scholarly journals SEMANTIC POLLUTION AND SYNTACTIC PURITY

2015 ◽  
Vol 8 (4) ◽  
pp. 649-661 ◽  
Author(s):  
STEPHEN READ
Keyword(s):  
Modal Logic ◽  
Formal System ◽  
Logical Constants ◽  
Deductive Systems ◽  
Labelled Deductive Systems

AbstractLogical inferentialism claims that the meaning of the logical constants should be given, not model-theoretically, but by the rules of inference of a suitable calculus. It has been claimed that certain proof-theoretical systems, most particularly, labelled deductive systems for modal logic, are unsuitable, on the grounds that they are semantically polluted and suffer from an untoward intrusion of semantics into syntax. The charge is shown to be mistaken. It is argued on inferentialist grounds that labelled deductive systems are as syntactically pure as any formal system in which the rules define the meanings of the logical constants.

Modality and folk psychology

2019 ◽  
Vol 28 (1) ◽  
pp. 19-27
Author(s):  
Ja. O. Petik
Keyword(s):  
Modal Logic ◽  
Brain Activity ◽  
Formal System ◽  
Philosophy Of Logic ◽  
Psychological States ◽  
Formal Systems ◽  
Modern Psychology ◽  
Philosophical Interpretation ◽  
Important Direction

The connection of the modern psychology and formal systems remains an important direction of research. This paper is centered on philosophical problems surrounding relations between mental and logic. Main attention is given to philosophy of logic but certain ideas are introduced that can be incorporated into the practical philosophical logic. The definition and properties of basic modal logic and descending ones which are used in study of mental activity are in view. The defining role of philosophical interpretation of modality for the particular formal system used for research in the field of psychological states of agents is postulated. Different semantics of modal logic are studied. The hypothesis about the connection of research in cognitive psychology (semantics of brain activity) and formal systems connected to research of psychological states is stated.

scholarly journals On a Formalism Which Makes any Sequence of Symbols Well-Formed

1968 ◽  
Vol 32 ◽  
pp. 1-4
Author(s):  
Shigeo Ōhama
Keyword(s):  
Formal System ◽  
Finite Sequence ◽  
Universal Quantifier ◽  
Logical Constants ◽  
System A

Any finite sequence of primitive symbols is not always well-formed in the usual formalisms. But in a certain formal system, we can normalize any sequence of symbols uniquely so that it becomes well-formed. An example of this kind has been introduced by Ono [2]. While we were drawing up a practical programming along Ono’s line, we attained another system, a modification of his system. The purpose of the present paper is to introduce this modified system and its application. In 1, we will describe a method of normalizing sentences in LO having only two logical constants, implication and universal quantifier, so that any finite sequence of symbols becomes well-formed. In 2, we will show an application of 1 to proof. I wish to express my appreciation to Prof. K. Ono for his significant suggestions and advices.

scholarly journals First-order modal logic in the necessary framework of objects

2016 ◽  
Vol 46 (4-5) ◽  
pp. 584-609 ◽  
Author(s):  
Peter Fritz
Keyword(s):  
Modal Logic ◽  
Model Theory ◽  
Consequence Relations ◽  
Possible World ◽  
Logical Constants ◽  
First Order ◽  
Timothy Williamson ◽  
Infinite Sets ◽  
Infinite Set ◽  
First Order Modal Logic

AbstractI consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson's argument.

Situation-Theoretic Account of Valid Reasoning with Venn Diagrams

1996 ◽  
Author(s):  
Sun-Joo Shin
Keyword(s):  
Semantic Analysis ◽  
Formal System ◽  
Venn Diagram ◽  
Formal Proofs ◽  
Representation System ◽  
Venn Diagrams ◽  
Linguistic Representation ◽  
Deductive Systems ◽  
Representation Systems ◽  
Mathematical Tradition

Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. Since elementary school we have been taught how to draw Venn diagrams for a problem, how to manipulate them, how to interpret the resulting diagrams, and so on. However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs. This is just a reflection of a general prejudice against visualization which resides in the mathematical tradition. With this bias for linguistic representation systems, little attempt has been made to analyze any nonlinguistic representation system despite the fact that many forms of visualization are used to help our reasoning. The purpose of this chapter is to give a semantic analysis for a visual representation system—the Venn diagram representation system. We were mainly motivated to undertake this project by the discussion of multiple forms of representation presented in Chapter I More specifically, we will clarify the following passage in that chapter, by presenting Venn diagrams as a formal system of representations equipped with its own syntax and semantics:. . . As the preceding demonstration illustrated, Venn diagrams provide us with a formalism that consists of a standardized system of representations, together with rules of manipulating them. . . . We think it should be possible to give an informationtheoretic analysis of this system, . . . . In the following, the formal system of Venn diagrams is named VENN. The analysis of VENN will lead to interesting issues which have their ana logues in other deductive systems. An interesting point is that VENN, whose primitive objects are diagrammatic, not linguistic, casts these issues in a different light from linguistic representation systems. Accordingly, this VENN system helps us to realize what we take for granted in other more familiar deductive systems. Through comparison with symbolic logic, we hope the presentation of VENN contributes some support to the idea that valid reasoning should be thought of in terms of manipulation of information, not just in terms of manipulation of linguistic symbols.

scholarly journals Proving infinitary formulas

2016 ◽  
Vol 16 (5-6) ◽  
pp. 787-799 ◽  
Author(s):  
AMELIA HARRISON ◽  
VLADIMIR LIFSCHITZ ◽  
JULIAN MICHAEL
Keyword(s):  
Propositional Logic ◽  
Formal System ◽  
Answer Set Programming ◽  
Logic Programs ◽  
Deductive Systems ◽  
Answer Set

AbstractThe infinitary propositional logic of here-and-there is important for the theory of answer set programming in view of its relation to strongly equivalent transformations of logic programs. We know a formal system axiomatizing this logic exists, but a proof in that system may include infinitely many formulas. In this note we describe a relationship between the validity of infinitary formulas in the logic of here-and-there and the provability of formulas in some finite deductive systems. This relationship allows us to use finite proofs to justify the validity of infinitary formulas.

Provability logic

2018 ◽  
Author(s):  
Albert Visser
Keyword(s):  
Modal Logic ◽  
Formal System ◽  
Formal Theory ◽  
Incompleteness Theorem ◽  
Provability Logic ◽  
Modal System ◽  
Modal Language ◽  
Modal Operators ◽  
Syntactic Approach

Central to Gödel’s second incompleteness theorem is his discovery that, in a sense, a formal system can talk about itself. Provability logic is a branch of modal logic specifically directed at exploring this phenomenon. Consider a sufficiently rich formal theory T. By Gödel’s methods we can construct a predicate in the language of T representing the predicate ‘is formally provable in T’. It turns out that T is able to prove statements of the form - (1) If A is provable in T, then it is provable in T that A is provable in T. In modal logic, predicates such as ‘it is unavoidable that’ or ‘I know that’ are considered as modal operators, that is, as non-truth-functional propositional connectives. In provability logic, ‘is provable in T’ is similarly treated. We write □A for ‘A is provable in T’. This enables us to rephrase (1) as follows: - (1′) □A →□□A. This is a well-known modal principle amenable to study by the methods of modal logic. Provability logic produces manageable systems of modal logic precisely describing all modal principles for □A that T itself can prove. The language of the modal system will be different from the language of the system T under study. Thus the provability logic of T (that is, the insights T has about its own provability predicate as far as visible in the modal language) is decidable and can be studied by finitistic methods. T, in contrast, is highly undecidable. The advantages of provability logic are: (1) it yields a very perspicuous representation of certain arguments in a formal theory T about provability in T; (2) it gives us a great deal of control of the principles for provability in so far as these can be formulated in the modal language at all; (3) it gives us a direct way to compare notions such as knowledge with the notion of formal provability; and (4) it is a fully worked-out syntactic approach to necessity in the sense of Quine.

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