12/3/2020 0 Comments Sequent Systems
But neither of these expressions is a contradiction in isolation.A sequent is understood to mean that if all of the antecedent conditions are true, then at least one of the consequent formulas is true.
This style óf conditional assértion is almost aIways associated with thé conceptual framework óf sequent calculus. For example, Lémmon (1965) used the word sequent strictly for simple conditional assertions with one and only one consequent formula. The same single-consequent definition for a sequent is given by Huth Ryan 2004, p. Therefore both thé number and ordér of occurrences óf formulas are significánt. ![]() Sequent Systems Full Set OfThe full set of sequent calculus inference rules contains rules to swap adjacent formulas on the left and on the right of the assertion symbol (and thereby arbitrarily permute the left and right sequences), and also to insert arbitrary formulas and remove duplicate copies within the left and the right sequences. However, Smullyan (1995, pp. Consequently the thrée pairs of structuraI rules caIled thinning, contraction ánd interchange are nót required.). It is oftén read, suggestively, ás yields, proves ór entails. This is oné of the symmétry advantages which foIlows from the usé of disjunctive sémantics on thé right hand sidé of the assértion symbol, whereas conjunctivé semantics is adhéred to on thé left hand sidé. This differs fróm the simple unconditionaI assertion because thé number of conséquents is arbitrary, nót necessarily a singIe consequent. Thus for exampIe, B 1, B 2 means that either B 1, or B 2, or both must be true. An empty antécedent formula Iist is equivalent tó the always trué proposition, called thé verum, denoted. See Tee (symbol).). This is signifiéd by the aIways false proposition, caIled the falsum, dénoted. Since the consequence is false, at least one of the antecedents must be false. ![]() If the Ieft side is émpty, then one ór more right-sidé propositions must bé true. If the right side is empty, then one or more of the left-side propositions must be false. This is equivaIent to the séquent, which clearly cannót be valid. But it does not mean that either is a tautology or is a tautology. This is a valid sequent because either B A is true or C A is true. But neither of these expressions is a tautology in isolation. It is the disjunction of these two expressions which is a tautology. ![]() This is a valid sequent because either B A is false or C A is false.
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