Paper - Department of Computer Science and Information Systems
... A closely related algorithmic problem for L is the admissibility problem for inference rules: given an inference rule ϕ1 , . . . , ϕn /ϕ, decide whether it is admissible in L, that is, for every substitution s, we have L ` s(ϕ) whenever L ` s(ϕi ), for 1 ≤ i ≤ n. It should be clear that if the admis ...
... A closely related algorithmic problem for L is the admissibility problem for inference rules: given an inference rule ϕ1 , . . . , ϕn /ϕ, decide whether it is admissible in L, that is, for every substitution s, we have L ` s(ϕ) whenever L ` s(ϕi ), for 1 ≤ i ≤ n. It should be clear that if the admis ...
A Simple Tableau System for the Logic of Elsewhere
... the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although E and the well-known propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense ...
... the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although E and the well-known propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense ...
full text (.pdf)
... We base S on + instead of ∗ because the resulting deductive system is cleaner—it contains no contraction rule1 . This is perhaps due to the fact that + can be viewed as a more primitive operation than ∗ . A test is either an atomic test, the symbol 0 representing falsity, or an expression b → c repr ...
... We base S on + instead of ∗ because the resulting deductive system is cleaner—it contains no contraction rule1 . This is perhaps due to the fact that + can be viewed as a more primitive operation than ∗ . A test is either an atomic test, the symbol 0 representing falsity, or an expression b → c repr ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... motivation for adopting this particular restriction will be given below. For α < ω, we get a logic with finitely many variables. Such logics have been studied in the literature, for purely logical reasons (Henkin [15], Henkin, Monk & Tarski [16], Tarski-Givant [41], Sain [37], Monk [24]) or because ...
... motivation for adopting this particular restriction will be given below. For α < ω, we get a logic with finitely many variables. Such logics have been studied in the literature, for purely logical reasons (Henkin [15], Henkin, Monk & Tarski [16], Tarski-Givant [41], Sain [37], Monk [24]) or because ...
First-order possibility models and finitary
... A propositional modal logic has a finitary completeness proof if it has a canonical model all of whose possibilities are finitely specified in this sense. This was one of Humberstone’s original motivations for considering possibility models. For many normal modal logics extending K with standard axi ...
... A propositional modal logic has a finitary completeness proof if it has a canonical model all of whose possibilities are finitely specified in this sense. This was one of Humberstone’s original motivations for considering possibility models. For many normal modal logics extending K with standard axi ...
A BRIEF INTRODUCTION TO MODAL LOGIC Introduction Consider
... Note the interconnections implied by this table. For example, any formula that is K-valid ought to be valid in all these systems, since its truth relies on no particular frame structure. Similarly, what is true in B will be true in S5, since the former is identical to the latter with the exception o ...
... Note the interconnections implied by this table. For example, any formula that is K-valid ought to be valid in all these systems, since its truth relies on no particular frame structure. Similarly, what is true in B will be true in S5, since the former is identical to the latter with the exception o ...
Admissible rules in the implication-- negation fragment of intuitionistic logic
... Although a logic may not be structurally complete, there may be well-behaved sets of formulas such that for rules whose premises form such a set, admissibility coincides with derivability. Let us fix L as a logic based on a language L containing a binary connective → for which modus ponens is deriva ...
... Although a logic may not be structurally complete, there may be well-behaved sets of formulas such that for rules whose premises form such a set, admissibility coincides with derivability. Let us fix L as a logic based on a language L containing a binary connective → for which modus ponens is deriva ...
Strong Completeness for Iteration
... Monads will be used in two different ways. One is related to the view that monads model computational effects [23]. The other is related to their role as abstract algebraic theories [21]. We briefly recall the basic definitions. A monad on Set is a triple T = (T, η, µ) where T is an Set-functor, and ...
... Monads will be used in two different ways. One is related to the view that monads model computational effects [23]. The other is related to their role as abstract algebraic theories [21]. We briefly recall the basic definitions. A monad on Set is a triple T = (T, η, µ) where T is an Set-functor, and ...
The unintended interpretations of intuitionistic logic
... Brouwer’s ideas about language did not prevent others from considering formalizations of parts of intuitionism. A. N. Kolmogorov [Kolmogorov 1925] gave an incomplete description of first-order predicate logic. Of particular interest is his description of the double negation translation. Although thi ...
... Brouwer’s ideas about language did not prevent others from considering formalizations of parts of intuitionism. A. N. Kolmogorov [Kolmogorov 1925] gave an incomplete description of first-order predicate logic. Of particular interest is his description of the double negation translation. Although thi ...
First-Order Intuitionistic Logic with Decidable Propositional
... Recently, research in the area of combining features of classical and intuitionistic logic has shifted towards logics containing two different variants of connectives. One of them is intuitionistic, and the other is classical [Kr] Fibring logics is the most noticeable technique in this research [Ga] ...
... Recently, research in the area of combining features of classical and intuitionistic logic has shifted towards logics containing two different variants of connectives. One of them is intuitionistic, and the other is classical [Kr] Fibring logics is the most noticeable technique in this research [Ga] ...
Formal Theories of Truth INTRODUCTION
... represents, so to speak, an ‘infinite logical product’ of those special theorems. But this does not at all mean that we can actually derive the principle of contradiction from the axioms or theorems mentioned by means of the normal modes of inference usually employed. On the contrary, by a slight mo ...
... represents, so to speak, an ‘infinite logical product’ of those special theorems. But this does not at all mean that we can actually derive the principle of contradiction from the axioms or theorems mentioned by means of the normal modes of inference usually employed. On the contrary, by a slight mo ...
