullary rule of inference. Classical propositional calculus typically uses the rule of modus ponens:
Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. [1] A sentence is said to be a logical consequence of a set of sentences, for a given language , if and only if , using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must ...
adult, since to state "John is a bachelor" implies John has each of those three additional predicates. Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime. Example 3 Consider thunder, the sound caused by lightning.
A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph G = (V, E) composed of vertex set V and directed edge set E. Each vertex in V represents the truth status of a Boolean literal, and each directed edge from vertex u to vertex v represents the material implication "If the literal u is true then the ...
In propositional logic, material implication [1] [2] is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not- P {\displaystyle P} or Q {\displaystyle Q} and that either form can replace the other in ...
A well-formed formula is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar. The language L {\displaystyle {\mathcal {L}}} , then, is defined either as being identical to its set of well-formed formulas, [ 48 ] or as containing that set (together with ...