How to tell the truth without knowing what you are talking about
... negation in Standard English (“I do not want nothing” means that I want something), but different from some other natural languages, such as Italian or French, where it is customary that a double negation negates (“Non voglio niente” and “Je ne veux rien” both mean that I do not want anything), or s ...
... negation in Standard English (“I do not want nothing” means that I want something), but different from some other natural languages, such as Italian or French, where it is customary that a double negation negates (“Non voglio niente” and “Je ne veux rien” both mean that I do not want anything), or s ...
EVERYONE KNOWS THAT SOMEONE KNOWS
... 4. Suppose that formula φ has the form ψ → χ. By Definition 6, statement (w, ρ[x 7→ α(c)]) ψ → χ is equivalent to the disjunction of statement (w, ρ[x 7→ α(c)]) 1 ψ and statement (w, ρ[x 7→ α(c)]) χ. By the induction hypothesis, this disjunction is in turn equivalent to the disjunction of statem ...
... 4. Suppose that formula φ has the form ψ → χ. By Definition 6, statement (w, ρ[x 7→ α(c)]) ψ → χ is equivalent to the disjunction of statement (w, ρ[x 7→ α(c)]) 1 ψ and statement (w, ρ[x 7→ α(c)]) χ. By the induction hypothesis, this disjunction is in turn equivalent to the disjunction of statem ...
Judgment and consequence relations
... Also, given (4), T is maximally consistent. In this definition we consider truth in the classical sense. A proposition is either true or false. If it is rejected, that is, if 0T ϕ this is because the proposition is false. So, no subjective element enters here. Truth is independent of whether we know ...
... Also, given (4), T is maximally consistent. In this definition we consider truth in the classical sense. A proposition is either true or false. If it is rejected, that is, if 0T ϕ this is because the proposition is false. So, no subjective element enters here. Truth is independent of whether we know ...
Written
... reflexive; (b) symmetric; (c) reflexive and symmetric; (d) reflexive and contain (1, 2); (e) symmetric and contain (1, 2); (f) anti-symmetric; (g) anti-symmetric and contain (1, 2); (h) symmetric and anti-symmetric; (i) reflexive, symmetric and anti-symmetric. a) Each of these relations must contain ...
... reflexive; (b) symmetric; (c) reflexive and symmetric; (d) reflexive and contain (1, 2); (e) symmetric and contain (1, 2); (f) anti-symmetric; (g) anti-symmetric and contain (1, 2); (h) symmetric and anti-symmetric; (i) reflexive, symmetric and anti-symmetric. a) Each of these relations must contain ...
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 ...
Lecture 1: Elements of Mathematical Logic
... The primary purpose of this course is to introduce you, most of whom are mathematics majors, to the most fundamental skills of a mathematician; the ability to read, write, and understand proofs. This is a course where proofs matter more than the material covered. That said, I should also stress that ...
... The primary purpose of this course is to introduce you, most of whom are mathematics majors, to the most fundamental skills of a mathematician; the ability to read, write, and understand proofs. This is a course where proofs matter more than the material covered. That said, I should also stress that ...
A Uniform Proof Procedure for Classical and Non
... In this paper we present a proof procedure which allows a uniform treatment of classical, intuitionistic, and modal logics. It is based on a unified representation of Wallen’s matrix characterizations and generalizes Bibel’s connection method [4, 5] for classical predicate logic accordingly. In orde ...
... In this paper we present a proof procedure which allows a uniform treatment of classical, intuitionistic, and modal logics. It is based on a unified representation of Wallen’s matrix characterizations and generalizes Bibel’s connection method [4, 5] for classical predicate logic accordingly. In orde ...
An Introduction to Löb`s Theorem in MIRI Research
... One (anachronistic) way of stating Gödel’s key insight is that you can use computer programs to search for proofs, and you can prove statements about computer programs. If we think about any conjecture in mathematics that can be stated in terms of arithmetic, you can write a rather simple program t ...
... One (anachronistic) way of stating Gödel’s key insight is that you can use computer programs to search for proofs, and you can prove statements about computer programs. If we think about any conjecture in mathematics that can be stated in terms of arithmetic, you can write a rather simple program t ...
A Resolution-Based Proof Method for Temporal Logics of
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
Notes on Modal Logic - Stanford University
... The modal invariance Lemma (Lemma 3.7) can be used to prove what can and cannot be expressed in the basic modal language. Fact 3.9 Let M = hW, R, V i be a relational structure. The universal operator is a unary operator Aϕ defined as follows: M, w |= Aϕ iff for all v ∈ W , M, v |= ϕ The universal o ...
