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CANONICAL CALCULI WITH (N,K)ARY QUANTIFIERS
, 806
"... Abstract. Propositional canonical Gentzentype systems, introduced in [2], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. In [2] a constructive coher ..."
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Abstract. Propositional canonical Gentzentype systems, introduced in [2], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. In [2] a constructive coherence criterion is provided for the nontriviality of such systems and shows that a system of this kind admits cutelimination iff it is coherent. The semantics of such systems is provided using twovalued nondeterministic matrices (2Nmatrices). In [27] these results are extended to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n, k)ary quantifiers, which bind k distinct variables and connect n formulas, and show that the coherence criterion remains constructive for such systems. Then we focus on the case of k ∈ {0, 1} and show that the following statements concerning a canonical calculus G are equivalent: (i) G is coherent, (ii) G has a strongly characteristic 2Nmatrix, and (iii) G admits strong cutelimination. We also show that coherence is not a necessary condition for standard cutelimination, and then characterize a subclass of canonical systems for which this property does hold.
A Triple Correspondence in Canonical Calculi: Strong CutElimination, Coherence, and Nondeterministic Semantics
"... Abstract. An (n, k)ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)ary quantifiers form a natural class of Gentzentype systems which in addition to the standard axioms and structural rules have only logical rules in w ..."
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Abstract. An (n, k)ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)ary quantifiers form a natural class of Gentzentype systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using twovalued nondeterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)ary quantifiers are equivalent: (i) G is coherent, (ii) G admits strong cutelimination, and (iii) G has a strongly characteristic twovalued generalized nondeterministic matrix. 1
On the Formal Semantics of IFlike Logics
, 2009
"... In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Information Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of the ..."
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In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Information Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of these logics but a defect in the way in which the compositional semantics given by Hodges for the regular fragment was generalized to arbitrary formulas. We fix this by proposing an alternative formalization, based on a variation of the classical notion of valuation. Basic metatheoretical results are proven. We present these results for Hodges' slash logic (from which these can be easily transferred to other IFlike logics) and we also consider the flattening operator, for which we give novel gametheoretical semantics.
Imperfect Information in Epistemic Logic
, 1998
"... . We argue that a mutual de dicto/de re knowledge between agents is not expressible in classical firstorder epistemic logic. It can be, however, captured in the epistemic logic of imperfect information, extended to the case of multiple agents. However, giving semantics for formulae involving imperf ..."
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. We argue that a mutual de dicto/de re knowledge between agents is not expressible in classical firstorder epistemic logic. It can be, however, captured in the epistemic logic of imperfect information, extended to the case of multiple agents. However, giving semantics for formulae involving imperfect information by means of semantic games poses difficulties. First, the modelling of mutual knowledge seems to give rise to cyclic dependencies, and it is therefore not clear how the games can be played on such quantifieroperator prefixes. Second, the semantic side of imperfect information in epistemic logic needs to be settled in general. We do not aim at solutions but rather suggestions and questions. We shall point out the need of a new inscoping device in addition to outscoping, and also put forward the question what the existence of winning strategies in epistemic logic of imperfect information means. 1 Introduction Although firstorder epistemic logic is an expressive and powerful ...
Strong CutElimination, Coherence, and Nondeterministic Semantics
"... Abstract. An (n, k)ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)ary quantifiers form a natural class of Gentzentype systems which in addition to the standard axioms and structural rules have only logical rules in w ..."
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Abstract. An (n, k)ary quantifier is a generalized logical connective, binding k variables and connecting n formulas. Canonical systems with (n, k)ary quantifiers form a natural class of Gentzentype systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a quantifier is introduced. The semantics for these systems is provided using twovalued nondeterministic matrices, a generalization of the classical matrix. In this paper we use a constructive syntactic criterion of coherence to characterize strong cutelimination in such systems. We show that the following properties of a canonical system G with arbitrary (n, k)ary quantifiers are equivalent: (i) G is coherent, (ii) G admits strong cutelimination, and (iii) G has a strongly characteristic twovalued generalized nondeterministic matrix. In addition, we define simple calculi, an important subclass of canonical calculi, for which coherence is equivalent to the weaker, standard cutelimination property. 1
Equivalence Checking for Partial Implementations Revisited
"... In this paper we consider the problem of checking whether a partial implementation can (still) be extended to a complete design which is equivalent to a given full specification. In particular, we investigate the relationship between the equivalence checking problem for partial implementations (PEC) ..."
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In this paper we consider the problem of checking whether a partial implementation can (still) be extended to a complete design which is equivalent to a given full specification. In particular, we investigate the relationship between the equivalence checking problem for partial implementations (PEC) and the validity problem for quantified Boolean formulae (QBF) with socalled Henkin quantifiers. Our analysis leads us to a sound and complete algorithmic solution to the PEC problem as well as to an exact complexity theoretical classification of the problem. 1.
ANCESTOR WORSHIP IN THE LOGIC OF GAMES HOW FOUNDATIONAL WERE ARISTOTLE’S
"... ABSTRACT: Notwithstanding their technical virtuosity and growing presence in mainstream thinking, game theoretic logics have attracted a sceptical question: “Granted that logic can be done game theoretically, but what would justify the idea that this is the preferred way to do it? ” A recent sugges ..."
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ABSTRACT: Notwithstanding their technical virtuosity and growing presence in mainstream thinking, game theoretic logics have attracted a sceptical question: “Granted that logic can be done game theoretically, but what would justify the idea that this is the preferred way to do it? ” A recent suggestion is that at least part of the desired support might be found in the Greek dialectical writings. If so, perhaps we could say that those works possess a kind of foundational significance. The relation of being foundational for is interesting in its own right. In this paper, I explore its ancient applicability to relevant, paraconsistent and nonmonotonic logics, before returning to the question of its ancestral tie, or want of one, to the modern logics of games. 1. LOGIC AND GAME THEORY Since its inception in the early 1940s (von Neumann & Morgenstern 1944),1 the mathematical theory of games has become something of
Complexity of twovariable Dependence Logic and IFLogic∗
, 2013
"... We study the twovariable fragments D2 and IF2 of dependence logic and independencefriendly logic. We consider the satisfiability and finite satisfiability problems of these logics and show that for D2, both problems are NEXPTIMEcomplete, whereas for IF2, the problems are Π 01 and Σ 0 1complete, ..."
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We study the twovariable fragments D2 and IF2 of dependence logic and independencefriendly logic. We consider the satisfiability and finite satisfiability problems of these logics and show that for D2, both problems are NEXPTIMEcomplete, whereas for IF2, the problems are Π 01 and Σ 0 1complete, respectively. We also show that D2 is strictly less expressive than IF2 and that already in D2, equicardinality of two unary predicates and infinity can be expressed (the latter in the presence of a constant symbol). This is an extended version of a publication in the proceedings of the 26th Annual