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851.
It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent
background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals
are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential
implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances
of CEM are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal
logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse
in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most
two worlds), which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered,
a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional
in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those
of propositional modal logic as applied to a special sort of conditional. 相似文献
852.
In Belnaps useful 4-valued logic, the set 2={T,F} of classical truth values is generalized to the set 4=(2)={,{T},{F},{T,F}}. In the present paper, we argue in favor of extending this process to the set 16=(4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR2 with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice SIXTEEN3 with an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN3 coincide with the logic of FOUR2, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arielis and Avrons notion of a logical bilattice and state a number of open problems for future research.Dedicated to Nuel D. Belnap on the occasion of his 75th Birthday 相似文献
853.
Hartry Field has recently examined the question whether our logical and mathematical concepts are referentially indeterminate. In his view, (1) certain logical notions, such as second-order quantification, are indeterminate, but (2) important mathematical notions, such as the notion of finiteness, are not (they are determinate). In this paper, I assess Fields analysis, and argue that claims (1) and (2) turn out to be inconsistent. After all, given that the notion of finiteness can only be adequately characterized in pure second-order logic, if Field is right in claiming that second-order quantification is indeterminate (see (1)), it follows that finiteness is also indeterminate (contrary to (2)). After arguing that Field is committed to these claims, I provide a diagnosis of why this inconsistency emerged, and I suggest an alternative, consistent picture of the relationship between logical and mathematical indeterminacy. 相似文献
854.
We consider two topological interpretations of the modal diamond—as the closure operator (C-semantics) and as the derived set operator (d-semantics). We call the logics arising from these interpretations C-logics and d-logics, respectively. We axiomatize a number of subclasses of the class of nodec spaces with respect to both semantics, and
characterize exactly which of these classes are modally definable. It is demonstrated that the d-semantics is more expressive than the C-semantics. In particular, we show that the d-logics of the six classes of spaces considered in the paper are pairwise distinct, while the C-logics of some of them coincide.
Mathematics Subject Classifications (2000): 03B45, 54G99.
Presented by Michael Zakharyaschev 相似文献
855.
-autonomous lattices are the algebraic exponentials and without additive constants. In this paper, we investigate the structure theory of this variety and some of its subvarieties, as well as its relationships with other classes of algebras. 相似文献
856.
An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic. 相似文献
857.
We prove that all finitely axiomatizable tense logics with temporal operators for ‘always in the future’ and ‘always in the
past’ and determined by linear fows time are coNP-complete. It follows, for example, that all tense logics containing a density
axiom of the form ■n+1F p → nF p, for some n ≥ 0, are coNP-complete. Additionally, we prove coNP-completeness of all ∩-irreducible tense logics. As these classes of tense
logics contain many Kripke incomplete bimodal logics, we obtain many natural examples of Kripke incomplete normal bimodal
logics which are nevertheless coNP-complete. 相似文献
858.
Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well.In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of entailment, hence the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, GödelΔ, and Gödel~ logics.There are several results (all for the countable set of propositional variables) concerning the compactness (based on satisfiability) of these logic by Cintula and Navara, and the question of compactness (based on entailment) for Gödel logic was fully answered by Baaz and Zach (see papers [3] and [2]).In this paper we give a nearly complete answer to the problem of compactness based on both concepts for all three logics and for an arbitrary cardinality of the set of propositional variables. Finally, we show a tight correspondence between these two concepts 相似文献
859.
860.
We show that it is possible to base fuzzy control on fuzzy logic programming. Indeed, we observe that the class of fuzzy Herbrand interpretations gives a semantics for fuzzy programs and we show that the fuzzy function associated with a fuzzy system of IF-THEN rules is the fuzzy Herbrand interpretation associated with a suitable fuzzy program. 相似文献