Modal Logic

... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...

... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...

valid - Informatik Uni Leipzig

... Proof for T and T. Let F be a frame from class T. Let I be an interpretation based on F and let w be an arbitrary world in I . If 2ϕ is not true in a world w, then axiom T is true in w. If 2ϕ is true in w, then ϕ is true in all accessible worlds. Since the accessibility relation is reflexive, w is a ...

... Proof for T and T. Let F be a frame from class T. Let I be an interpretation based on F and let w be an arbitrary world in I . If 2ϕ is not true in a world w, then axiom T is true in w. If 2ϕ is true in w, then ϕ is true in all accessible worlds. Since the accessibility relation is reflexive, w is a ...

PROVING UNPROVABILITY IN SOME NORMAL MODAL LOGIC

... first one (in theorem 2 below) is syntactic, based on some uniform presentation of the formulas, and is suitable for particular cases. Similar idea is used in [7] when an L-complete system for S5 is presented. The second method (theorem 3) is semantic and is applicable in more general situation. Its ...

... first one (in theorem 2 below) is syntactic, based on some uniform presentation of the formulas, and is suitable for particular cases. Similar idea is used in [7] when an L-complete system for S5 is presented. The second method (theorem 3) is semantic and is applicable in more general situation. Its ...

mj cresswell

... of course is up to the model, since the value of 0 will deliver in each world the set o f things which satisfy 0 . (it might be tempting to require that i f (u,w) E V (0 ) then u E D „ , but although this would make 0 x false fo r every atomic wf f when x has a value not in D, it would make every — ...

... of course is up to the model, since the value of 0 will deliver in each world the set o f things which satisfy 0 . (it might be tempting to require that i f (u,w) E V (0 ) then u E D „ , but although this would make 0 x false fo r every atomic wf f when x has a value not in D, it would make every — ...

An Independence Result For Intuitionistic Bounded Arithmetic

... 1 Introducing Classical and Intuitionistic Bounded Arithmetic We first briefly describe the first-order theories of bounded arithmetic introduced by Samuel Buss [B1]. The language of these theories extends the usual language of first-order arithmetic by adding function symbols x x2 y (= x2 rounded d ...

... 1 Introducing Classical and Intuitionistic Bounded Arithmetic We first briefly describe the first-order theories of bounded arithmetic introduced by Samuel Buss [B1]. The language of these theories extends the usual language of first-order arithmetic by adding function symbols x x2 y (= x2 rounded d ...

pdf

... S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiability problem for S5 is NP-complete. What causes the gap between NP and PSPACE here? We show that, in a precise sense, it is the negative introspection axiom: ¬Kϕ ⇒ K¬Kϕ. It easily follows f ...

... S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiability problem for S5 is NP-complete. What causes the gap between NP and PSPACE here? We show that, in a precise sense, it is the negative introspection axiom: ¬Kϕ ⇒ K¬Kϕ. It easily follows f ...

Normalised and Cut-free Logic of Proofs

... [2002]). Although simple and cut-free, these sequent calculi fail to satisfy certain properties that are standardly required from a “good" sequent calculus (in Poggiolesi [2010] and Wansing [1998], one can find a precise description of everything that is required from a “good" sequent calculus), nam ...

... [2002]). Although simple and cut-free, these sequent calculi fail to satisfy certain properties that are standardly required from a “good" sequent calculus (in Poggiolesi [2010] and Wansing [1998], one can find a precise description of everything that is required from a “good" sequent calculus), nam ...

comments on the logic of constructible falsity (strong negation)

... Görnemann’s result suggests the conjecture that a classical model theory for the logic I have described may be obtained by allowing the domain to “grow with time”. This is in fact true. We may define a Nelson model structure as a triple (K, R, D), where K is a non-empty set of “stages of investigat ...

... Görnemann’s result suggests the conjecture that a classical model theory for the logic I have described may be obtained by allowing the domain to “grow with time”. This is in fact true. We may define a Nelson model structure as a triple (K, R, D), where K is a non-empty set of “stages of investigat ...

Modal_Logics_Eyal_Ariel_151107

... It means that A knows that if B doesn’t knows whether his forehead is muddy then A knows that it is possible in B’s knowledge that A’s forehead is muddy! Remember that: [i]A *A
...
*

... It means that A knows that if B doesn’t knows whether his forehead is muddy then A knows that it is possible in B’s knowledge that A’s forehead is muddy! Remember that: [i]A

Intuitionistic Logic

... those that would make S false. If truth is not epistemic, it might be elusive; there might be truths such that no amount of investigation we could ever do would put us in a position to know whether they are true. Consider In the decimal representation of π, for each n, somewhere there occurs in n-fo ...

