—implicational, adj./im'pli kay"sheuhn/, n.1. something implied or suggested as naturally to be inferred or understood: to resent an implication of dishonesty.2. the act of implying: His implication of immediate changes surprised us.3. the state of being implied: to know only by implication.4. Logic. the relation that holds between two propositions, or classes of propositions, in virtue of which one is logically deducible from the other.5. the act of implicating: the implication of his accomplices.6. the state of being implicated: We heard of his implication in a conspiracy.7. Usually, implications. relationships of a close or intimate nature; involvements: the religious implications of ancient astrology.[1400-50; late ME implicacio(u)n < L implication- (s. of implicatio) an interweaving, equiv. to implicat(us) (see IMPLICATE) + -ion- -ION]Syn. 7. associations, connections.
* * *In logic, a relation that holds between two propositions when they are linked as antecedent and consequent of a true conditional proposition.Logicians distinguish two main types of implication, material and strict. Proposition p materially implies proposition q if and only if the material conditional p ⊃ q (read "if p then q") is true. A proposition of the form p ⊃ q is false whenever p is true and q is false; it is true in the other three possible cases (i.e., p true and q true; p false and q true; p false and q false). It follows that whenever p is false, p ⊃ q is automatically true: this is a peculiarity that makes the material conditional inadequate as an interpretation of the meaning of conditional sentences in ordinary English. On the other hand, proposition p strictly implies proposition q if and only if it is impossible for p to be true without q also being true (i.e., if the conjunction of p and not-q is impossible).
* * *▪ logicin logic, a relationship between two propositions in which the second is a logical consequence of the first. In most systems of formal logic, a broader relationship called material implication is employed, which is read “If A, then B,” and is denoted by A ⊃ B or A → B. The truth or falsity of the compound proposition A ⊃ B depends not on any relationship between the meanings of the propositions but only on the truth-values of A and B; A ⊃ B is false when A is true and B is false, and it is true in all other cases. Equivalently, A ⊃ B is often defined as ∼(A·∼B) or as ∼A∨B (in which ∼ means “not,” · means “and,” and ∨ means “or”). This way of interpreting ⊃ leads to the so-called paradoxes of material implication: “grass is red ⊃ ice is cold” is a true proposition according to this definition of ⊃.In an attempt to construct a formal relationship more closely akin to the intuitive notion of implication, Clarence Irving Lewis (Lewis, C.I.), known for his conceptual pragmatism, introduced in 1932 the notion of strict implication. Strict implication was defined as ∼♦(A·∼B), in which ♦ means “is possible” or “is not self-contradictory.” Thus A strictly implies B if it is impossible for both A and ∼B to be true. This conception of implication is based upon the meanings of the propositions, not merely upon their truth or falsity.Finally, in intuitionistic mathematics and logic, a form of implication is introduced that is primitive (not defined in terms of other basic connectives): A ⊃ B is true here if there exists a proof (q.v.) that, if conjoined to a proof of A, would produce a proof of B. See also deduction; inference.
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