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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 be true if every sentence in the set is true.
The hearer can now draw the contextual implications that +> Susan needs to be cheered up. +> Peter wants me to ring Susan and cheer her up. If Peter intended the hearer to come to these implications, they are implicated conclusions. Implicated premises and conclusions are the two types of implicatures in the relevance theoretical sense. [51]
An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense that hearing thunder, and unambiguously recognizing it as such, justifies concluding that there has been a lightning bolt. Example 4 If the U.S. Congress passes a bill, the president's signing of the bill is sufficient to make it law.
An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false. For example a valid argument might run: If it is raining, water exists (1st premise) It is raining (2nd premise) Water exists (Conclusion)
A is the premise and B is the conclusion of the implication A→B . A set C respects the implication A → B when ¬( C ⊆ A ) or C ⊆ B . A formal context is a triple (G,M,I) , where G and M are sets (of objects and attributes , respectively), and where I ⊆ G × M is a relation expressing which objects have which attributes.
Tautological consequence can also be defined as ∧ ∧ ... ∧ → is a substitution instance of a tautology, with the same effect. [2]It follows from the definition that if a proposition p is a contradiction then p tautologically implies every proposition, because there is no truth valuation that causes p to be true and so the definition of tautological implication is trivially satisfied.
In a Hilbert system, the premises and conclusion of the inference rules are simply formulae of some language, usually employing metavariables.For graphical compactness of the presentation and to emphasize the distinction between axioms and rules of inference, this section uses the sequent notation instead of a vertical presentation of rules.
A rule of inference allowing the conclusion that something exists with a certain property, based on the existence of a particular example. existential import The implication that something exists by the assertion of a particular kind of statement, especially relevant in traditional syllogistic logic. existential instantiation