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By
Malinowski, Grzegorz; Michalczyk, Marek
Proofs of two interpolation theorems for Sentential Calculus with Identity as well as some general comments on sentential interpolation are given.
By
Smoryński, Craig
7 Citations
In a recent paper, Montagna proved the undecidability of the firstorder theory of diagonalisable algebras. This result is here refined — the set of finitely refutable sentences is shown effectively inseparable from the set of theorems. The proof is quite simple.
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By
Ouellet, Roch
3 Citations
It is shown that a locally finite polyadic algebra on an infinite set V of variables is a Booleanalgebra object, endowed with some internal supremum morphism, in the category of locally finite transformation sets on V. Then, this new categorical definition of polyadic algebras is used to simplify the theory of these algebras. Two examples are given: the construction of dilatations and the definition of terms and constants.
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By
Wolniewicz, Bogusław
1 Citations
A generalized Wittgensteinian semantics for propositional languages is presented, based on a lattice of elementary situations. Of these, maximal ones are possible worlds, constituting a logical space; minimal ones are logical atoms, partitioned into its dimensions. A verifier of a proposition α is an elementary situation such that if real it makes α true. The reference (or objective) of a proposition is a situation, which is the set of all its minimal verifiers. (Maximal ones constitute its locus.) Situations are shown to form a Boolean algebra, and the Boolean set algebra of loci is its representation. Wittgenstein's is a special case, admitting binary dimensions only.
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By
Czelakowski, Janusz; Dziobiak, Wiesław
2 Citations
Let q(K) denote the least quasivariety containing a given class K of algebraic structures. Mal'cev [3] has proved that q(K) = ISP_{r}(K)^{(1)}. Another description of q(K) is given in Grätzer and Lakser [2], that is, q(K) = ISPP_{u}(K)^{2}. We give here other proofs of these results. The method which enables us to do that is borrowed from prepositional logics (cf. [1]).
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By
Sorbi, Andrea
In this paper we study the reducibility order ⩽ _{m} (defined in a natural way) over ∑_{n}^{0}
equivalence relations. In particular, for every n> 0 we exhibit ∑
_{n}^{0}
equivalence relations which are complete with respect to ⩽_{m} and investigate some consequences of this fact (see Introduction).
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By
Bugajski, Sławomir
3 Citations
The paper describes in detail the procedure of identification of the inner language and an inner logico of a physical theory. The procedure is a generalization of the original ideas of J. von Neuman and G. Birkhoff about quantum logic.
By
Engdahl, Elisabet
1 Citations
Conclusion
The restrictive grammatical format suggested in GPSG provides an extremely interesting alternative to transformational approaches to grammar. However, we have seen that the way the grammar is currently organized, it will in certain cases fail to give the correct interpretation to sentences with displaced constituents. Whenever a left or rightward displaced constituent contains an element that can stand in an anaphoric relation with some other element in the sentence, i.e. contains a quantifier or a pronoun, the semantic rules as given do not provide the intended reading. It should be made clear, however, that this shortcoming pertains to one particular type of GPSG, namely one whose interpretation procedures make use of lambda abstraction. The problematic cases addressed here do not by themselves invalidate the syntactic framework of a GPSG. It seems quite plausible that one could construct a semantics that does not depend on lambda conversion, in which case the problem addressed in this note is presumably no longer relevant. One such proposal is put forward in Cooper (forthcoming) where a nonmovement syntax is combined with interpretation procedures that do not involve translation into a lambda calculus.
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By
Czelakowski, Janusz
23 Citations
The main result of the present paper — Theorem 3 — establishes the equivalence of the interpolation and amalgamation properties for a large family of logics and their associated classes of matrices.
By
Dziobiak, Wlesław
5 Citations
In classes of algebras such as lattices, groups, and rings, there are finite algebras which individually generate quasivarieties which are not finitely axiomatizable (see [2], [3], [8]). We show here that this kind of algebras also exist in Heyting algebras as well as in topological Boolean algebras. Moreover, we show that the lattice join of two finitely axiomatizable quasivarieties, each generated by a finite Heyting or topological Boolean algebra, respectively, need not be finitely axiomatizable. Finally, we solve problem 4 asked in Rautenberg [10].
