M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing, NorthHolland Publ. Co., Amsterdam 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Hilbert system for PLL [FM97] The natural deduction sequent rules for QLL are a first order (universal and existential) quantificational extension to the natural deduction sequent system for CL logic [BBdP97] For readers familiar with intuitionistic quantified logic IQC (if not see, for example, [Fit69]) QLL is a conservative extension of IQC. That is to say, by prohibiting the use of QLL specific proof rules, it is not possible to introduce new theorems of IQC. The language of QLL has individual constant symbols (c 1 ; c 2 ; c 3 ; individual variable symbols (x 1 ; x 2 ; x 3 ; ....

....The device [c=x]ae is defined to mean that the assignment of c (assumed to exist in jffj) to x over rides existing assignments to x in ae . The assignments to variables given by ae can be lifted to the assignment ae of entire terms (Term) to elements of worlds in the usual manner (see, e.g. [Fit69]) Using the above terminology, therefore, ae is used to denote the function ae : Term jffj The validity of entire formulae may now be defined, based upon the above definitions. Definition 2.7 (Validity) Let C be a QLL Kripke constraint structure. A formula, OE, valid at a world ff in C ....

M. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969. 33

....F[##] FH[C1 ##A(p) ##Ch #D] with p new FH[#xA(x) F[#] FH[A(y) ##xA(x) FH[C1 ###xA(x) ##Ch #D] F[##] FH[C1 ##A(y) ##xA(x) ##Ch #D] FH[#xA(x) F[#] FH[A(p) with p new Fig. 1. The tableau calculus CDt definitions and conventions concerning the tableau calculi (see, for instance, [1, 2, 6, 15]) In this framework, a configuration is a finite sequence # 1 # 2 . #n (with n # 1) such that every # j (1 # j # n) is a set of signed formulas; a CDt proof table is a finite sequence of applications of the rules of the calculus CDt starting from some configuration. A set of signed ....

....if the latter hypothesis holds we can apply the induction hypothesis to G 2 and (ii) follows also in this case. 4 Completeness Our aim is to realize finite consistent sets of signed formulas # fin in Kripke models with constant domain. In line with standard completeness proofs (see for instance [6]) the starting point is the following definition. Definition 4.1 Let C be a collection of sets of signed formulas and let # be a nonempty set of individual variables. We say that C is a CD collection with respect to # if and only if, for every # #C, all the elements of # have the form ....

M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....as programs together with their verification. Consequently intuitionistic logic is a method for developing correct programs. To demonstrate the advantage of this approach, we construct two theorem provers for the propositional part by extracting them from a decidability proof. Taking up Fitting s [2] completeness proof, Underwood [12] outlined how a decidability proof could be implemented in a formal system like Nuprl. For this, she proved that each set of sequents either contains a provable one or has a Kripke model so that each sequent is refuted at a certain node. The sequents themselves ....

Melvin C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, Amsterdam, 1969.

....logic, resolution, constraint propagation, flow analysis, looping, model checking modal logics. 1 Introduction Deciding the validity of formulas in intuitionistic propositional logic is a PSPACE complete problem [7] Any non valid formula of that logic can be refuted by a finite Kripke tree [1]. Various tools for deciding validity have been introduced. We mention (1) a constructive version of the completeness proof for Kripke models, which either produces a Nuprl type, representing a proof of the formula, or a Kripke model with a world that refutes the formula in question [9] 2) a ....

M. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, Amsterdam, 1969.

....inference has also been carried out, largely by the logic programming community. Most of this work focuses on the hypothetical insertion of atoms into a database. One reason for this focus is that hypothetical insertion fits neatly into a well known logical system: intuitionistic logic [29]. Gabbay was the first to show that intuitionistic logic models hypothetical insertion [32] Working independently, McCarty and Miller extended this result to operations that create new constant symbols during inference, and they developed fixpoint semantics based on intuitionistic logic [47, 50] ....

....for Horn rulebases and atomic queries. That is, if i denotes entailment in first order intuitionistic logic, then R i A iff R c A (2.4) for any Horn rulebase R, and any ground atomic formula A. However, intuitionistic and classical logic have different semantics and different theorems [29]. For example, in intuitionistic logic, implication is not defined in terms of disjunction and negation, but has an independent semantic definition. Thus, the formula ff fi is not equivalent to ff fi intuitionistically. One consequence is that intuitionistic implication has the following ....

