Labeled Natural Deduction for Temporal Logics

... is far from being concluded. In particular, a satisfactory proof-theoretical analysis for temporal logics is still lacking. This is especially true in the case of branchingtime logics, as shown by the fact that for one of the most important of such logics, CTL∗ , even the problem of finding a comple ...

... is far from being concluded. In particular, a satisfactory proof-theoretical analysis for temporal logics is still lacking. This is especially true in the case of branchingtime logics, as shown by the fact that for one of the most important of such logics, CTL∗ , even the problem of finding a comple ...

Goal-directed Proof Theory

... Although we generally speak about deduction, there are namely several dierent tasks/problems which can be quali ed as deductive. These dierent tasks might be theoretically reducible one to each other, but a method or an algorithm to solve one does not necessarily apply well to another. To make thi ...

... Although we generally speak about deduction, there are namely several dierent tasks/problems which can be quali ed as deductive. These dierent tasks might be theoretically reducible one to each other, but a method or an algorithm to solve one does not necessarily apply well to another. To make thi ...

SEQUENT SYSTEMS FOR MODAL LOGICS

... of relational terms. These terms may contain subterms representing the accessibility relation in possible-worlds models, so that semantic information is available at the same level as syntactic information. The derivation rules in relational proof systems manipulate finite sequences of relational fo ...

... of relational terms. These terms may contain subterms representing the accessibility relation in possible-worlds models, so that semantic information is available at the same level as syntactic information. The derivation rules in relational proof systems manipulate finite sequences of relational fo ...

On the Complexity of Qualitative Spatial Reasoning: A Maximal

... When describing a spatial configuration or when reasoning about such a configuration, often it is not possible or desirable to obtain precise, quantitative data . In these cases, qualitative reasoning about spatial configurations may be used . One particular approach in this context has been develop ...

... When describing a spatial configuration or when reasoning about such a configuration, often it is not possible or desirable to obtain precise, quantitative data . In these cases, qualitative reasoning about spatial configurations may be used . One particular approach in this context has been develop ...

a semantic perspective - Institute for Logic, Language and

... perspective provided by modal languages) whether or not a formula is satisfied depends on where (or in this example, when) it is evaluated. For example, the formulas 3(p∧q) is satisfied at points t1 , t2 and t3 (because all these points are to the left of t4 where both p and q are true together) but ...

... perspective provided by modal languages) whether or not a formula is satisfied depends on where (or in this example, when) it is evaluated. For example, the formulas 3(p∧q) is satisfied at points t1 , t2 and t3 (because all these points are to the left of t4 where both p and q are true together) but ...

Everything Else Being Equal: A Modal Logic for Ceteris Paribus

... which we can evaluate our own results. In Section 3, we present a basic modal logic of weak and strict preference interpreted in ordered models of possible worlds, we discuss its expressive power and we provide a complete axiomatization. In particular, we show how this language can define global pre ...

... which we can evaluate our own results. In Section 3, we present a basic modal logic of weak and strict preference interpreted in ordered models of possible worlds, we discuss its expressive power and we provide a complete axiomatization. In particular, we show how this language can define global pre ...

Combinaison des logiques temporelle et déontique pour la

... In order to formally specify a security policy, it is natural to reason about time on the one hand, and obligations, permissions, and prohibitions on the other hand. Indeed, we have to express for instance the permission to access a resource for a certain period, the obligation to release a resource ...

... In order to formally specify a security policy, it is natural to reason about time on the one hand, and obligations, permissions, and prohibitions on the other hand. Indeed, we have to express for instance the permission to access a resource for a certain period, the obligation to release a resource ...

The Herbrand Manifesto

... Let's look at transitive closure first. Let's say that we have a binary relation p and we want to axiomatize the transitive closure q. The typical approach in Relational Logic would be to write the definition shown here. It is easy to see that q contains the transitive closure of p. The problem is t ...

... Let's look at transitive closure first. Let's say that we have a binary relation p and we want to axiomatize the transitive closure q. The typical approach in Relational Logic would be to write the definition shown here. It is easy to see that q contains the transitive closure of p. The problem is t ...

MoL-2013-07 - Institute for Logic, Language and Computation

... techniques which modularise the process of calculating the modal logics of set-theoretic constructions. In addition to this, in [HL08], various relativisations of modal logic of forcing were also considered. For example, if we fix a definable class of partial orders P, and a definition for it, we ma ...

... techniques which modularise the process of calculating the modal logics of set-theoretic constructions. In addition to this, in [HL08], various relativisations of modal logic of forcing were also considered. For example, if we fix a definable class of partial orders P, and a definition for it, we ma ...