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By
Fukmanowski, Tomasz
The aim of this paper is to give a general background and a uniform treatment of several notions of mutual interpretability. Sentential calculi are treated as preorders and logical invariants of adjoint situations, i.e. Galois connections are investigated. The class of all sentential calculi is treated as a quasiordered class.
Some methods of the axiomatization of the Mcounterparts of modal systems are based on particular adjoints. Also, invariants concerning adjoints for calculi with implication are pointed out. Finally, the notion of interpretability is generalized so that it may be applied to closure spaces as well.
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By
Larson, Richard K.
2 Citations
Conclusion
By accepting the above proposals for translating tenses it appears possible to achieve a very general account of the interpretation of Warlpiri adjoined clauses. Moreover, if the analysis is correct it would provide an interesting example of natural language generalizing across tenses and NPs, since what we would have is a single syntactic construction whose interpretation varied according to whether an NP or a tense were translated with a distinguished variable. These results thus serve to pose once again the question of where precisely the common features of tenses and NPs reside. Recent work applying modeltheoretic techniques to natural language semantics may well provide an answer. Thus in Dowty (1979) and Larson and Cooper (1980) NPs and tenses both denote the same sort of settheoretic object, viz., sets of sets. Within generalized quantification theory this is just to say that both NPs and tenses denote quantifiers (cf. Barwise and Cooper, 1981, for much illuminating information on quantifiers and natural language). It may thus be possible to view the interpretation of Warlpiri adjoined clauses as a case of natural language generalizing across the semantic type of quantifiers.
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By
Ades, Anthony E.; Steedman, Mark J.
112 Citations
Conclusions
There is no doubt that the model presented here is incomplete. Many important categories, particularly negation and the adverbials, have been entirely ignored, and the treatment of Tense and the affixes is certainly inadequate. It also remains to be seen how the many constructions that have been ignored here are to be accommodated within the framework that has been outlined. However, the fact that a standard categorial lexicon, plus the four rule schemata, seems to come close to exhaustively specifying the main clause constructions of English, and also seems to explain a number of major constraints on transformations, encourages us to compare the theory with certain alternatives, and to examine its broader implications.
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By
Pullum, Geoffrey K.; Gazdar, Gerald
78 Citations
Conclusions
Notice that this paper has not claimed that all natural languages are CFL's. What it has shown is that every published argument purporting to demonstrate the noncontextfreeness of some natural language is invalid, either formally or empirically or both.^{18} Whether noncontextfree characteristics can be found in the stringset of some natural language remains an open question, just as it was a quarter century ago.
Whether the question is ultimately answered in the negative or the affirmative, there will be interesting further questions to ask. If it turns out that natural languages are indeed always CFL's, it will be reasonable to ask whether this helps to explain why speakers apparently recognize so quickly whether a presented utterance corresponds to a grammatical sentence or not, and to associate structural and semantic details with it. It might also be reasonable to speculate about the explanation for the universally contextfree character of the languages used by humans, and to wonder whether evolutionary biological factors are implicated in some way (Sampson (1979) could be read in this light). And naturally, it will be reasonable to pursue the program put forward by Gazdar (1981, in press) to see to what extent CFLinducing grammatical devices can be exploited to yield insightful descriptions of natural languages that capture generalizations in revealing ways.
If a human language that is not a CFL is proved to exist, on the other hand, a different question will be raised: give the noncontextfree character of human languages in general, why has this property been so hard to demonstrate that it has taken over twentyfive years to bring it to light since the issue was first explicitly posed? If human languages do not have to be CFL's, why do so many (most?) of them come so close to having the property of contextfreeness? And, since the CFL's certainly constitute a very broad class of mathematically natural and computationally tractable languages, what property of human beings or their communicative or cognitive needs is it that has caused some linguistic communities to reach beyond the boundaries of this class in the course of evolving a linguistic system?