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M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

.... there a subtraction analogue ffl Functional Completeness: Is there a sense in which the connectives of intuitionistic propositional logic capture all of the propositional connectives on some domain of propositions ffl Forcing: It is known that intuitionistic logic has a connection with forcing [4]. Does subtraction add anything to our understanding of this connection ffl Topos Theory: Intuitionistic logic is modelled in toposes, which are naturally occurring category theoretic structures. Does subtraction have a place in these structures Work in closed set logic in categories is no ....

Melvin Fitting. Intuitionistic Logic, Model Theory and Forcing. North Holland, Amsterdam, 1969.

.... Hilbert system for PLL [FM97] The natural deduction sequent rules for QLL are a first order (universal and existential) quantificational extension to the natural deduction sequent system for CL logic [BBdP97] For readers familiar with intuitionistic quantified logic IQC (if not see, for example, [Fit69]) QLL is a conservative extension of IQC. That is to say, by prohibiting the use of QLL specific proof rules, it is not possible to introduce new theorems of IQC. 2.1 Language The language of QLL has individual constant symbols (c 1 ; c 2 ; c 3 ; individual variable symbols (x 1 ; x 2 ; ....

....[c=x]ae ff is defined to mean that the assignment of c (assumed to exist in jffj) to x over rides existing assignments to x in ae ff . The assignments to variables given by ae can be lifted to the assignment ae of entire terms (Term) to elements of worlds in the usual manner (see, e.g. [Fit69]) Using the above terminology, therefore, ae ff is used to denote the function ae : Term jffj The validity of entire formulae may now be defined, based upon the above definitions. Definition 2.7 (Validity) Let C be a QLL Kripke constraint structure. A formula, OE, valid at a world ff in ....

M. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....to Wallen s matrix characterization. While originally we were interested only in constructing a matrix proof which should be translated into a sequent proof afterwards our investigations have shown that it is helpful to exploit the close relation between a matrix proof and a proof in Fitting s [7] sequent calculus LJNS 1 and to consider the structure of the corresponding sequent proof already during the proof search and design our proof procedure as a hybrid method combining the connection method with the sequent calculus. The result of our proof search procedure will be a matrix proof ....

....also applies to predicate logic if the connected formulae can be shown to be complementary , i.e. if all the terms contained in connected formulae can be made identical by some (first order quantifier) substitution oe Q . In sequent calculi like Gentzen s LK and LJ [8] or Fitting s calculi [7] the difference between classical and intuitionistic reasoning is expressed by certain restrictions on the intuitionistic rules. If rules are applied in a top down fashion these restrictions cause formulae to be deleted from a sequent. Applying a rule (i.e. reducing a subformula) too early may ....

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M. C. Fitting. Intuitionistic logic, model theory and forcing. North--Holland, 1969.

....46] Theoretical work on hypothetical inference has also been carried out, largely by the logic programming community. Most of this work focuses on the hypothetical insertion of atoms into a database. These updates, it turns out, fit neatly into a well known logical system, intuitionistic logic [23]. Most of this work has been semantic, first showing that hypothetical insertion is indeed intuitionistic, and then casting the semantics in terms of a least fixpoint theory, in the logic programming tradition. Gabbay first showed that hypothetical insertion is intuitionistic [25] Working ....

....intuitionistic model theory, specializing the presentation for the special case of embedded implications. Note that the syntax of the logic is firstorder, but its model theory is modal, i.e. is based on a set of possible worlds. A complete development of intuitionistic logic can be found in [23, 35]. Definition 4.1 (Structures) An intuitionistic structure is a triple M = hS; i, where ffl S is a non empty set, ffl is a transitive, reflexive relation on S, ffl is a mapping from elements of S to sets of ground atomic formulas, ffl for any two elements s 1 and s 2 in S, if s 1 s 2 ....

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M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....sequent calculus. 1991 Mathematics Subject Classification: 03B20, 03B35, 03C25, 03F03, 68T15 1. Introduction It is well known that IPL (Intuitionistic Propositional Logic) has the finite model property; in fact, any non theorem of IPL can be invalidated by means of a finite Kripke tree [2]. The standard method (see [9] for a formal treatment) for constructing such counter models requires a loop checker. Here, we present a method for constructing counter models not requiring a loop checker, based on the contractionfree sequent calculi LJT and LJT [1] LJT provides a very simple ....

