Dirk van Dalen. Intuitionistic Logic. In D. Gabbay and F. Gunthner, editors, Handbook of Philosophical Logic; Vol. III: Alternatives in Classical Logic, pages 225--339. D. Reidel, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Indeed, the studied problem can be further traced back to the invalidity of the Law of the Excluded Middle. To overcome the problem, we interpret Statecharts, relative to a given system state, as intuitionistic formulas. These are given meaning as specific intuitionistic Kripke structures [18], namely linear increasing sequences of event sets, called stabilization sequences, which encode interactions between Statecharts and environments. In this domain, which is also characterized via semi lattices and in which Pnueli and Shalev s semantics may be explained by considering a ....

....due to interactions with the environment, in this case event b in C 16 fa; bg. Consider further that C 79 = b=a b=a is also step equivalent to C 12 = Delta=a. Hence, a compositional macrostep semantics does not validate the Law of the Excluded Middle b :b = true. Since intuitionistic logic [18] differs from classic logic by refuting the Law of the Excluded Middle, it is a good candidate framework for analyzing Statecharts semantics. It should be stressed, however, that the compositionality defect is mainly an issue of operator k and not of , as we will see below. Our goal is to ....

[Article contains additional citation context not shown here]

D. van Dalen. Intuitionistic logic. In Handbook of Philosophical Logic, vol. III, chap. 4, pp. 225--339. Reidel, 1986.

....a relevant analogue of the intuitionistic disjunction property. One of the consequences of cut elimination in sequent systems for propositional intuitionistic logic is the disjunction property, if A B is provable then so is A or B, which re ects the constructive character of the logic [29, 33]. The property holds also under hypotheses, if A B is provable then so is A or B, provided that the hypotheses in are Harrop formulas [33] where the class H of Harrop formulas is inductively de ned by: i) p 2 H for p a propositional variable, ii) A B 2 H if A; B 2 H, and (iii) A B ....

.... logic is the disjunction property, if A B is provable then so is A or B, which re ects the constructive character of the logic [29, 33] The property holds also under hypotheses, if A B is provable then so is A or B, provided that the hypotheses in are Harrop formulas [33], where the class H of Harrop formulas is inductively de ned by: i) p 2 H for p a propositional variable, ii) A B 2 H if A; B 2 H, and (iii) A B 2 H if B 2 H, where is intuitionistic implication. Since C D 62 H, the restriction to Harrop formulas ensures that there is no disjunctive ....

D. van Dalen. Intuitionistic logic. In D. M. Gabbay and F. Guenthner, eds., Handbook of Philosophical Logic III, pages 225-339. Reidel, 1986.

....stronger than classical negation. Along with the principle of the excluded middle, the other classical principles of double negation, de Morgan, etc have also to be partially dismissed from intuitionistic logic. For example, it is known that intuitionistic logic has the disjunction property (see [102]) p q is provable iff p is provable or q is provable. But the following does not hold for intuitionistic logic: p q) is provable iff :p is provable or :q is provable, which seems necessary for a truly constructive negation. Because of this unsymmetric characteristic, we may say that ....

.... various approaches to the universal quantifier in the following table, where INT is intuitionistic predicate logic, H is Gurevich s intuitionistic logic with strong negation [54] and CF is Thomason s first order logic [98] For a unifying exposition of both Kripke and Beth models, see van Dalen [102]. Logics Quantifiers 8x Models Domains INT, H dynamic Kripke models expanding INT static Beth models constant CF static Kripke models constant CF 0 static Kripke models expanding 3. First Order Logic CF 0 36 In the following, we shall first introduce the logical system CF 0 , and then ....

D. van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, Vol III: Alternatives in Classical Logic, pages 225--339. D. Reidel, Dordrecht, 1986.

....Indeed, the studied problem can be further traced back to the invalidity of the Law of the Excluded Middle. To overcome the problem, we interpret Statecharts, relative to a given system state, as intuitionistic formulas. These are given meaning as specific intuitionistic Kripke structures [18], namely linear increasing sequences of event sets, called stabilization sequences, which encode interactions between Statecharts and environments. In this domain, which is also characterized algebraically via semilattices, we develop a fully abstract macro step semantics in two steps. First, we ....

....will never arise. This eliminates the possibility that events may be generated due to interactions with the environment, in this case event b in C 16 fa; bg. In short, a compositional macro step semantics does not validate the Law of the Excluded Middle b :b = true. Since intuitionistic logic [18] differs from classic logic by refuting the Law of the Excluded Middle, it is a good candidate framework for analyzing Statecharts semantics. It should be stressed that the compositionality defect is mainly an issue of operator k and not of , as we will see below. Our goal is to characterize the ....