Either way, we shall be interested to see our initial question resolved, and further questions raised. One cautionary word should be said, however, about the implications (or lack of them) that the answer will have for grammatical studies. Chomsky has repeatedly stated that he does not see weak generative capacity as a theme of central importance in the theory of grammar, and we agree. It is very far from being the case that the recent resurgence of interest in exploring the potential of CFPSG or equivalent systems will, or should, be halted dead in its tracks by the discovery (if it is ever forthcoming) that some natural language is not a CFL. In the area of parsing, for instance, it seems possible that natural languages are not only parsed on the basis of constituent structure such as a CFPSG would assign, but are parsed as if they were finite state languages (see Langendoen (1975) and Church (1980) for discussion along these lines). That is, precisely those constructiontypes that figure in the various proofs that English is not an FSL appear to cause massive difficulty in the human processing system; the sentences crucial to the proofs are for the most part unprocessable unless they are extremely short (yet the arguments for English not being and FSL only go through if length is not an issue). This means that in practice properties of finite state grammars are still of great potential importance to linguistic theory despite the fact that they do not provide the framework for defining the total class of grammatical sentences. The same would almost certannly be true of CFPSG's if they were shown to be inadequate in a similar sense. It is highly unlikely that the advances made so far in far in phrase structure description could be nullified by a discovery about weak generative capacity. Moreover, there are known to be numerous ways in which the power of CFPSG's can be marginally enhanced to permit, for example, xx languages to be generated without allowing anything like the full class of recursively enumerable or even contextsensitive languages (see Hopcroft and Ullmann (1979), Chapter 14) for an introduction to this topic, noting especially Figure 14.7 on p. 393. The obvious thing to do if natural languages were ever shown not to be CFL's in the general case would be to start exploring such minimal enhancements of expressive power to determine exactly what natural languages call for in this regard and how it could be effectively but parsimoniously provided in a way that closely modelled human linguistic capacities.
In the meantime, it seems reasonable to assume that the natural languages are a proper subset of the infinitecardinality CFL's until such time as they are validly shown not to be.
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By
Nuessel, Frank H., Jr.; Iivonen, Antti; Radday, Y. T.; Glenn, Jerry; Willée, Gerd; Dubrocard, Michel; Braun, Stephan; Beauchemin, Normand; Lansdown, John
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No abstract available
By
Paijmans, J. J.; VerrijnStuart, A. A.
Conclusions
(1) Microcomputers may render the same services to the individual museums as mainframes and/or networks. When a sizable amount of data is entered in several museums they may be integrated in ‘loosely coupled systems’, that share data as and when required.
(2) The museum staff is motivated more readily by working with a small inhouse system that they may control themselves than when they have to work with a system, however extended the help and facilities it provides, that is outside their control.
(3) Data for an automated system may correspond in format to the traditional ones for manual documentation.
(4) The input format should be easily manipulable, both at programming level and from the user's point of view. This ‘user friendliness’ should provide an efficient use of manpower resources, e.g., by concentrating on a few selected fields, by copying fields with identical entries, etc.
(5) If the retrieval program incorporates three essential components—AND/OR/NOT operators, full field control and inverted file searching—a high precisionrecall ratio may be expected, even when relatively few fields are used.
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By
Chierchia, Gennaro
31 Citations
Conclusions
We started from the fact that type theory, in the way it was implemented in IL, makes it costly to deal with nominalization processes. We have also argued that the type hierarchy as such doesn't play any real role in a grammar; the classification it provides for different semantic objects is already contained, in some sense, in the categorial structure of the grammar itself. So, on the basis of a theory of properties (Cocchiarella's HST^{*}) we have tried to build a language (IL^{*}) whose syntax does not contain any explicit typing of expressions. Some of the consequences that this move brings about in the overall organization of the grammar can be summarized as follows:
(a)
it allows for a simple treatment of infinitives, gerunds, factives and, in general, all those phenomena which might be analyzed as cases of nominalization;
(b)
it provides a simpler and more constrained semantics than IL, since IL^{*} doesn't go beyond second order, and its nonmodal basis is axiomatizable;
(c)
it suggests that the role of logical form in a theory of grammar could be that of a family of theories of semantic objects;
(d)
it eliminates the extrinsic limitations of a type hierarchy on the choice of the system of syntactic categories.
I think that it is interesting to notice how having an explicit semantic framework helps to provide a sense in which it is legitimate to regard the syntax of a language as ‘autonomous’.
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By
Fodor, Janet Dean; Sag, Ivan A.
