Arnon Avron. Simple consequence relations. Information and Computation, 92:105--139, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and from additive conjunctions ( and and disjunctions (# and #) We have not included negation as a first class citizen of our logical system. One reason being that the system does not admit such an operation which would fulfill conditions of being a logical negation in the sense of Avron ([2]) Yet, in order to retain the expressive power of (quantifier free) classical logic the usual trick is applied, i.e. one keeps negation on the level of atomic formul. In our case it is safe to apply the trick to platonic atomic formul, i.e. to predicates. Thus, for each predicate symbol R of ....

Avron, A. Simple consequence relations. Information & Computation, Vol 92, No 1, pp. 105--139, May 1991.

....l does not imply that l Body is true in every four valued valuation (Consider, e.g. the case in which Body= l) We therefore consider an alternative de nition for the implication connective, according to which it does function as an entailment in the four valued setting: De nition 2. 4 [5,8]. Let x; y 2 FOUR. De ne: x y = x if y2D t otherwise Note that on ft; fg the material implication and the new implication are identical, and both of them are generalizations of the classical implication. However, unlike the material implication, the implication connective de ned in 2.4 ....

A.Avron, Simple consequence relations, Information and Computation 92 (1991) 105-139.

....change anything when truth values 0 and 1 are considered, only, so the connective can be considered as one of the possible generalizations of the implication connective of classical logic. It is the right generalization of classical implication, because is the internal implication connective [3] for the de ned inference relation in the sense that a deduction (meta)theorem holds for it: j= i j= An interesting feature of the inference relation j= is that inconsistency cannot occur in the f: g fragment (this is not the case when the full language is considered ....

A. Avron. Simple consequence relations. Information and Computation, 92:105{ 139, 1991.

....of the Logical Framework. With this interpretation, a judgement is viewed as a type whose inhabitants correspond to proofs of the judgement. It is worthwhile noticing that Logical Frameworks based on type theory directly give rise to proof systems in Natural Deduction style in the sense of [1, 12]. This follows from the fact that the typing systems of the underlying # calculi are in Natural Deduction style, and rules and proofs are represented by # terms. As it is well known, Natural Deduction style systems are more suited to the practical usage, since they allow for developing proofs the ....

A. Avron. Simple Consequence Relations. Information and Computation, 92:105--139, 1991.

....of an editor for a given object logic, to look for a presentation of the logic which can take best advantage of the possibility of manipulating assumptions. The crucial concept involved in discussing the notion of assumption for a given logic is that of consequence relation (CR) [2]. CR s are abstract representations of logical dependencies between assumptions and conclusions. They play a crucial role in stating and proving adequacies of encodings in Logical Frameworks. Usually, a logic gives rise to more than one CR. For instance, in FOL we have the validity CR and the ....

.... by adding either the convergence rule or the equivalent dual induction principle rule: Conver # # p x 1 x # #c# p # # p t x # # #c # # p 0 x x ## FV(c) Induc # # [c] p # p x 1 x # # [c # ] p 0 x # # p t x x ## FV(c) Both rules are impure in the sense of Avron [2], and are proof rules, since the first premise is a theorem. One can easily see that # S a ND (DL) #(x : x 1) # # (x = 0) and # S a ND (DL) #while x 0 do x : x 1# (x = 0) Indeed, S a ND (DL) is sound and complete with respect to the standard model of integers. Theorem ....

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A. Avron. Simple consequence relations. Inform. Comput., 92:105--139, Jan. 1991.

....of the NOS style In this section we try to convey briefly to the reader the main features of operational semantics in N.D. style. Recall that a N.D. style rule can be viewed as a concise description of a special kind of rule for deriving sequents, i.e. metapropositions of the form # # A ([8, 2]) # 1 ) # k ) A 1 . A k A corresponds to #, # 1 # A 1 . #, # k # A k # # A where A, A 1 , A k are propositions and # 1 , # k sets of propositions. # is any set of proposition. This rule means that, in order to prove that A is a consequence ....

A. Avron. Simple consequence relations. Information and Computation, 92:105-- 139, Jan. 1991.

....style typing system (Section 5) the fundamental notion is that of consequence: which assertions follow from a given set of hypotheses. This is also one of the reasons for using other styles of proof systems, such as Natural Deduction or Sequent Calculus; for a deeper discussion we refer to [6, 10]) For defining the notion of logical consequence, we need to introduce contexts: Definition 10 (Ctxt) The set Ctxt of contexts is defined as follows: # # Ctxt (the empty context) if # # Ctxt, x # Nam does not appear in #, and A # PT is a type, then #, x : A # Ctxt; if # # ....

A. Avron. Simple consequence relations. Information and Computation, 92:105--139, Jan. 1991.

....is another possibility. 3 Officially we should have written fAg ) fAg. We shall, however, follow tradition and omit the curly brackets from both sides of ) Also, we shall usually write Gamma; Delta for Gamma [ Delta, etc. 4 This is a variant of a notion which was first introduced in [Av]. 2 inferred by it from Gamma i ; Gamma 0 ) Delta i ; Delta 0 (i = 1; n; Gamma 0 ; Delta 0 arbitrary sets of formulae) A Gentzen type system is called pure if all its rules are pure. A Gentzen type system, G, directly defines a Consequence Relation (C.R. G between ....