[Article contains additional citation context not shown here]

D. van Dalen. Intuitionistic logic. In Handbook of Philosophical Logic, vol. III, chap. 4, pp. 225--339. Reidel, 1986.

....for converting constructions of A into constructions of B. There is no construction of #. A construction of #A is a technique for converting constructions of A into constructions of #. This is the Brouwer, Heyting and Kolmogorov (BHK) interpretation of the intuitionistic connectives [3, 4, 5], and it plays an important role especially in the formalisation of mathematical theories where the notion of a proof of a proposition or of a construction of an object can be rigourously defined. In more general settings, the notion of proof or construction is perhaps better replaced by the ....

DIRK VAN DALEN. "Intuitionistic Logic". In DOV M. GABBAY AND FRANZ G UNTHNER, editors, Handbook of Philosophical Logic, volume III. Reidel, Dordrecht, 1986.

.... than those they were originally intended for individually [6] In this paper we investigate the problem of knowledge sharing among resource sensitive systems (also called substructural logic systems [9] The class of substructural logics, encompassing, for instance, intuitionistic logics [8], relevance logics [2] and linear logics [15] employ in their inferences structural rules which take into account the structure of premises in a deduction [9] Substructural logics differ from each other by virtue of the structural rules allowed in their proofs: the set of structural rules ....

D. Van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosoph. Log., volume III, 1984.

....5) that C j= a = 1 oe b = 1 is equivalent to 8D: C v D D j= a = 1 ) 9ffi: D ffi j= b = 1; where D ffi is obtained from D by time shifting all W 2 D by an amount of ffi, D ffi = f W ffi j W 2 D g. This formulation has the structure essentially of an intuitionistic implication (see e.g. [52]) interpreted on the Kripke frame induced by the set of circuits C S N B ordered by the v relation. Using this Kripke model for LJ obtains a fullblooded intuitionistic semantics based on j= in which non trivial stabilisation behaviour can be expressed and verified. To finish off this section ....

.... if then but gives rise to in bounded time, where the stress is on bounded. We will elaborate on the connection between this Kripke semantics and bounded stabilisation in the next section. The semantic clauses above defining j= are almost exactly the standard clauses (see e.g. [52]) for intuitionistic validity in Kripke models. Compared to the standard setting there are two important things to note, though. i) We are considering a particular class of Kripke models generated from sets of waveforms under the v ordering. This means that our interpretation of LJ is rather 6 ....

D. van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume III, chapter 4, pages 225--339. Reidel, 1986.

....such a proposal, in [13] some intuitionistic modal logics and Kripke type semantics are given. It is further shown that IS5 is equivalent to Bull s MIPC (see [2, 3, 21] The forcing relations for both modal operators are analogous to those of the intuitionistic quantifiers on Kripke 2 frames ([5], see Definition 3.6) Two connecting properties for the Kripke type frames are given to characterize the intuitionistic modal logics (see also [7] We show that IK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IK45 and IS5 have the Disjunction Property. A similar path in tense logic has ....

....but this follows from the above remark. Proof of (16) Suppose k 32A: we have to show jA. k 32A implies k 2A and thusj2A , i.e.k A. Part B) is K, K4, K45, K5, KB, KB4. We show the Disjunction Property for via a semantic proof, based on modal intuitionistic Kripke models 1 (see Theorem 4. 1 [5]) Consider two models M 1 = W 1 ; 1 ; R 1 ; j= 1 and M 2 = W 2 ; 2 ; R 2 ; j= 2 s.t. for some w 1 2 W 1 and w 2 2 W 2 , w 1 6j= 1 A and w 2 6j= 2 B. Construct the new Kripke model M by taking the disjoint union of M 1 and M 2 and add w, s.t. w 62 W 1 S W 2 , w w 1 and w w 2 . We ....

D.van Dalen, Intuitionistic Logic, in D.Gabbay and F.Guenthner (eds.), Handbook of Philosophical Logic, Vol. III, Reidel, Dordrecht 1986, pp. 225-339.

....an explicit link (already mentioned by other authors) between constructive negation and intuitionistic logic that may be worth to study. In particular, it could serve as a basis for extending with negation those approaches to modularity based on the use of an intuitionistic implication (e.g. see [27]) The paper is organized as follows: in the next section we introduce some basic notions and notation; in section 3, we sketch the propositional case to provide some intuition about the proposed solutions. In sections 4, 5 and 6, we present the semantics for the first order case, including a ....