Completeness theorems and lambda

... What happens if one restricts system F to the subsystem F0 where one allows only to form ΠX.T (X) if T (X) is built only with X and →?? Can one use the techniques of proof theory and give a predicative normalisation proof for this fragment? I. Takeuti gave such a proof in 1993, following G. Takeuti, ...

... What happens if one restricts system F to the subsystem F0 where one allows only to form ΠX.T (X) if T (X) is built only with X and →?? Can one use the techniques of proof theory and give a predicative normalisation proof for this fragment? I. Takeuti gave such a proof in 1993, following G. Takeuti, ...

? A Unified Semantic Framework for Fully

... family of basic systems. For example, this includes standard sequent calculi for modal logics, as well as the usual multiple-conclusion systems for intuitionistic logic, its dual, and bi-intuitionistic logic. On the semantic side, we consider a generalization of Kripke-style semantics for modal and ...

... family of basic systems. For example, this includes standard sequent calculi for modal logics, as well as the usual multiple-conclusion systems for intuitionistic logic, its dual, and bi-intuitionistic logic. On the semantic side, we consider a generalization of Kripke-style semantics for modal and ...

Modal Consequence Relations

... culminated in the notion of a consequence relation, which is a relation between sets of formulae and a single formula. A consequence relation ` specifies which arguments are valid; the argument from a set Σ to a formula δ is valid in ` iff hΣ, δi ∈ `, for which we write Σ ` δ. δ is a tautology of ` ...

... culminated in the notion of a consequence relation, which is a relation between sets of formulae and a single formula. A consequence relation ` specifies which arguments are valid; the argument from a set Σ to a formula δ is valid in ` iff hΣ, δi ∈ `, for which we write Σ ` δ. δ is a tautology of ` ...

John Nolt – Logics, chp 11-12

... quantifiers over a domain of possible worlds. So, for example, to say that it is necessary that 2 + 2 = 4 is to say that in all possible worlds 2 + 2 = 4; and to say that it is possible for the earth to be destroyed by an asteroid is to say that there is at least one possible world (universe) in whi ...

... quantifiers over a domain of possible worlds. So, for example, to say that it is necessary that 2 + 2 = 4 is to say that in all possible worlds 2 + 2 = 4; and to say that it is possible for the earth to be destroyed by an asteroid is to say that there is at least one possible world (universe) in whi ...

1 Non-deterministic Phase Semantics and the Undecidability of

... — on the one hand, we think that the faithful embedding of the elementary fragment of ILL into BBI is a key point here. Strictly speaking, the detour through ILL and (trivial) phase semantics is not absolutely necessary and we could have implemented the encoding of Minsky machines directly into BBI ...

... — on the one hand, we think that the faithful embedding of the elementary fragment of ILL into BBI is a key point here. Strictly speaking, the detour through ILL and (trivial) phase semantics is not absolutely necessary and we could have implemented the encoding of Minsky machines directly into BBI ...

Insights into Modal Slash Logic and Modal Decidability

... such that the domain M 0 = {(x0 , x1 , . . . , xn ) ∈ M n+1 : x0 = w and n < ω and Rxi xi+1 for all 0 ≤ i < n}, the accessibility relation R0 satisfies h(w, x1 , . . . , xn ), (w, y1 , . . . , ym )i ∈ R0 iff (m = n + 1 and xi = yi for all 1 ≤ i ≤ n and Rxn ym ), and the valuation V 0 satisfies (w, x ...

... such that the domain M 0 = {(x0 , x1 , . . . , xn ) ∈ M n+1 : x0 = w and n < ω and Rxi xi+1 for all 0 ≤ i < n}, the accessibility relation R0 satisfies h(w, x1 , . . . , xn ), (w, y1 , . . . , ym )i ∈ R0 iff (m = n + 1 and xi = yi for all 1 ≤ i ≤ n and Rxn ym ), and the valuation V 0 satisfies (w, x ...

Modal Languages and Bounded Fragments of Predicate Logic

... What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has b ...

... What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has b ...

Intuitionistic and Modal Logic

... • Theorem 12. Γ ` IPC ϕ iff ϕ is valid in all Kripke models of Γ for IPC. • For the Completeness side ( ⇐ ) we show: if Γ 0 IPC ϕ, then ϕ ∈/ ∆ for some ∆ containing Γ in the canonical model. • First show by induction on ψ that Θ ² ψ ⇔ ψ ∈ Θ. • Most cases easy: it is for example necessary to show tha ...