....contradictory. In case L has internal negation : and internal falsehood 6 this can be formally expressed as: t G A iff :A v G . 5 It is enough to assume monotonicity and purity on the l.h.s. Hence, the proposition is true, e,g. also for the intuitionistic Gentzen calculus. 6 See [Av] for definitions. 3 III. Classical Propositional Calculus We turn now to investigate the special case of GCPL the Gentzen type system for Classical Propositional Logic (CPL) in which most of the connections between tableaux and resolution are already reflected. The version we use here is ....

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Avron A., Simple Consequence Relations , Information and Computation, 92 (1991), pp. 105-139.

....# Dipartimento di Matematica e Informatica, Universita di Udine. Via delle Scienze 206, I 33100 Udine, Italy. mailto: honsell,miculan,paravano dimi.uniud.it, http: www.dimi.uniud.it SLP 1 i.e. capable of referring to their own proofs. 1 generally impure rules in the sense of [2]. Such rules, in fact, cannot be applied uniformly to any set of premises, but are subject to various forms of restrictions, e.g. the premises depend on no assumption; or depend only on assumptions of a certain shape (boxed, essentially boxed, etc. or even, the premises have been derived only ....

....and a state s, we define when # is true in s (s = M #) inductively on the structure of the formula, as usual. In particular, s = M ## ## #s # .s # s # # s # = M #. If # is true in every state of M, we say that # is valid in M ( M #) 1. 2 Consequence Relations According to [2, 24, 31], the semantic interpretation of formul gives rise to (at least) two (logical) consequence relations (CR s) Definition 1.1 (Truth and Validity Consequence Relations) Given # # #, # # #, and M class of models, we say that . # is true in # w.r.t. M (# = M #) if #M # M.#s # M.s = M ....

A. Avron. Simple consequence relations. Infor. Comp., 92:105--139, Jan. 1991.

....its set of axioms includes the standard axioms: for all ) and its set of rules includes the standard structural rules, and the following weakening and cut rules: 0 ; 0 1 ) 1 ; 2 ) 2 1 ; 2 ) 1 ; 2 Definition 2. 3 A rule of a Gentzen type system G is called pure ([11]) or multiplicative ( 33] if whenever ) can be inferred by it from i ) i (i = 1; n) then, for arbitrary 0 1 ; 0 n ; 0 1 ; 0 n , the sequent ; 0 1 ; 0 n ) 0 1 ; 0 n can also be inferred by it from i ; 0 i ) i ; 0 i (i ....

A. Avron. Simple consequence relations. Information and Computation, 92(1):105--139, 1991.

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Avron A., Simple Consequence relations, Information and Computation, vol 92 (1991), pp. 105-139.

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Avron A., Simple Consequence relations, Information and Computation, vol 92 (1991), pp. 105-139.

....It is possible to remove it with the expense of complicating somewhat the definitions and propositions. It is preferable instead to employ (whenever necessary) the propositional constants t and f to represent the empty l.h.s. and the empty r.h.s. respectively. 12 This definition is taken from [7]. Definitions 2.4 and 2.21 are obvious adaption of it. 13 It easily follows from (a) above and from the properties of # in # that the order according to which ## is taken has no importance here. 14 This property is dual to the property of internal conjunction reduction (TICR, see ....

....consider only the # k minimal models of #. However, as Proposition 3.22(b) shows, in the general case # L,F #k is not equivalent to # L,F . The next proposition (3.24) is another evidence for that. Its proof easily provides an example for the note after Proposition 2.16: Definition 3. 23 [7, 2] Let (L, F) be a logical [bi ]lattice. Define: a # b = b if a #F, and a#b= t otherwise. 25 Note It is well known that in multiple valued semantics it is usually no longer true that every classical tautology remains valid. For instance, in Kleene three valued logic logic [22] as well as in ....

A.Avron. Simple consequence relations. Journal of Information and Computation, Vol.92, pages 105--139, 1991.

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Arnon Avron. Simple consequence relations. Information and Computation, 92:105--139, 1991.

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Arnon Avron. Simple consequence relations. Information and Computation, 92:105--139, 1991.

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Avron, A. (1991) Simple consequence relations, in Information and Computation Vol.92, pp.105--139.

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Arnon Avron. Simple consequence relations. Information and Computation, 92:105--139, 1991.

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A. Avron, Simple consequence relations, Information and Computation, 92 (1991), 105-139.

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Arnon Avron. Simple consequence relations. Information and Computation, 92:105--139, 1991.

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A. Avron. Simple consequence relations. Information and Computation, 92(1):105-139, 1991.

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A. Avron. Simple consequence relations, Information and Computation 92: 105--139, 1991.

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Arnon Avron. Simple consequence relations. Information and Computation, 92:105--139, 1991.

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A. Avron. Simple Consequence Relations. Information and Computation, 92:105-139, 1991.

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Arnon Avron. Simple consequence relations. Information and Computation, 92(1):105-140, 1991.

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A. Avron. Simple consequence relations. Information and Computation, 91(1): 105-139, 1991.

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