The rules are —6 : Rules for S1 Uniform Substitution A valid formula remains valid if a formula is uniformly substituted in it for a propositional variable. Substitution of Strict Equivalents Either of two strictly equivalent formulas can be substituted for one another. By , Lewis has come to prefer system S2. In Lewis is not sure that the questionable theorem is not derivable in S2. Should it be, he would then adjudicate S1 as the proper system for strict implication. From B9 Lewis proceeds to deduce the existence of at least four logically distinct propositions: one true and necessary, one true but not necessary, one false and impossible, one false but not impossible —9.
Lewis concludes Appendix II by noting that the study of logic is best served by focusing on systems weaker than S5 and not exclusively on S5. Paradoxes of strict implication similar to those of material implication arise too. Lewis argues that this is as it ought to be. Since impossibility is taken to be logical impossibility, i. See, for example, Nelson , Strawson , and Bennett See also the SEP entry on relevance logic. System S3, an extension of S2, is not contained in M.
Nor is M contained in S3. Von Wright finds S3 of little independent interest, and sees no reason to adopt S3 instead of the stronger S4. In general, the Lewis systems are numbered in order of strength, with S1 the weakest and S5 the strongest, weaker systems being contained in the stronger ones.
System P4, equivalent to S4, employs PC , rule a , and axioms 2 and 1. Lemmon considers also some systems weaker than S1. Of particular interest is system S0. Lemmon interprets system S0. System K is the smallest normal system. System T adds axiom T to system K. For the relationship between these and other systems, and the conditions on frames that the axioms impose, consult the SEP entry on modal logic.
Only a few of the many extensions of the Lewis systems that have been discussed in the literature are mentioned here. First, he was thinking in algebraic terms, rather than syntactically, concerning himself not so much with the construction of new systems, but with the evaluation of the systems relatively to sets of values. Ironically, later work employing his original matrix method will show that the hope of treating modal logic as a three-valued system cannot be realized.
See also the SEP entry on many-valued logic. Matrices are typically used to show the independence of the axioms of a system as well as their consistency. A proper modality is of degree higher than zero.
Parry proves that S3 has 42 distinct modalities, and that S4 has 14 distinct modalities. It was already known that system S5 has only 6 distinct modalities since it reduces all modalities to modalities of degree zero or one.
Parry introduces system S4. Therefore the number of modalities does not uniquely determine a system. Systems S1 and S2, as well as T and B, have an infinite number of modalities Burgess , chapter 3 on Modal Logic, discusses the additional systems S4.
A characteristic matrix for a system L is a matrix that satisfies all and only the theorems of L. A matrix is finite if its set K of truth-values is finite. A finite characteristic matrix yields a decision procedure, where a system is decidable if every formula of the system that is not a theorem is falsified by some finite matrix this is the finite model property.
Yet Dugundji shows that none of S1—S5 has a finite characteristic matrix. Later, Scroggs will prove that every proper extension of S5 that preserves detachment for material implication and is closed under substitution has a finite characteristic matrix. Despite their lack of a finite characteristic matrix, McKinsey shows that systems S2 and S4 are decidable.
The proof employs three steps. M is a trivial matrix whose domain is the set of formulas of the system, whose designated elements are the theorems of the system, and whose operations are the connectives themselves. A similar proof is given for S4. A matrix is a special kind of algebra. An algebra is a matrix without a set D of designated elements.
Boolean algebras correspond to matrices for propositional logic. According to Bull and Segerberg 10 the generalization from matrices to algebras may have had the effect of encouraging the study of these structures independently of their connections to logic and modal systems.
The set of designated elements D in fact facilitates a definition of validity with respect to which the theorems of a system can be evaluated.
Without such a set the most obvious link to logic is severed. A second generalization to classes of algebras, rather than merely to individual algebras, was also crucial to the mathematical development of the subject matter. Tarski is the towering figure in such development.
Lemmon b: attributes to Dana Scott the main result of his second paper. Kripke a is already explicit on this connection. In The Lemmon Notes , written in collaboration with Dana Scott and edited by Segerberg, the technique is transformed into a purely model theoretic method which yields completeness and decidability results for many systems of modal logic in as general a form as possible See also the SEP entry on the algebra of logic tradition.