236 Citations
Conclusion
The formal semantics that we have proposed for definite and indefinite descriptions analyzes them both as variablebinding operators and as referring terms. It is the referential analysis which makes it possible to account for the facts outlined in Section 2, e.g. for the purely ‘instrumental’ role of the descriptive content; for the appearance of unusually wide scope readings relative to other quantifiers, higher predicates, and island boundaries; for the fact that the islandescaping readings are always equivalent to maximally wide scope quantifiers; and for the appearance of violations of the identity conditions on variables in deleted constituents. We would emphasize that this is not a random collection of observations. They cohere naturally with each other, and with facts about other phrases that are unambigously referential.
We conceded at the outset of this paper that the referential use of an indefinite noun phrase does not, by itself, motivate the postulation of a referential interpretation. Our argument has been that the behavior of indefinites in complex sentences cannot be economically described, and certainly cannot be explained, unless a referential interpretation is assumed. It could be accounted for in pragmatic terms only if the whole theory of scope relations and of conditions on deletion could be eliminated from the semantics and incorporated into a purely pragmatic theory. But this seems unlikely.
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By
Goldblatt, Robert
4 Citations
Hoare's Iteration Rule is a principle of reasoning that is used to derive correctness assertions about the effects of implementing a whilecommand. We show that the propositional modal logic of this type of command is axiomatised by Hoare's rule in conjunction with two additional axioms. The proof also establishes decidability of the logic. The paper concludes with a discussion of the relationship between the logic of “while” and Segerberg's axiomatisation of propositional dynamic logic.
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By
Cresswell, M. J.
1 Citations
Urn models were developed by Veikko Rantala to provide a nonstandard semantics for firstorder logic in which the domains, over which the quantifiers range, are allowed to vary. Rantala uses gametheoretical semantics in his presentation, and the present paper is a study of urn models from a more classical, truthconditional point of view. An axiomatic system for urn logic is set out and completeness is proved by the method of maximal consistent sets.
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By
Slater, B. H.
Fregean logic has difficulty with certain arguments in which there is crossreference between premises and conclusion. In this paper I describe a method of handling arguments of the troublesome kind: It involves replacing standard quantifiers with explicit existential statements, and turns standard logic into a free one. A validation procedure is provided for the logic.
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By
Segerberg, Krister
28 Citations
The formal language studied in this paper contains two categories of expressions, terms and formulas. Terms express events, formulas propositions. There are infinitely many atomic terms and complex terms are made up by Boolean operations. Where α and β are terms the atomic formulas have the form α=β (α is the same as β), Forb α (α is forbidden) and Perm α (α is permitted). The formulae are truth functional combinations of these. An algebraic and a model theoretic account of validity are given and an axiomatic system is provided for which they are characteristic.
The ‘closure principle’, that what is not forbidden is permitted is shown to hold at the level of outcomes but not at the level of events. In the two final sections some other operators are considered and a semantics in terms of action games.
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By
Bunder, M. W.
1 Citations
The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.
In this paper we look in detail at special cases of the Gentzen rule for ⊢ and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as well as not well known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E_{→},R_{→}, T, PW and PWI.
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By
Harris, J. H.
14 Citations
Intuitionists and classical logicians use in common a large number of the “logical” axioms, even though they supposedly “mean different things” by the logical connectives and quantifiers — conquans for short. But Wittgenstein says “The meaning of a word is its use in the language”. We prove that in a definite sense the intuitionistic axioms do indeed characterize the “logical” conquans, both for the intuitionist and the classical logician.
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By
Zalta, Edward N.
1 Citations
In this paper I propose a fundamental modification of standard type theory, produce a new kind of type theoretic language, and couch in this language a comprehensive theory of abstract individuals and abstract properties and relations of every type. I then suggest how to employ the theory to solve the four following philosophical problems: (A) the identification and ontological status of Frege's Senses; (B) the deviant behavior of terms in propositional attitude contexts; (C) the nonidentity of necessarily equivalent propositions, and (D) the “paradox” of analysis. We can roughly describe these solutions as follows: (A) the senses of English names and descriptions which denote individuals will be modelled as abstract individuals; the senses of English relation denoting expressions of a given type will be modelled as abstract relations of that type. (B) Inside de dicto attitude contexts, these English expressions denote (the abstract objects which serve as) their senses. (C) Relations and propositions will not be identified with their extensions, nor with functions or sets of any kind. They will be taken as primitive, and precise “being” and identity conditions will be proposed consistent with the view that necessarily equivalent relations and propositions may be distinct. (D) With the modelling described in (A), the expressions “being a brother” and “being a male sibling” may both denote the same property, though (the abstract properties which serve as) their senses may differ. Just as Frege predicts, “being a brother just is being a male sibling” is an informative identity statement because the terms flanking the identity sign have the same denotation, though distinct senses.