....we have shown that for any pair ( Gamma; Delta) either LJT Gamma ) Delta or CRIP Gamma 6) Delta. Also, it is routine to show by induction (on the size of the LJT proof) that only one of these two can hold. ut 4. Construction of counter models We assume familiarity with Kripke semantics [4, 2, 7]. In brief, we are interested in finite Kripke trees, i.e. sets K with a reflexive transitive binary relation on K, with a least element w 0 such that Knw 0 is a disjoint union of Kripke trees, together with a monotone relation j= from K to the set of atoms of IPL, extendable in Loop free ....

Fitting, M., Intuitionistic Logic, Model Theory and Forcing, North-Holland, Amsterdam, 1969.

....or the partial circumscription axiom. We will discuss these results further in Section 7. A further variant of this example shows how to combine the base predicate with other base predicates such as On and Green . For this, it is necessary to use intuitionistic rather than minimal negation [17]. We thus stipulate that every proposition is entailed by . To do this, using Kripke semantics, as presented in Section 3, we simply insist that is never forced at any state in any Kripke model. Suppose we adopt the definition of Covered in rule (32) and then define: Clear(x ) ....

....s 2 K a subset of the n fold Cartesian product of t(u(s) with itself, subject to the requirement that R(s 1 ) R(s 2 ) whenever s 1 s 2 . We then define h to be a mapping from the predicate constants in L to the set of intuitionistic relations on K. The atomic clause of the forcing relation [17] is thus: s; K j= P (t 1 ; t n ) iff ht 1 ; t n i 2 h(P ) s) for P a predicate constant of arity n. The remaining clauses are defined as usual. Since we are only using embedded implications in this paper, we only need the following: s; K j= A B iff s; K j= A and s; K j= B, s; K ....

M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....Other researchers in the logic programming community have investigated the semantics of hypothetical inference. This work has focussed on the hypothetical insertion of facts into a database, since such updates fit neatly into a well established logical framework intuitionistic logic [17]. In intuitionistic logic, hypothetical insertion arises from formulas called embedded implications [36] These are rules of the form A (B C) which informally means that A can be inferred if assuming C allows B to be inferred. The assumption of C is a hypothetical insertion into the ....

....logic is that of first order predicate calculus. It includes three infinite, enumerable sets: a set of variables x; y; z; a set of constant symbols a; b; c; and a set of predicate symbols A; B; C; More extensive treatment and discussion of intuitionistic logic can be found in [17, 32]. Definition 3.1. Structures] A first order intuitionistic structure is a quadruple M = hS; di, where 1. S is a non empty set. 2. is a partial order on S. 3. is a monotonic mapping from elements of S to sets of ground atomic formulas. Thus, if s 1 s 2 then (s 1 ) s 2 ) 4. d is a ....

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M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....above are non classical. Indeed, they were introduced precisely to overcome the shortcomings of classical logic described in section 3.2. The next section shows that they have an intuitionistic semantics. This section provides a brief development of propositional intuitionistic logic adapted from [4] and [10] Definition 4 Suppose L is a finite or countably infinite set of propositional atoms. A substate is a subset of L, and an intuitionistic structure is a set of substates. Furthermore, if s 1 and s 2 are substates, then s 1 s 2 iff s 1 s 2 . Definition 5 (Satisfaction) Suppose is a ....

M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....how tableau structures (not only proofs ) can be translated into resolution structures. This translation leads to an improvement of the given algorithm for optimization. 1. Introduction The Beth s tableau calculus is used in the form of the deductive system described by Smullyan [5] and Fitting [2]. Tableaux are interpreted as trees, the contents of the nodes in these trees are signed formulae. The tree should also contain information about the closing pairs, i.e. about the nodes that close up the branch they lie in, and about the substitutions used for that closing up. So it is possible to ....