... those that would make S false. If truth is not epistemic, it might be elusive; there might be truths such that no amount of investigation we could ever do would put us in a position to know whether they are true. Consider In the decimal representation of π, for each n, somewhere there occurs in n-fo ...

Distributed Knowledge

... The information that a has in w is given by the singleton set f(K; v)g, and the information of b is given by the singleton set f(K; u)g. Since the two worlds are dierent, the intersection of state of a and that of b is empty, and hence, (K; w) j= Dfa;bg?: the distributed knowledge of the two agents ...

... The information that a has in w is given by the singleton set f(K; v)g, and the information of b is given by the singleton set f(K; u)g. Since the two worlds are dierent, the intersection of state of a and that of b is empty, and hence, (K; w) j= Dfa;bg?: the distributed knowledge of the two agents ...

Modal Logics Definable by Universal Three

... for all universal Horn formulas. Our second contribution is confirming this conjecture for the case of formulas with at most three-variables, UHF3 . ...

... for all universal Horn formulas. Our second contribution is confirming this conjecture for the case of formulas with at most three-variables, UHF3 . ...

A Propositional Modal Logic for the Liar Paradox Martin Dowd

... 1. Introduction. The paradox of the liar is the statement “this statement is false”. In various forms it has puzzled logicians and philosophers of natural language since the time of the Greeks. Within the last decade, the tools of mathematical logic have been brought to bear on this paradox. It is ...

... 1. Introduction. The paradox of the liar is the statement “this statement is false”. In various forms it has puzzled logicians and philosophers of natural language since the time of the Greeks. Within the last decade, the tools of mathematical logic have been brought to bear on this paradox. It is ...

Unification in Propositional Logic

... By exploiting the above mentioned effect of transformations of the kind (θaA)∗ on Kripke models, one can show that a carefully built iteration θA of substitutions of the kind θaA can in fact always act as the contraction transformation of a contractible A∗ (that is, either such θA unifies A and cons ...

... By exploiting the above mentioned effect of transformations of the kind (θaA)∗ on Kripke models, one can show that a carefully built iteration θA of substitutions of the kind θaA can in fact always act as the contraction transformation of a contractible A∗ (that is, either such θA unifies A and cons ...

An Introduction to Modal Logic VII The finite model property

... A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not valid; consider M∼ , it is differentiat ...

... A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not valid; consider M∼ , it is differentiat ...

proceedings version

... Put ϕ+ in conjunctive normal form, and let κ = ( P) be some clause of that CNF, for some P ⊆ Pϕ+ , ∅. (Observe that P , ∅ by the definition of positive formulas.) Let Pw = P ∩ Vw . We have Pw , ∅ because M, w |= κ. Let Q = Vw ∩ Pψ . As M satisfies the constraint (negatable), there is a u ∈ W such th ...

... Put ϕ+ in conjunctive normal form, and let κ = ( P) be some clause of that CNF, for some P ⊆ Pϕ+ , ∅. (Observe that P , ∅ by the definition of positive formulas.) Let Pw = P ∩ Vw . We have Pw , ∅ because M, w |= κ. Let Q = Vw ∩ Pψ . As M satisfies the constraint (negatable), there is a u ∈ W such th ...

Multi-Agent Only

... I Similarly, Bob’s epistemic state is again a set of affairs where a state of affairs consists of a world and Alice’s epistemic state (that determines her beliefs at this ...

... I Similarly, Bob’s epistemic state is again a set of affairs where a state of affairs consists of a world and Alice’s epistemic state (that determines her beliefs at this ...

A Brief Introduction to the Intuitionistic Propositional Calculus

... the possibility that there may be other formula, not provable in this system, which are nevertheless intuitionistically valid. As time as passed, and no such formula has been forthcoming, this possibility has seemed increasingly remote. In the effort to pin the intuitionists down, a number of formal ...

... the possibility that there may be other formula, not provable in this system, which are nevertheless intuitionistically valid. As time as passed, and no such formula has been forthcoming, this possibility has seemed increasingly remote. In the effort to pin the intuitionists down, a number of formal ...

A General Proof Method for ... without the Barcan Formula.*

... This paper generalizes the proof method for modal predicate logic first described in Jackson [1987] and axiomatized in Jackson & Reichgelt [1987]. As before, the inference rules are identical for each system; different systems differ only with respect to the definition of complementarity between for ...