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By
Shaw, J. L.
3 Citations
The aim of this paper is to present the Nyāya concept of number in the light of contemporary philosophy and to show that the FregeRussell concept of number does not contradict the Nyāya concept of number but rather supplements it.
By
Oddie, Graham; Tichý, Pavel
2 Citations
The aim of this paper is to offer a rigorous explication of statements ascribing ability to agents and to develop the logic of such statements. A world is said to be feasible iff it is compatible with the actual pastandpresent. W is a Pworld iff W is feasible and P is true in W (where P is a proposition). P is a sufficient condition for Q iff every P world is a Q world. P is a necessary condition for Q iff Q is a sufficient condition forP. Each individual property S is shown to generate a rule for an agent X. X heeds S iff X makes all his future choices in accordance with S. (Note that X may heed S and yet fail to have it). S is a Pstrategy for X iff X's heeding S together with P is a necessary and sufficient condition for X to have S. (Pstrategies are thus rules which X is able to implement on the proviso P).Provisional opportunity: X has the opportunity to A provided P iff there is an S such that S is a Pstrategy for X and X's implementing S is a sufficient condition for X's doing A. P is etiologically complete iff for every event E which P reports P also reports an etiological ancestry of E, and P is true. Categorical opportunity: X has the opportunity to A iff there is a P such that P is etiologically complete and X has the opportunity to A provided P. For X to have the ability to A there must not only be an appropriate strategy, but X must have a command of that strategy. X steadfastly intends A iff X intends A at every future moment at which his doing A is not yet inevitable. X has a command of S w.r.t. A and P iff X's steadfastly intending A together with P is a sufficient condition for X to implement S. Provisional ability: X can A provided P iff there is an S such that S is a Pstrategy for X, X's implementing S is a sufficient condition for X's doing A, and X has a command of S w.r.t. A and P. Categorical ability: X can A iff there is a P such that P is etiologically complete and X can A provided P. X is free w.r.t. to A iff X can A and X can non A. X is free iff there is an A such that X is free w.r.t. A.
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By
Kalman, J. A.
1 Citations
In the first part of this paper we indicate how Meredith's condensed detachment may be used to give a new proof of Belnap's theorem that if every axiom x of a calculus S has the “twoproperty” that every variable which occurs in x occurs exactly twice in x, then every theorem of S is a substitution instance of a theorem of S which has the twoproperty. In the remainder of the paper we discuss the use of mechanical theoremprovers, based either on condensed detachment or on the resolution rule of J. A. Robinson, to investigate various calculi whose axioms all have the twoproperty. Particular attention is given to “Dgroupoids”, i.e. sets of formulae which are closed under condensed detachment.
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By
Liddell, G. F.
In the first part of this paper a logic is defined for propositions whose probability of being true may not be known. A speaker's beliefs about which propositions are true are still interesting in this case. The meaning of propositions is determined by the consequences of asserting them: in this logic there are debates which incur certain costs for the protagonists.
The second part of the paper describes the mathematics of the resulting logic which displays several novel features.
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By
Crossley, John N.
1 Citations
The paper presents a brief survey of recent work by Metakides, Nerode and others in the area of effective algebra and makes some comments on the relation between formal presentations, characterizations, etc. of sets and of algebraic structures and their practical presentations.
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By
Priest, Graham
12 Citations
The paper concerns time, change and contradiction, and is in three parts. The first is an analysis of the problem of the instant of change. It is argued that some changes are such that at the instant of change the system is in both the prior and the posterior state. In particular there are some changes from p being true to ℸp being true where a contradiction is realized. The second part of the paper specifies a formal logic which accommodates this possibility. It is a tense logic based on an underlying paraconsistent prepositional logic, the logic of paradox. (See the author's article of the same name Journal of Philosophical Logic 8 (1979).) Soundness and completeness are established, the latter by the canonical model construction, and extensions of the basic system briefly considered. The final part of the paper discusses Leibniz's principle of continuity: “Whatever holds up to the limit holds at the limit”. It argues that in the context of physical changes this is a very plausible principle. When it is built into the logic of the previous part, it allows a rigorous proof that change entails contradictions. Finally the relation of this to remarks on dialectics by Hegel and Engels is briefly discussed.
