Saul A. Kripke. A completeness theorem in modal logic. The journal of symbolic logic, vol. 24 no. 1 (1959), pp. 1–14.

1966 ◽  
Vol 31 (2) ◽  
pp. 276-277 ◽  
Author(s):  
Arnould Bayart
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Symbolic Logic

scholarly journals Interpretability over peano arithmetic

1999 ◽  
Vol 64 (4) ◽  
pp. 1407-1425
Author(s):  
Claes Strannegård
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Peano Arithmetic ◽  
Embedding Theorem ◽  
Compactness Theorem ◽  
Partial Answer ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

On finite and infinite modal systems

1938 ◽  
Vol 3 (2) ◽  
pp. 77-82 ◽  
Author(s):  
C. West Churchman
Keyword(s):  
Modal Logic ◽  
Symbolic Logic ◽  
Modal Operator ◽  
Image Position ◽  
Modal Systems ◽  
Complex Modal

In Oskar Becker's Zur Logik der Modalitäten four systems of modal logic are considered. Two of these are mentioned in Appendix II of Lewis and Langford's Symbolic logic. The first system is based on A1–8 plus the postulate,From A7: ∼◊p⊰∼p we can prove the converse of C11 by writing ∼◊p for p, and hence deriveThe addition of this postulate to A1–8, as Becker points out, allows us to “reduce” all complex modal functions to six, and these six are precisely those which Lewis mentions in his postulates and theorems: p, ∼p, ◊p, ∼◊p, ∼◊∼p, and ◊∼p This reduction is accomplished by showingwhere ◊n means that the modal operator ◊ is repeated n times; e.g., ◊3p = ◊◊◊p. Then it is shown thatBy means of (1), (2), and (3) any complex modal function whatsoever may be reduced to one of the six “simple” modals mentioned above.It might be asked whether this reduction could be carried out still further, i.e., whether two of the six “irreducible” modals could not be equated. But such a reduction would have to be based on the fact that ◊p = p which is inconsistent with the set B1–9 of Lewis and Langford's Symbolic logic and independent of the set A1–8. Hence for neither set would such a reduction be possible.

scholarly journals LOGIC OF RELATIVITY

2021 ◽  
pp. 42-52
Author(s):  
Ihor Ohirko ◽  
Zinovii Partyko
Keyword(s):  
Modal Logic ◽  
Symbolic Logic ◽  
Logical Theory ◽  
Relative Truth ◽  
Time Space ◽  
Different Types ◽  
Logical Method ◽  
Standard Values ◽  
The Difference ◽  
Modal Values

The problem of the truth of statements is considered. This study had the goal to develop a logical theory that would allow considering the context (the paradigm) from which would depend on the truth of the statement. For the development of such a theory, called the logic of relativity, the following methods of research are used as abstraction, analysis (traditional), synthesis, deduction, formalisation, axiomatisation, logical method. In order to develop the logic of relativity, it is expedient to use the achievements in the area of situational logic. Under the situation, it is proposed to understand two circumstances (time and space) and a condition that creates a context (paradigm) statement. Specifies the modal values that these three parameters can acquire and examines different types of situations. In order to write statements in the logic of relativity, a form of the statement of statements is proposed in the language of extended symbolic logic. For the theory of the logic of relativity, a set of four axioms is proposed and a series of laws. In particular, it is indicated that the values of the assertions in the logic of relativity are the following five estimates: truth, relative truth, relative is absurd, unclear, uncertain. Some theorems of the logic of relativity are proposed. A number of examples of texts in the natural language are given to interpret the statements of the logic of relativity. It is indicated that the proposed apparatus of the logic of relativity should be regarded as a kind of modal logic. The difference in the logic of relativity from situational logic is that it considers the factor of movement (motion) of statements in time, space and environment conditions, which was not considered by situational logic. The logic of relativity should be used wherever it is necessary to take into account the possibility of moving allegations regarding time, space and environment of conditions. One of the most important conclusions of the study is that in the logic to the standard values of truth (true, probably true, false, uncertain), it is expedient to add another value: relatively true (and accordingly: relatively false).

scholarly journals EVERYONE KNOWS THAT SOMEONE KNOWS: QUANTIFIERS OVER EPISTEMIC AGENTS

2019 ◽  
Vol 12 (2) ◽  
pp. 255-270 ◽  
Author(s):  
PAVEL NAUMOV ◽  
JIA TAO
Keyword(s):  
Modal Logic ◽  
Epistemic Logic ◽  
Possible Worlds ◽  
Completeness Theorem ◽  
Kripke Semantics ◽  
Possible Worlds Semantics ◽  
Distributed Knowledge ◽  
First Order ◽  
Soundness And Completeness ◽  
First Order Modal Logic

AbstractModal logic S5 is commonly viewed as an epistemic logic that captures the most basic properties of knowledge. Kripke proved a completeness theorem for the first-order modal logic S5 with respect to a possible worlds semantics. A multiagent version of the propositional S5 as well as a version of the propositional S5 that describes properties of distributed knowledge in multiagent systems has also been previously studied. This article proposes a version of S5-like epistemic logic of distributed knowledge with quantifiers ranging over the set of agents, and proves its soundness and completeness with respect to a Kripke semantics.