Notes on Modal Logic - Stanford University
... definable (why?). However, note that even in finite relational structures, not all subsets may be definable. A problem can arise if states cannot be distinguished by modal formulas. For example, if the reflexive arrow is dropped in the relational structure above, then w2 and w3 cannot be distinguish ...
... definable (why?). However, note that even in finite relational structures, not all subsets may be definable. A problem can arise if states cannot be distinguished by modal formulas. For example, if the reflexive arrow is dropped in the relational structure above, then w2 and w3 cannot be distinguish ...
Belief closure: A semantics of common knowledge for
... partitions, and then explains that this definition can be rephrased into more intuitive terms, using the notion of a 'reachable' state of the world. Very roughly speaking, definition 1 is of the circular kind, and definition 2 of the iterate kind. However, in view of the immediate mathematical equiv ...
... partitions, and then explains that this definition can be rephrased into more intuitive terms, using the notion of a 'reachable' state of the world. Very roughly speaking, definition 1 is of the circular kind, and definition 2 of the iterate kind. However, in view of the immediate mathematical equiv ...
Modal Logic - Web Services Overview
... • Semantics is given in terms of Kripke Structures (also known as possible worlds structures) • Due to American logician Saul Kripke, City University of NY • A Kripke Structure is (W, R) – W is a set of possible worlds – R : W W is an binary accessibility relation over W – This relation tells us h ...
... • Semantics is given in terms of Kripke Structures (also known as possible worlds structures) • Due to American logician Saul Kripke, City University of NY • A Kripke Structure is (W, R) – W is a set of possible worlds – R : W W is an binary accessibility relation over W – This relation tells us h ...
Intuitionistic Logic - Institute for Logic, Language and Computation
... various sides of intuitionistic logic. In no way we strive for a complete overview in this short course. Even though we approach the subject for the most part only formally, it is good to have a general introduction to intuitionism. This we give in section 2 in which also natural deduction is introd ...
... various sides of intuitionistic logic. In no way we strive for a complete overview in this short course. Even though we approach the subject for the most part only formally, it is good to have a general introduction to intuitionism. This we give in section 2 in which also natural deduction is introd ...
34-2.pdf
... the parallel counterpart which often can solve the problem in a shorter amount of time. Finally, it is important to note the second printing of the book (which I have) has undergone quite a big change because of the relatively large number of small (mostly typographyical) errors found in the first p ...
... the parallel counterpart which often can solve the problem in a shorter amount of time. Finally, it is important to note the second printing of the book (which I have) has undergone quite a big change because of the relatively large number of small (mostly typographyical) errors found in the first p ...
Quantified Equilibrium Logic and the First Order Logic of Here
... present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternative axiom set for first-order here-and-there. The new system appears to be ...
... present a slightly modified version of QEL where the so-called unique name assumption or UNA is not assumed from the outset but may be added as a special requirement for specific applications. We also consider here an alternative axiom set for first-order here-and-there. The new system appears to be ...
Bounded Proofs and Step Frames - Università degli Studi di Milano
... principle and elements from Γ as well as modus ponens, necessitation and inferences from Ax (again notice that uniform substitution cannot be applied to members of Γ ). We need some care when replacing a logic L with an inference system Ax, because we want global consequence relation to be preserve ...
... principle and elements from Γ as well as modus ponens, necessitation and inferences from Ax (again notice that uniform substitution cannot be applied to members of Γ ). We need some care when replacing a logic L with an inference system Ax, because we want global consequence relation to be preserve ...
Intuitionistic Logic
... Negation is also defined by means of proofs: p : ¬A says that each proof a of A can be converted by the construction p into a proof of an absurdity, say 0 = 1. A proof of ¬A thus tells us that A has no proof! The most interesting propositional connective is the implication. The classical solution, i ...
... Negation is also defined by means of proofs: p : ¬A says that each proof a of A can be converted by the construction p into a proof of an absurdity, say 0 = 1. A proof of ¬A thus tells us that A has no proof! The most interesting propositional connective is the implication. The classical solution, i ...
article in press - School of Computer Science
... account for conditions which involve more than one guard relation. We believe that this method is particularly promising for intuitionistic modal logic, where there exists a variety of systems, most of them semantically defined, with various conditions connecting the intuitionistic and modal accessi ...
... account for conditions which involve more than one guard relation. We believe that this method is particularly promising for intuitionistic modal logic, where there exists a variety of systems, most of them semantically defined, with various conditions connecting the intuitionistic and modal accessi ...
Topological Completeness of First-Order Modal Logic
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
Intuitionistic modal logic made explicit
... Justification logics are explicit modal logics in the sense that they unfold the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics f ...
... Justification logics are explicit modal logics in the sense that they unfold the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics f ...
Strong Completeness and Limited Canonicity for PDL
... modal harmony. We show that, to our surprise, the canonical model of PDLω fails to have modal harmony, while at the same time it does have formula harmony. We first prove a similar disharmony result for ancestral logic (the logic with modalities and its reflexive transitive closure ∗ ). This proo ...
... modal harmony. We show that, to our surprise, the canonical model of PDLω fails to have modal harmony, while at the same time it does have formula harmony. We first prove a similar disharmony result for ancestral logic (the logic with modalities and its reflexive transitive closure ∗ ). This proo ...
Reaching transparent truth
... with the additional feature that the value assigned to an atomic sentence T hAi is always the same as the value assigned to A itself. Call any model with these features a KK model (for ‘Kleene-Kripke’).3 The models produced by this construction have two main features that make them interesting for o ...
... with the additional feature that the value assigned to an atomic sentence T hAi is always the same as the value assigned to A itself. Call any model with these features a KK model (for ‘Kleene-Kripke’).3 The models produced by this construction have two main features that make them interesting for o ...
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).