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By
Kobayashi, Mitsunori; Ishimoto, Arata
5 Citations
The propositional fragment L_{1} of Leśniewski's ontology is the smallest class (of formulas) containing besides all the instances of tautology the formulas of the forms: ɛ(a, b) ⊃ ɛ(a, a), ɛ(a, b) ∧ ɛ(b,).⊃ ɛ(a, c) and ɛ(a, b) ∧ ɛ(b, c). ⊃ ɛ(b, a) being closed under detachment. The purpose of this paper is to furnish another more constructive proof than that given earlier by one of us for:
Theorem A is provable in L
_{1}
iff TA is a thesis of firstorder predicate logic with equality, where T is a translation of the formulas of L
_{1}
into those of firstorder predicate logic with equality such that Tɛ(a, b) = F_{blx}F_{a}x (Russeltiantype definite description), TA ∨ B = TA ∨ TB, T ∼ A = ∼TA, etc.
For the proof of this theorem use is made of a tableau method based upon the following reduction rules:
$$\begin{gathered} \frac{{G\left[ {A \vee B} \right]}}{{G\left[ {A \vee B{\text{\_}}} \right] \vee \sim AG[A \vee B\_] \vee \sim B,}}{\text{ }}\frac{{G[\varepsilon (a,b)\_]}}{{G[\varepsilon (a,b)\_] \vee \sim \varepsilon (a,a),}} \hfill \\ \frac{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_]}}{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_], \vee \sim \varepsilon (a,c),}}{\text{ }}\frac{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_]}}{{G[\varepsilon (a,b)\_,\varepsilon (b,c)\_] \vee \sim \varepsilon (b,a),}} \hfill \\ \end{gathered} $$
where F[A_{+}] (G[A_{−}]) means that A occurs in F[A_{+}] (G[A_{−}]) as its positive (negative) part in accordance with the definition given by Schütte.
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By
Goebl, Hans; Selberherr, S.; Rase, W. D.; Pudlatz, H.
Show all (4)
The subject of this article is a brief survey of dialectometry, which is defined according to the following equation: linguistic geography+numerical taxonomy=dialectometry. First dealt with are problems concerning the processing of raw data, the coding, scaling, measurement and compilation of the data matrix (atlas points×atlas maps). After the description of the similarity coefficient used and the compilation of the similarity matrix it will be shown how the individual vectors of the similarity matrix can be visualized in the form of choropleth maps and threedimensional surfaces. These visualized vectors of the similarity matrix are described as similarity maps. After consideration of the usefulness of such similarity maps for genuine linguistic geography there is a discussion of other possible fields which will be opened up as a result of intensified dialectometrical research (e.g. graph theory, network theory, game theory). This article is supplemented by 9 figures, 2 tables and an extensive bibliography.
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By
Bloom, Stephen L.
A signed ∑equation is an expression of the form t ≈ t′ or t ≉t′, where t and t′ are ∑terms (for some ranked set ∑). We characterize those classes of ∑algebras which are models of a set of signed ∑equations. Further we consider the problem of finding a complete deductive system analogous to equational logic for the logical consequence operation restricted to signed equations.
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By
Dziobiak, Wiesław
The aim of this note is to show (Theorem 1.6) that in each of the cases: ψ= {→, ∨ }, or {→, ∨, ∧ }, or {→, ∨, ℸ } there are uncountably many ψintermediate logics which are not finitely approximable. This result together with the results known in literature allow us to conclude (Theorem 2.2) that for each ψ: either all ψintermediate logics are finitely approximate or there are uncountably many of them which lack the property.
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By
Rantala, Veikko
21 Citations
One way to obtain a comprehensive semantics for various systems of modal logic is to use a general notion of nonnormal world. In the present article, a general notion of modal system is considered together with a semantic framework provided by such a general notion of nonnormal world. Methodologically, the main purpose of this paper is to provide a logical framework for the study of various modalities, notably prepositional attitudes. Some specific systems are studied together with semantics using nonnormal worlds of different kinds.
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By
Heny, Frank
10 Citations
Summary
In Section 1, we questioned the evidence for iteration of tenses, even with abstraction. To permit abstraction would in any case risk neutralizing our distinction between tensed and untensed sentences. Sequence of tense phenomena, far from supporting iteration, were incompatible with it. Instead, we argued, tense always retains its full deictic character; tenses never have scope over each other. The future modal WILL is exceptional (Section 2), but abstraction is not required to deal with this.
An important suggestion, first made in Section 1 and developed further in Section 3, would restrict quantification induced by the interpretation of Past to a domain determined by pragmatic factors. When Past serves as a reference time, e.g. when it forms part of would, as in
(42)
Sam believed that he would be in Boston
(44)
James bought a stamp that would be priceless then a previous occurrence of Past serves to define the domain relevant to the quantificational element of Past. Without semantic binding or reference, the two cases of Past in such sentences will thus be satisfied by the same interval in the past.
The extension of this idea in Section 3, permits an account of the supposed incompleteness of the past tense without invoking a hidden THEN in e.g. I turned the lights off. In this case, restriction of the relevant domain does not proceed via an earlier occurrence of Past but depends solely on discourse factors.
In Section 4, the analysis of have given in Part I is modified to introduce an element of ‘pastness’. The relevant subinterval must be nonfinaland perhaps need not even be a subinterval of the evaluation interval itself. Empirically necessary, this change makes the incompatibility of Pres,have and a definite adverb like at 6 o'clock in a single sentence even harder to explain semantically. However, the vague restrictions on domain imposed by the present tense and relevant to the subinterval required by an embedded have below it seem clearly pragmatically incompatible with the explicit restriction of that same interval to a very specific interval in the past — a restriction which, we argue, is imposed by a definite time adverb.
Thus, Part II represents a consistent, though entirely informal plea for the embedding of the semantics of Part I in a framework in which pragmatic considerations can interact freely with the semantics, to restrict the domains within which quantification is permitted. It is in effect a plea for the introduction of partial models along the lines suggested by Barwise and Perry (1981). The advantages of the basic distinctions between tense, aspect and adverbs introduced in Part I would, we have suggested in the second part, be yet more obvious in such a framework. Formalization in a system of partial models must await further work.
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By
Kabziński, Jacek K.
2 Citations
In this paper we investigate some basic semantic and syntactic conditions characterizing the equivalence connective. In particular we define three basic classes of algebras: the class of weak equivalential algebras, the class of equivalential algebras and the class of regular equivalential algebras (see [12]).
Weak equivalential algebras can be used to study purely equivalential fragments of relevant logics and strict equivalential fragments of some modal logics (for investigations of strict implicational fragments of modal logics see [20]). Equivalential algebras are suitable to study purely equivalential fragment of BCI and BCK logic (see [21], p. 316). A subclass of the class of regular equivalential algebras is suitable to study equivalential fragments of Łukasiewicz logics. Some subvarieties of the class of regular equivalential algebras provide natural semantics for equivalential fragments of the intuitionistic prepositional logic and various intermediate logics (see [13]).
The last chapter is a selection of research problems which in the author's opinion are worth to be solved.
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By
Cignoli, Roberto
21 Citations
Proper nvalued Łukasiewicz algebras are obtained by adding some binary operators, fulfilling some simple equations, to the fundamental operations of nvalued Łukasiewicz algebras. They are the salgebras corresponding to an axiomatization of Łukasiewicz nvalued propositional calculus that is an extention of the intuitionistic calculus.
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By
Dowty, David R.
43 Citations
Summary
I might summarize this section by saying that the English tenses, according to this analysis, form quite a motley group. PAST, PRES and FUT serve to relate reference time to speech time, while WOULD and USEDTO behave like Priorian operators, shifting the point of evaluation away from the reference time. HAVE also shifts the point of evaluation away from the reference time, but in a more complicated way. And FUT, in contrast to PRES and PAST, is a substitution operator, putting the reference time of its clause in the plate of the speech time of subordinate clauses.
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