... • Theorem 12. Γ ` IPC ϕ iff ϕ is valid in all Kripke models of Γ for IPC. • For the Completeness side ( ⇐ ) we show: if Γ 0 IPC ϕ, then ϕ ∈/ ∆ for some ∆ containing Γ in the canonical model. • First show by induction on ψ that Θ ² ψ ⇔ ψ ∈ Θ. • Most cases easy: it is for example necessary to show tha ...

S. P. Odintsov “REDUCTIO AD ABSURDUM” AND LUKASIEWICZ`S

... general class of paraconsistent logics and pointing out some special property distinguishing extensions of minimal logic in the latter class. We suggest also that negation in logics from the above mentioned class should preserve the most essential property of intuitionistic negation, namely, that th ...

... general class of paraconsistent logics and pointing out some special property distinguishing extensions of minimal logic in the latter class. We suggest also that negation in logics from the above mentioned class should preserve the most essential property of intuitionistic negation, namely, that th ...

MODAL LANGUAGES AND BOUNDED FRAGMENTS OF

... Modal Logic is traditionally concerned with the intensional operators "possibly" and "necessary", whose intuitive correspondence with the standard quantifiers "there exists" and "for all" comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from prop ...

... Modal Logic is traditionally concerned with the intensional operators "possibly" and "necessary", whose intuitive correspondence with the standard quantifiers "there exists" and "for all" comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from prop ...

Master Thesis - Yoichi Hirai

... We apply formal constructive reasoning to asyncyhronous communication. After deﬁning a general-purpose logic called intuitionistic epistemic logic (IEC in short), we solve a motivating example problem, characterising waitfree communication logically in response to the abstract simplicial topological ...

... We apply formal constructive reasoning to asyncyhronous communication. After deﬁning a general-purpose logic called intuitionistic epistemic logic (IEC in short), we solve a motivating example problem, characterising waitfree communication logically in response to the abstract simplicial topological ...

THE SEMANTICS OF MODAL PREDICATE LOGIC II. MODAL

... into existence. Next, let us assume (again simplifying things considerably) that the trace of an object is just the collection of atoms of which it consists. Then, while the object b continued to exist, its trace has changed from one day to the other. In order not to get confused with the problem of ...

... into existence. Next, let us assume (again simplifying things considerably) that the trace of an object is just the collection of atoms of which it consists. Then, while the object b continued to exist, its trace has changed from one day to the other. In order not to get confused with the problem of ...

Taming method in modal logic and mosaic method in temporal logic

... PALH ’s universe is any binary relation satisfying the conditions in H, and it’s the relativized version of PAL. PAL∅ is the completely relativized version of PAL. In PALSQ transitivity of the universe ensures that composition is associative and that causes Hilbert incompleteness and undecidability ...

... PALH ’s universe is any binary relation satisfying the conditions in H, and it’s the relativized version of PAL. PAL∅ is the completely relativized version of PAL. In PALSQ transitivity of the universe ensures that composition is associative and that causes Hilbert incompleteness and undecidability ...

possible-worlds semantics for modal notions conceived as predicates

... This kind of quantification often is called substitutional quantification, sometimes even if it does not come with any semantics but only with axioms and rules. This terminology is surely confusing, but it has become almost standard terminology after Kripke [27] had shown how to apply a substitution ...

... This kind of quantification often is called substitutional quantification, sometimes even if it does not come with any semantics but only with axioms and rules. This terminology is surely confusing, but it has become almost standard terminology after Kripke [27] had shown how to apply a substitution ...

Introduction to Modal Logic - CMU Math

... then w1 “knows about” w2 and must consider it in making decisions about whether something is possible or necessary. V is a function mapping the set of propositional variables P to P(W ). The interpretation is the if P is mapped into a set contain w then w thinks that the variable P is true. ...

... then w1 “knows about” w2 and must consider it in making decisions about whether something is possible or necessary. V is a function mapping the set of propositional variables P to P(W ). The interpretation is the if P is mapped into a set contain w then w thinks that the variable P is true. ...

Proof theory for modal logic

... possibility to obtain direct completeness proofs without artificial Henkin-set constructions, and their use in the solution of problems that usually involve complex model-theoretic constructions such as negative results in correspondence theory and modal embeddings among different logics. All these ...

... possibility to obtain direct completeness proofs without artificial Henkin-set constructions, and their use in the solution of problems that usually involve complex model-theoretic constructions such as negative results in correspondence theory and modal embeddings among different logics. All these ...

# 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).