A completeness theorem in modal logic

1959 ◽  
Vol 24 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Saul A. Kripke
Keyword(s):  
Modal Logic ◽  
Completeness Theorem ◽  
Propositional Variable ◽  
First Order ◽  
Individual Variables ◽  
Predicate Variable

The present paper attempts to state and prove a completeness theorem for the system S5 of [1], supplemented by first-order quantifiers and the sign of equality. We assume that we possess a denumerably infinite list of individual variables a, b, c, …, x, y, z, …, xm, ym, zm, … as well as a denumerably infinite list of n-adic predicate variables Pn, Qn, Rn, …, Pmn, Qmn, Rmn,…; if n=0, an n-adic predicate variable is often called a “propositional variable.” A formula Pn(x1, …,xn) is an n-adic prime formula; often the superscript will be omitted if such an omission does not sacrifice clarity.

scholarly journals ERRATUM

2008 ◽  
Vol 1 (3) ◽  
pp. 393-393
Keyword(s):  
Modal Logic ◽  
Production Process ◽  
Symbolic Logic ◽  
First Order ◽  
Topological Interpretation ◽  
Final Equation ◽  
First Order Modal Logic

Steve Awodey and Kohei Kishida (2008). Topology and Modality: The Topological Interpretation of First-Order Modal Logic. The Review of Symbolic Logic 1(2): 146-166.On page 148 of this article an error was introduced during the production process. The final equation in the displayed formula 8 lines from the bottom of the page should read,[0, 1) ≠ [0, 1]The publisher regrets this error.

S. K. Thomason. Noncompactness in propositional modal logic. The journal of symbolic logic, vol. 37 no. 4 (for 1972, pub. 1973), pp. 716–720. - Kit Fine. An incomplete logic containing S4. Theoria, vol. 40 (1974), pp. 23–29. - S. K. Thomason. An incompleteness theorem in modal logic. Theoria, vol. 40 (1974), pp. 30–34. - Martin Gerson. The inadequacy of the neighbourhood semantics for modal logic. The journal of symbolic logic, vol. 40 (1975), pp. 141–148. - Martin Sebastian Gerson. An extension of S4 complete for the neighbourhood semantics but incomplete for the relational semantics. Studio logica, vol. 34 (1975), pp. 333–342. - Martin Gerson. A neighbourhood frame for T with no equivalent relational frame. Zeitschrift für mathematische Logik und Grundlugen der Mathematik, vol. 22 (1976), pp. 29–34. - V. B. Šehtman. On incomplete propositional logics. Soviet mathematics, vol. 18 (1977), pp. 985–989. (English translation by B. F. Wells of O népolnyh logikah vyskazyvanij, Doklady Akadémii Nauk SSSR, vol. 235 (1977), pp. 542–545.) - J. F. A. K. van Benthem. Two simple incomplete modal logics. Theoria, vol. 44 (1978), pp. 25–37. - J. F. A. K. van Benthem and W. J. Blok. Transitivity follows from Dummett's axiom. Theoria, vol. 44 (1978), pp. 117–118. - J. F. A. K. van Benthem. Syntactic aspects of modal incompleteness theorems. Theoria, vol. 44 (1978), vol. 45 (1979). pp. 63–77.

1983 ◽  
Vol 48 (2) ◽  
pp. 488-495 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Modal Logic ◽  
English Translation ◽  
Nauk Sssr ◽  
Symbolic Logic ◽  
Relational Semantics ◽  
Akademii Nauk Sssr ◽  
Doklady Akademii Nauk Sssr ◽  
Propositional Modal Logic ◽  
Relational Frame ◽  
Incompleteness Theorems

Hegel’s Subjective Logic as a Logic for (Hegel’s) Philosophy of Mind

2016 ◽  
Vol 39 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Paul Redding
Keyword(s):  
Modal Logic ◽  
Philosophy Of Mind ◽  
Continental Philosophy ◽  
Symbolic Logic ◽  
Arthur Prior ◽  
Subjective Logic ◽  
Science Of Logic ◽  
The 1950S ◽  
Josiah Royce

AbstractIn the 1930s, C. I. Lewis, who was responsible for the revival of modal logic in the era of modern symbolic logic, characterized ‘intensional’ approaches to logic as typical of post-Leibnizian ‘continental philosophy’, in contrast to the ‘extensionalist’ approaches dominant in the British tradition. Indeed Lewis’s own work in this area had been inspired by the logic of his teacher, the American ‘Absolute Idealist’, Josiah Royce. Hegel’s ‘Subjective Logic’ in Book III of hisScience of Logic, can, I suggest, be considered as an intensional modal logic, and this paper explores parallels between it and a later variety of modal logic—tenselogic, as developed by Arthur Prior in the 1950s and 60s. Like Lewis, Prior too had been influenced in this area by a teacher with strong Hegelian leanings—John N. Findlay. Treated as anintensional(with an ‘s’) logic, Hegel’s subjective logic can be used as a framework for addressing issues ofintentionality(with a ‘t’)—the mind’s capacity to be intentionally directed to objects. In this way, I suggest that the structures of his subjective logic can clarify what is at issue in the ‘Psychology’ section of theEncyclopaediaPhilosophy of Subjective Spirit.

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