Alan Ross Anderson. Improved decision procedures for Lewis's calculus S4 and von Wright's calculus M. The journal of symbolic logic, vol. 19 (1954), pp. 201–214. - Alan Ross Anderson. Correction to a paper on modal logic. The journal of symbolic logic, vol. 20 (1955), p. 150. - Alan Ross Anderson. On alternative formulations of a modal system of Feysvon Wright. The journal of computing systems, vol. 1 no. 4 (1954), pp. 211–212.

1955 ◽  
Vol 20 (3) ◽  
pp. 302-303
Author(s):  
Naoto Yonemitsu
Keyword(s):  
Modal Logic ◽  
Decision Procedures ◽  
Symbolic Logic ◽  
Modal System ◽  
Computing Systems

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).

New foundations for Lewis modal systems

1957 ◽  
Vol 22 (2) ◽  
pp. 176-186 ◽  
Author(s):  
E. J. Lemmon
Keyword(s):  
Modal Logic ◽  
Propositional Calculus ◽  
Modal System ◽  
New Foundations ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Modal Systems

The main aims of this paper are firstly to present new and simpler postulate sets for certain well-known systems of modal logic, and secondly, in the light of these results, to suggest some new or newly formulated calculi, capable of interpretation as systems of epistemic or deontic modalities. The symbolism throughout is that of [9] (see especially Part III, Chapter I). In what follows, by a Lewis modal system is meant a system which (i) contains the full classical propositional calculus, (ii) is contained in the Lewis system S5, (iii) admits of the substitutability of tautologous equivalents, (iv) possesses as theses the four formulae:We shall also say that a system Σ1 is stricter than a system Σ2, if both are Lewis modal systems and Σ1 is contained in Σ2 but Σ2 is not contained in Σ1; and we shall call Σ1absolutely strict, if it possesses an infinity of irreducible modalities. Thus, the five systems of Lewis in [5], S1, S2, S3, S4, and S5, are all Lewis modal systems by this definition; they are in an order of decreasing strictness from S1 to S5; and S1 and S2 alone are absolutely strict.

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

scholarly journals Symbolic WS1S

10.29007/t28j ◽  
2018 ◽  
Author(s):  
Loris D'Antoni ◽  
Margus Veanes
Keyword(s):  
Blow Up ◽  
Finite Automata ◽  
Decision Procedures ◽  
Second Order ◽  
Order Logic ◽  
Symbolic Logic ◽  
First Order ◽  
Symbolic Automata ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We extend weak monadic second-order logic of one successor (WS1S) to symbolic alphabets byallowing character predicates to range over decidable first order theories and not just finite alphabets.We call this extension symbolic WS1S (s-WS1S). We then propose two decision procedures for such alogic: 1) we use symbolic automata to extend the classic reduction from WS1S to finite automata toour symbolic logic setting; 2) we show that every s-WS1S formula can be reduced to a WS1S formulathat preserves satisfiability, at the price of an exponential blow-up.

Sign in / Sign up

Export Citation Format

Share Document