D. van Dalen, Intuitionistic Logic, in Handbook of Philosophical Logic, D. Gabbay, F. Guenthner (eds), Vol III, 1986, pp. 225-339.

....has some idea both of the formal properties of intuitionistic logic, and some motivating philosophical principles which inform the development of intuitionistic logic. Readers wanting such an introduction can do no better than look at some of the excellent, extensive literature on intuitionism [3, 5, 8, 12]. Other work has been done on extending intuitionistic propositional logic with new connectives. Gabbay [6] considers extending the logic with propositional connectives. Our new connective is not one he considers. De Jongh too considers extending intuitionistic logic by adding arbitrary ....

Dirk van Dalen. Intuitionistic logic. In Dov M. Gabbay and Franz Gunthner, editors, Handbook of Philosophical Logic, volume III. D. Reidel, Dordrecht, 1986. 1 Thanks to Rob Goldblatt, Ed Mares and the others in an audience at the Victoria University of Wellington for comments on an earlier draft of this paper.

....388 the if part and then part of the rule. Therefore, CML does not satisfy the essential requirements for the logic system to be used as the fundamental theory underlying reasoning rule generation and verification. All logic systems (including modal logic systems [3,11] intuitionistic logic [16], and those logic systems developed in recent years for nonmonotonic reasoning [7,15] where the entailment is directly or indirectly represented by the material implication have the similar implicational paradox problem as that in CML. Therefore, in order to solve our problems, we have to ....

D. Van Dalen, "Intuitionistic Logic," in D. Gabbay and F. Guenthner (eds.), "Handbook of Philosophical Logic," Vol.III, D. Reidel, pp.225-339, 1986.

....to the modal Heyting algebra induced by the constraint frame 4 Theta 4 Theta 2. Theorem 4.5 implies that PST(a) is a finite algebra. Contrast this with intuitionistic logic for which the Lindenbaum algebra in one atomic proposition, known as the Rieger Nishimura lattice, is infinite (see e.g. [38]) 4.3 Relationship with Intermediate Logics of Kreisel Putnam, Dummett, and Medvedev Let us take a closer look at the intuitionistic base, i.e. the fl free fragment of PST. We call this modal free fragment PST I. We show that PST I is related to two well known intuitionistic intermediate ....

....to other constructive logics. First of all, one can show that the standard part S(PST I) strictly extends intuitionistic propositional logic IPC but is properly included in classical propositional logic CPC. As to strictness we note that S(PST I) contains the KreiselPutnam axiom scheme (see e.g. [38]) oe ( 1 2 ) oe ( oe 1 ) oe 2 ) which is not a theorem of IPC. At the other end S(PST I) refutes the CPC axiom of the Excluded Middle : Thus, IPC ( KP S(PST I) CPC, where KP is the logic obtained from extending IPC with the Kreisel Putnam scheme. This means that ....

D. van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume III, chapter 4, pages 225--339. Reidel, 1986.

....may think that our semantics is more complicated than needed, in the sense of dealing with complex ranked structures instead of just three valued ones. We disagree with that: ranked structures are just a special case of Beth structures, used to provide semantics to intuitionistic logic (e.g. see [38]) In this sense, our semantics establishes an explicit link (already mentioned by other authors) between constructive negation and intuitionistic logic that may be worth to study. In particular, it could serve as a basis for extending with negation those approaches to modularity based on the use ....

....problem, in terms of general Beth structures. In this sense, we believe that our semantics could also be valuable for knowledge representation considering the intuition behind Beth (and also Kripke) structures where each world in a model represents the knowledge one has at a given moment (see e.g. [38]) The motivation for this new semantics was the definition of a specification frame of normal logic programs that could be used for defining compositional semantics to a variety of program units. In this sense, we have shown that the proposed semantics defines indeed a specification frame with ....

van Dalen, D., Intuitionistic Logic, in: Gabbay, D., Guenthner, F. (eds), Handbook of Philosophical Logic Vol III, 1986.

....and lemmas, and make it possible to hide trivial fragments of proofs from the user. see e.g. 17, 18] Building a system that does this in an efficient and useful way is as much or more of an engineering challenge as devising the right formal system is a mathematical one. 7 If not, we recommend [21] for a quick introduction[114, 23] for a more thorough one, and for further references, and [24] for a detailed discussion of the philosophy and metamathematics of intuitionism. 8 Brouwer, the originator of intuitionism, certainly never felt that formal systems had anything to do with ....

