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... 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 ...
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 ...
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 — ...
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 ...
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 ...
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 ...
Saul Kripke
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).