For a more comprehensive treatment, see chapter 5 of Blackburn, de Rijke, and Venema See also Goldblatt The Model Theoretic Tradition 3. At the same time, he recognized that the many syntactical advances in modal logic from on were still not accompanied by adequate semantic considerations.
Carnap instead thought of necessity as logical truth or analyticity. The idea of quantified modal systems occurred to Ruth Barcan too. Though the strategies are closely related, there are two important distinctions to be made between them: The underlying mathematical model of the logic-based approach are Kripke semantics , while the event-based approach employs the related Aumann structures.
In the event-based approach logical formulas are done away with completely, while the logic-based approach uses the system of modal logic. Typically, the logic-based approach has been used in fields such as philosophy, logic and AI, while the event-based approach is more often used in fields such as game theory and mathematical economics. In the logic-based approach, a syntax and semantics have been built using the language of modal logic, which we will now describe.
Syntax[ edit ] The basic modal operator of epistemic logic, usually written K, can be read as "it is known that," "it is epistemically necessary that," or "it is inconsistent with what is known that not.
Symbolic Logic 4, — Schilpp ed. Google Scholar Friedman, H. Symbolic Logic 40, — Google Scholar Gerson, M. Google Scholar Goldblatt, R. Symbolic Logic 40, Crossley ed. Symbolic Logic 14, — Google Scholar Hansson, B. Google Scholar Hilpinen, R. Google Scholar Hintikka, J. Acta Philosophica Fennica 8, 11— Revised version reprinted in Hintikka . Reprinted in Hintikka . Hintikka ed. Google Scholar Hofstadter, A. Second edition Google Scholar Jeffrey, R.
Google Scholar Kanger, S. Reprinted in Hilpinen. Google Scholar Kaplan, D. Symbolic Logic 35, Google Scholar Kneale, W. Google Scholar Kripke, S. Symbolic Logic 24, 1— Addison et al. Google Scholar Kuhn, S. Google Scholar Leivant, D. Symbolic Logic 46, — Symbolic Logic 22, —Symbolic Logic 8, 24— Moore, R. Standard possible worlds model[ edit ] Most attempts at modeling knowledge have been based on the possible worlds model. Without such a set the most obvious link to logic is severed. It was already known that system S5 has only 6 distinct modalities since it reduces all modalities to modalities of degree zero or one. CrossRef Google Scholar. Therefore the number of modalities does not uniquely determine. Kripke a is already explicit on this connection. So although on balance that support is not as a vast experience in writing quality academic essays. Thus, essays have become an inseparable part of academics Lld thesis 2012 nissan no means a complete description of either the. Google Scholar Hartshorne, C.
First, the maximal notion of validity must be replaced by a new universal notion. Systems S1 and S2, as well as T and B, have an infinite number of modalities Burgess , chapter 3 on Modal Logic, discusses the additional systems S4. Prior and K. In Lewis is not sure that the questionable theorem is not derivable in S2. Lewis, C.
Symbolic Logic 6, — Google Scholar Goldblatt, R.
Symbolic Logic 11, Lemmon b: attributes to Dana Scott the main result of his second paper.
Philosophical Logic 7, —, , 10, Copeland ed. Moore, R. Though the strategies are closely related, there are two important distinctions to be made between them: The underlying mathematical model of the logic-based approach are Kripke semantics , while the event-based approach employs the related Aumann structures. Since impossibility is taken to be logical impossibility, i.
A matrix is a special kind of algebra.
Of particular interest is system S0. System T adds axiom T to system K.
In the logic-based approach, a syntax and semantics have been built using the language of modal logic, which we will now describe. Google Scholar Hartshorne, C. Formal Logic 6,—
First, he shows that every satisfiable formula has a connected model, i. Google Scholar Meredith, D. Google Scholar Segerberg, K. Kripke reconstructs a proof of the converse Barcan formula in quantified T and shows that the proof goes through only by allowing the necessitation of a sentence containing a free variable.