... This paper generalizes the proof method for modal predicate logic first described in Jackson [1987] and axiomatized in Jackson & Reichgelt [1987]. As before, the inference rules are identical for each system; different systems differ only with respect to the definition of complementarity between for ...

De Jongh`s characterization of intuitionistic propositional calculus

... is the only intermediate logic1 having the disjunction property, i.e., if ` φ ∨ ψ, then ` φ or ` ψ. However, Kreisel and Putnam [13] disproved this conjecture by constructing a proper extension of intuitionistic logic satisfying the disjunction property. Later, Wronski [17] showed that there are in ...

... is the only intermediate logic1 having the disjunction property, i.e., if ` φ ∨ ψ, then ` φ or ` ψ. However, Kreisel and Putnam [13] disproved this conjecture by constructing a proper extension of intuitionistic logic satisfying the disjunction property. Later, Wronski [17] showed that there are in ...

Bisimulation and public announcements in logics of

... EBK language. Fitting models for the EBK language are obtained from the Fitting models defined above by adding a reflexive relation Ri corresponding to each modal Ki . Thus a general Fitting model is a tuple M = (G, {Ri }ni=1 , Re , E, V ). In single-agent logics, where n = 1, the subscript on both ...

... EBK language. Fitting models for the EBK language are obtained from the Fitting models defined above by adding a reflexive relation Ri corresponding to each modal Ki . Thus a general Fitting model is a tuple M = (G, {Ri }ni=1 , Re , E, V ). In single-agent logics, where n = 1, the subscript on both ...

A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part

... 2A → 2(∀x)A.4 This axiom plays a role in making Gödel completeness work (cf. Lemma 6.3 in Part II). Motivation for keeping the two modal axioms of K4 is straightforward: Axiom (M1) in Section 3 simulates classical modus ponens (cf. 4.10). Axiom (M2), of less obvious intuitive value, is technically ...

... 2A → 2(∀x)A.4 This axiom plays a role in making Gödel completeness work (cf. Lemma 6.3 in Part II). Motivation for keeping the two modal axioms of K4 is straightforward: Axiom (M1) in Section 3 simulates classical modus ponens (cf. 4.10). Axiom (M2), of less obvious intuitive value, is technically ...

On Provability Logic

... (which is provable in PA) can be read the number three is a prime. The term S(S(S(0))) is denoted 3. More generally, the n-th numeral is defined as the term S(S . . (0) . .) with n occurrence of the symbol S. As an exercise we suggest the reader to formulate the fact that there are infinitely many p ...

... (which is provable in PA) can be read the number three is a prime. The term S(S(S(0))) is denoted 3. More generally, the n-th numeral is defined as the term S(S . . (0) . .) with n occurrence of the symbol S. As an exercise we suggest the reader to formulate the fact that there are infinitely many p ...

Normal modal logics (Syntactic characterisations)

... Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilbert systems: given a set of formulas called axioms and a set of rules of proof, a formula A is a theorem of the system when it is the last formula of a sequ ...

... Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilbert systems: given a set of formulas called axioms and a set of rules of proof, a formula A is a theorem of the system when it is the last formula of a sequ ...

Identity in modal logic theorem proving

... construct proofs within one of these proof theories - - by which I mean both that the result generated would be recognized as a proof in [say] Whitehead Russell's axiom system and also that the "machine internal" strategies and methods are applications of what it is legal to do within the proof theo ...

... construct proofs within one of these proof theories - - by which I mean both that the result generated would be recognized as a proof in [say] Whitehead Russell's axiom system and also that the "machine internal" strategies and methods are applications of what it is legal to do within the proof theo ...

Saul Aaron Kripke (/sɔːl ˈkrɪpki/; born November 13, 1940) is an American philosopher and logician. He is a Distinguished Professor of Philosophy at the City University of New York and emeritus professor at Princeton University. Since the 1960s Kripke has been a central figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and set theory. Much of his work remains unpublished or exists only as tape-recordings and privately circulated manuscripts. Kripke was the recipient of the 2001 Schock Prize in Logic and Philosophy. A recent academic poll ranked Kripke among the top ten most important philosophers of the past 200 years.Kripke has made influential and original contributions to logic, especially modal logic. His work has profoundly influenced analytic philosophy, with his principal contribution being a semantics for modal logic, involving possible worlds as described in a system now called Kripke semantics. Another of his most important contributions is his argument that necessity is a 'metaphysical' notion, which should be separated from the epistemic notion of a priori, and that there are necessary truths which are a posteriori truths, such as ""Water is H2O."" He has also contributed an original reading of Wittgenstein, referred to as ""Kripkenstein."" His most famous work is Naming and Necessity (1980).