....is usually denoted [ x = y ] satisfying symmetry and transitivity: x = y ] y = x ] and [ x = y ] y = z ] x = z ] 17 This definition, in the case of topological cHa s, predates Kripke s semantics by more than a decade. It was proposed by Tarski and McKinsey s in the 1940 s [82, 21] It will be convenient to define the extent of a member a of A to be [ a = a ] and a singleton to be a map s : A Omega satisfying s(a) a = b ] s(b) and s(a) s(b) a = b ] There is a natural induced notion of weak equality, or equivalence [ a j b ] defined by (Ea Eb) a = ....

Van Dalen, D. [1986] "Intuitionistic Logic", in The Handbook of Philosophical Logic, vol.III , D. Reidel, Dordrecht.

....:a immediately we must see if we can entail either a or :a before evaluating the truth of the statement. Frequently, it may be the case that neither disjunct can be entailed from the current state of the constraint 5 [28] p124. 6 For an excellent introduction to intuitionistic logic, see [37]. network, and so we are left in a middle, undecided state. The rule for operational assertion is much more interesting: we need to create a set of worlds, one for each of the disjunct, to execute independently of the others. This is a fairly expensive operation to perform, which is why ....

D. van Dalen. Intuitionistic logic. In Logic and Structure (second edition), chapter 5. Springer-Verlag, ?

....9xA iff for some object b in s, s j= A(xjb) then what is it for s j= 9xA to be true This is quite a tricky question, and it s one in which situation theorists don t agree [3] We will come back to it later. 2. Intuitionistic Logic. Consider the Kripke style semantics for intuitionistic logic [12, 18, 20, 39]. Under the received interpretation, you have a set of points, each of which represents the state of knowledge of a mathematical reasoner. These points are ordered under the relationship of possible extension. A point is a possible extension of another point if you can get from the first to the ....

.... regions (for a technical reason we ll come to later) so the domain L(r) of the function 7 For examples of formal semantic structures in which disjunction is not treated with the usual clause, consider the Beth semantics for intuitionistic logic, and the Fine semantics for relevant logics [17, 12, 39]. r (which we might call the region s location) will be a subset of Z Theta Z. Then a region r is a part of the world w iff r w. We can graphically represent regions by picking out the coordinates of one point of the region, as follows. h4; 5i Xi Xi Xi Xi Xi Xi Xi Xi The coordinate ....

Dirk van Dalen. Intuitionistic Logic. In D. Gabbay and D. Guenther, editors, Handbook of Philosophical Logic, volume III. D. Reidel, 1986.

....8v w:9u:vRu and u j= A. Computational Types from a Logical Perspective 15 We also require that the two relations are hereditary ffl If w j= p and w v then v j= p, ffl If w j= A and wRv then v j= A. Theorem 16 A iff 8w:w j= A. Proof By standard Henkin constructions (see, for example, Van Dalen (1986)) It is an interesting question to ask how these Kripke models are related to the categorical models of the previous section. Indeed there appears to be more than one way to approach the question. One approach is given by Alechina et al. 1997) who demonstrate a natural method of finding a ....

Van Dalen, D. (1986). Intuitionistic Logic. Chap. 4, pages 225--339 of: Gabbay, D., & Guenthner, F. (eds), Handbook of philosophical logic. Volume 3.

....meaning of the connectives and quantifiers not in terms of classical truth conditions but in terms of constructions acting on proofs. There are many variations on this explanation, which, when made sufficiently precise, lead to logical calculi equivalent to Heyting s (see any standard text, e.g. [3]) Intuitionistic logic can readily be presented in any of the usual deductive styles, e.g. as a tableau system, a natural deduction system or as a Gentzenstyle sequent calculus. For example, the sequent calculus for H restricts sequents on the right to single formulas; the natural deduction ....

....ff) ff :ff. N is a conservative extension of H in the sense that any formula without strong negation is a theorem of N if and only if it is a theorem of H. Notice that Nelson s negation is aptly termed strong , since in N, is a theorem, for all (not only atomic ) See e.g. [15, 3]) The derivability relation for N is denoted by N . A semantics for N can be obtained by a straightforward generalisation of the Kripke semantics for H discussed earlier. One may take the same Kripke frames as for intuitionistic logic; what changes is the nature of the assignments or ....

van Dalen, D, Intuitionistic Logic, in D Gabbay & F Guenthner (eds), Handbook of Philosophical Logic, Vol II, Kluwer, Dordrecht, 1986.

