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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 ...
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. may mean the same as
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 ...
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no counterexample, the argument is valid. But one could construct an argument in which the premises are inconsistent.
13, then/if, Converse implication; 14, OR, Logical disjunction; 15, true, Tautology. Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples:
For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics ), in the sense that if the premises are true (under ...
An example of a conventional implicature is "Donovan is poor but happy", where the word "but" implicates a sense of contrast between being poor and being happy. [ 7 ] Later linguists introduced refined and different definitions of the term, leading to somewhat different ideas about which parts of the information conveyed by an utterance are ...
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