Fitting, M. C., Intuitionistic Logic, Model Theory And Forcing, North-Holland, Amsterdam, 1969

.... You are eligible for citizenship if your father would be eligible if he were still alive [13] Rules such as these are called embedded implications [20] Embedded implications have an intuitionistic semantics. That is, their models are Kripke structures, consisting of many possible substates [11]. McCarty has developed an intuitionistic fixpoint semantics for embedded implications and has shown that they have interesting semantical properties analogous to the unique minimal model property of Horn clauses [20, 3] In particular, he has shown that they have a unique maximal Kripke model, ....

M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....theorems are stated without proofs. Full proofs will be included in an expanded version of the paper. 2 Foundations: Minimal Logic We begin with some technical background. We assume that the reader is generally familiar with the Kripke semantics for first order intuitionistic logic, as given in [18, 8], and we simply review our notation here. Minimal logic [17] differs from intuitionistic logic only in the treatment of negation, as we will see below. We also define in this section the concept of a final Kripke model, and we present enough of the rudiments of second order intuitionistic logic ....

....s 2 K a subset of the n fold Cartesian product of u(s) with itself, subject to the requirement that R(s 1 ) R(s 2 ) whenever s 1 s 2 . We then define h to be a mapping from the predicate constants in L to the set of intuitionistic relations on K. The atomic clause of the forcing relation [8] is thus: s; K j= P (c 1 ; c n ) iff hc 1 ; c n i 2 h(P ) s) for P a predicate constant of arity n. The remaining clauses are defined as usual. The most important, for our purposes, are the following: s; K j= B ( A iff s 0 ; K j= A implies s 0 ; K j= B for every s 0 s ....

M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.

....of intuitionistic semantics. The following sections then show how this semantics can be modified to account for negation as failure. 4.1 Intuitionistic Semantics The following is a simplified development of the Kripke semantics of intuitionistic logic. A complete development may be found in [9, 13]. Definition 4.1 (Structures) An intuitionistic structure is a triple M = hS; i, where ffl S is a non empty set, ffl is a transitive, reflexive relation on S, ffl is a mapping from elements of S to sets of ground atomic formulas, ffl for any two elements s 1 and s 2 in S, if s 1 s 2 ....

....j= 2 1 iff r; M j= 1 implies r; M j= 2 for all r s Note that unlike classical logic, intuitionistic implication is not defined in terms of disjunction and negation. Rather, it has an independent semantic definition. An intuitive interpretation of this semantics may be found in [13] and [9]. Definition 4.3 (Models) M j= iff s; M j= for all substates s of M . If M j= then M is a model of . Definition 4.4 (Validity) A formula is valid iff M j= for all intuitionistic structures M . Definition 4.5 (Entailment) Suppose 1 and 2 are formulas. Then 1 j= 2 iff the ....

M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. NorthHolland, 1969.

....that it is intuitionistic; that is, the inference system is both sound and complete with respect to intuitionistic semantics. 4.1 Intuitionistic Logic This section defines the semantics of first order intuitionistic logic in the function free case. A more extensive treatment may be found in [9]. Recall that the syntax of the logic is first order and that it includes three infinte sets: a set of variables x; y; z; a set of constant symbols a; b; c; and a set of predicate symbols A; B; C; Each of these sets is called a universe. Definition 3 An intuitionistic structure is ....

....s; M j= b) for some b 2 dom(s) and 9 x (x) is in F(s) Note that unlike classical logic, intuitionistic implication is not defined in terms of disjunction and negation. Rather, it has an independent semantic definition. An intuitive interpretation of this semantics may be found in [13] and [9]. The following basic result is an immediate consequence of the above definitions. Lemma 5 s; M j= iff r; M j= for all r s. Definition 6 (Models) M j= iff s; M j= for all substates s of M such that 2 F(s) If M j= then we say that M is a model of . Definition 7 (Validity) A ....

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M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. NorthHolland, 1969.

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M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing, NorthHolland Publ. Co., Amsterdam 1969.

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M. C. Fitting, Intuitionistic Logic, Model Theory and Forcing (North Holland, 1969).

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M. C. Fitting. Intuitionistic logic, model theory and forcing. North--Holland, 1969.

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Melvin Fitting. Intuitionistic Logic, Model Theory and Forcing. North--Holland, Amsterdam, 1969.

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M. C. Fitting. Intuitionistic logic, model theory and forcing. North--Holland, 1969.

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