.... various approaches to the universal quantifier in the following table, where INT is intuitionistic predicate logic, H is Gurevich s intuitionistic logic with strong negation [18] and CF is Thomason s firstorder logic [26] For a unifying exposition of both Kripke and Beth models, see van Dalen [10]. Logics Quantifiers 8x Models Domains INT , H dynamic Kripke models expanding INT static Beth models constant CF static Kripke models constant CF 0 static Kripke models expanding In the following, we shall first introduce the logical system CF 0 , and then prove its soundness and ....

Van Dalen, D. "Intuitionistic logic", pp. 225-339 in Handbook of Philosophical Logic, Vol. III: Alternatives in Classical Logic, edited by D. Gabbay, and F. Guenthner, D. Reidel, Dordrecht, 1985.

....variables among elements of X . Here the terms are freely generated by the grammar terms(X) x(2 X) j j j t t 0 j t t 0 j t t 0 ; and the equivalence relation is provability in intuitionistic propositional logic, that is, t and t 0 are equivalent if i t t 0 . We refer to [6] for a possible axiomatisation. We use the usual abbreviations :t for t and t t 0 for (t t 0 ) t 0 t) the latter we already used above. The free Heyting algebra on a set of generators X is denoted HA[X ] In case X is a nite set (a tuple x) we write as well HA[ x] We ....

....set of generators) As is well known, this holds more generally for all nitely presented Heyting algebras. The usual proof uses the completeness theorem of intuitionistic propositional logic with respect to Kripke models, and then one has to show that nite Kripke models su ce (see for example [6]) We will give a simple algebraic proof in the next section. 2 Finite Heyting algebras and the nite model property In this section our xed data is a representation : HA[ x] H of some nitely presented Heyting algebra H . Associated we have the sequence of nite distributive lattices E n ....

D. van Dalen. Intuitionistic logic. In: Handbook of Philosophical Logic. Vol. III: Alternatives to Classical Logic. D. Gabbay and F. Guenther (eds.). D. Reidel Publishing Company, Dordrecht 1985.

....thus guaranteeing that any two epistemic states that agree on their objective sentences also agree on all other sentences. The constructive interpretation of existential quantifiers within belief has an intuitionistic flavor. However, the differences between our logic and intuitionistic logic [24] are considerable, since in our case disjunction and negation retain their classical interpretation. 3.1 Syntax and Semantics The language of KL Gamma is the same as for KL except that we add a new primitive called f term. An f term is a term other than a variable or a standard name followed ....

van Dalen, D., Intuitionistic Logic, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. III, D. Reidel, 1986, pp. 225--339.

....as an extension of intuitionistic logic, H . So terms and formulas are built up in the usual manner, using the logical constants of H : and the additional negation . Intuitionistic negation is actually definable in N by : The axioms and rules of N are those of H (see e.g. [5]) together with the following axiom schemata involving strong negation (where ff fi abbreviates (ff fi) fi ff) 1 N1. ff fi) ff fi N2. fffi) ff fi N3. fffi) ff fi N4. ff ff. N5. ff ff N6. for atomic ff) ff :ff N is a conservative extension of H in the ....

....atomic ff) ff :ff N is a conservative extension of H in the sense that any formula without strong negation is a theorem of N if and only if it is a theorem of H . Notice that Nelson s negation is termed strong , since in N , is a theorem, for all (not only atomic ) See e.g. [12, 5]) The derivability relation for N is denoted by N , that for H by H . Gentzen style sequent systems for N can be found in [4, 12] cf. also [29, 30] A tableau proof system for N is discussed in [27] 1 For present purposes we can restrict attention to propositional logic 2.2 Kripke Models ....

[Article contains additional citation context not shown here]

van Dalen, D, Intuitionistic Logic, in D Gabbay, & F Guenthner (eds), Handbook of Philosophical Logic, Vol. III, Kluwer, Dordrecht, 1986.

No context found.

Dirk van Dalen. Intuitionistic Logic. In D. Gabbay and F. Gunthner, editors, Handbook of Philosophical Logic; Vol. III: Alternatives in Classical Logic, pages 225--339. D. Reidel, 1986.

No context found.

D. van Dalen. Intuitionistic logic. In Handbook of Philosophical Logic, vol. III, chap. 4, pp. 225#339. Reidel, 1986.

No context found.

van Dalen, D, Intuitionistic Logic, in D Gabbay & F Guenthner (eds), Handbook of Philosophical Logic, Vol II, Kluwer, Dordrecht, 1986.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC