Richard Montague. Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability. Proceedings of a Colloquium on Modal and Many-valued Logics, Helsinki, 23-26 August, 1962, Acta philosophica Fennica, no. 16, Helsinki 1963, pp. 153–167.

1975 ◽  
Vol 40 (4) ◽  
pp. 600-601
Author(s):  
Perry Smith
Keyword(s):  
Finite Axiomatizability ◽  
Richard Montague

Formal Philosophy: Selected Papers of Richard Montague

1975 ◽  
Vol 72 (7) ◽  
pp. 196-203 ◽  
Author(s):  
Terence Parsons ◽  
Keyword(s):  
Formal Philosophy ◽  
Richard Montague

scholarly journals On the Decidability of Intuitionistic Tense Logic without Disjunction

2020 ◽  
Author(s):  
Fei Liang ◽  
Zhe Lin
Keyword(s):  
Proof Theory ◽  
Tense Logic ◽  
Heyting Algebras ◽  
Finite Model ◽  
Algebraic Proof ◽  
Model Property ◽  
Algebraic Models ◽  
Finite Axiomatizability ◽  
Finite Model Property ◽  
Modal Operators

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

On the finite axiomatizability of ∀Σ̂1b(R̂21)

2018 ◽  
Vol 64 (1-2) ◽  
pp. 6-24
Author(s):  
Chris Pollett
Keyword(s):  
Finite Axiomatizability

Modal logic, philosophical issues in

2018 ◽  
Author(s):  
Thomas J. McKay
Keyword(s):  
Modal Logic ◽  
Natural Language ◽  
Possible Worlds ◽  
Possible World ◽  
Modal Claim ◽  
Knowing That ◽  
Modal Semantics ◽  
Bull's Eye ◽  
Modal Systems ◽  
Richard Montague

In reasoning we often use words such as ‘necessarily’, ‘possibly’, ‘can’, ‘could’, ‘must’ and so on. For example, if we know that an argument is valid, then we know that it is necessarily true that if the premises are true, then the conclusion is true. Modal logic starts with such modal words and the inferences involving them. The exploration of these inferences has led to a variety of formal systems, and their interpretation is now most often built on the concept of a possible world. Standard non-modal logic shows us how to understand logical words such as ‘not’, ‘and’ and ‘or’, which are truth-functional. The modal concepts are not truth-functional: knowing that p is true (and what ‘necessarily’ means) does not automatically enable one to determine whether ‘Necessarily p’ is true. (‘It is necessary that all people have been people’ is true, but ‘It is necessary that no English monarch was born in Montana’ is false, even though the simpler constituents – ‘All people have been people’ and ‘No English monarch was born in Montana’– are both true.) The study of modal logic has helped in the understanding of many other contexts for sentences that are not truth-functional, such as ‘ought’ (‘It ought to be the case that p’) and ‘believes’ (‘Alice believes that p’); and also in the consideration of the interaction between quantifiers and non-truth-functional contexts. In fact, much work in modern semantics has benefited from the extension of modal semantics introduced by Richard Montague in beginning the development of a systematic semantics for natural language. The framework of possible worlds developed for modal logic has been fruitful in the analysis of many concepts. For example, by introducing the concept of relative possibility, Kripke showed how to model a variety of modal systems: a proposition is necessarily true at a possible world w if and only if it is true at every world that is possible relative to w. To achieve a better analysis of statements of ability, Mark Brown adapted the framework by modelling actions with sets of possible outcomes. John has the ability to hit the bull’s-eye reliably if there is some action of John’s such that every possible outcome of that action includes John’s hitting the bull’s-eye. Modal logic and its semantics also raise many puzzles. What makes a modal claim true? How do we tell what is possible and what is necessary? Are there any possible things that do not exist (and what could that mean anyway)? Does the use of modal logic involve a commitment to essentialism? How can an individual exist in many different possible worlds?

Type-logical semantics

2018 ◽  
Author(s):  
Reinhard Muskens
Keyword(s):  
Type Theory ◽  
Predicate Logic ◽  
Logical Form ◽  
Ordinary Language ◽  
Linguistic Meaning ◽  
Lexical Meaning ◽  
Logical Language ◽  
Competing Hypotheses ◽  
Richard Montague ◽  
Logical Semantics

Type-logical semantics studies linguistic meaning with the help of the theory of types. The latter originated with Russell as an answer to the paradoxes, but has the additional virtue that it is very close to ordinary language. In fact, type theory is so much more similar to language than predicate logic is, that adopting it as a vehicle of representation can overcome the mismatches between grammatical form and predicate logical form that were observed by Frege and Russell. The grammatical forms of ordinary language sentences consequently may be taken to be much less misleading than logicians in the first half of the twentieth century often thought them to be. This was realized by Richard Montague, who used the theory of types to translate fragments of ordinary language into a logical language. Semantics is commonly divided into lexical semantics, which studies the meaning of words, and compositional semantics, which studies the way in which complex phrases obtain a meaning from their constituents. The strength of type-logical semantics lies with the latter, but type-logical theories can be combined with many competing hypotheses about lexical meaning, provided these hypotheses are expressed using the language of type theory.

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