... The modal invariance Lemma (Lemma 3.7) can be used to prove what can and cannot be expressed in the basic modal language. Fact 3.9 Let M = hW, R, V i be a relational structure. The universal operator is a unary operator Aϕ defined as follows: M, w |= Aϕ iff for all v ∈ W , M, v |= ϕ The universal o ...
Notes on Propositional Logic
... In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets. We have no straightforward way of formulating propositions such as: ...
... In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets. We have no straightforward way of formulating propositions such as: ...
Paper - Christian Muise
... disjunction, we follow Fan et al. (2015) by modelling knowing whether as its own modal operator, ∆i , defined as ∆i φ ≡ i φ ∨ i ¬φ. After providing the necessary background for the paper in the following section, we show how to compile an extended PEKB into a prime implicate normal form in exponen ...
... disjunction, we follow Fan et al. (2015) by modelling knowing whether as its own modal operator, ∆i , defined as ∆i φ ≡ i φ ∨ i ¬φ. After providing the necessary background for the paper in the following section, we show how to compile an extended PEKB into a prime implicate normal form in exponen ...
A Resolution Method for Modal Logic S5
... the second approach by using the notions of nominals and satisfaction connectives stemming from hybrid logics to define a simple and elegant system for S5. Hybrid logics were mainly introduced to explicitly express the relativity of truth in modal logics [5, 3]. This relativity is obtained by adding ...
... the second approach by using the notions of nominals and satisfaction connectives stemming from hybrid logics to define a simple and elegant system for S5. Hybrid logics were mainly introduced to explicitly express the relativity of truth in modal logics [5, 3]. This relativity is obtained by adding ...
An Introduction to Modal Logic VII The finite model property
... Let Λ be a normal modal logic and M = hW , R, V i be a finite differentiated model of Λ; we want to show that every formula ϕ ∈ Λ is valid in the frame M = hW , Ri. In search of a contradiction, suppose that there is α ∈ Λ, a model M0 = hW , R, V 0 i and w ∈ W such that M0 , w 2 α, since M is finite ...
... Let Λ be a normal modal logic and M = hW , R, V i be a finite differentiated model of Λ; we want to show that every formula ϕ ∈ Λ is valid in the frame M = hW , Ri. In search of a contradiction, suppose that there is α ∈ Λ, a model M0 = hW , R, V 0 i and w ∈ W such that M0 , w 2 α, since M is finite ...
6. Truth and Possible Worlds
... listener’s epistemic state. We can now see more clearly what that change is. Suppose Fred tells Betty: “Your cat has shredded my logic assignment”. Let the proposition expressed by this sentence be A. Now consider the minimal state where A is fully believed, which we will call KA. This is actually t ...
... listener’s epistemic state. We can now see more clearly what that change is. Suppose Fred tells Betty: “Your cat has shredded my logic assignment”. Let the proposition expressed by this sentence be A. Now consider the minimal state where A is fully believed, which we will call KA. This is actually t ...
Bisimulation and public announcements in logics of
... Definition 3.1. Given the model M = (G, {Ri }ni=1 , Re , E, V ), a world Γ ∈ G, and a formula ϕ in the EBK language, to say that ϕ is knowable at Γ means that M, ∆ |= ϕ whenever ΓRe ∆. Definition 3.2. Given models M1 = (G1 , {Ri }ni=1 , Re , E1 , V1 ) and M2 = (G2 , {Si }ni=1 , Se , E2 , V2 ), a non ...
... Definition 3.1. Given the model M = (G, {Ri }ni=1 , Re , E, V ), a world Γ ∈ G, and a formula ϕ in the EBK language, to say that ϕ is knowable at Γ means that M, ∆ |= ϕ whenever ΓRe ∆. Definition 3.2. Given models M1 = (G1 , {Ri }ni=1 , Re , E1 , V1 ) and M2 = (G2 , {Si }ni=1 , Se , E2 , V2 ), a non ...
From Syllogism to Common Sense Normal Modal Logic
... ‣ More generally, in a fixed language, the class of all normal modal logics is defined as any set of formulae that ‣ (1) contains K (2) is closed under substitution and (3) Modus Ponens ‣ In particular, any normal extension of K contains the Axiom of Box-Distribution: ...
... ‣ More generally, in a fixed language, the class of all normal modal logics is defined as any set of formulae that ‣ (1) contains K (2) is closed under substitution and (3) Modus Ponens ‣ In particular, any normal extension of K contains the Axiom of Box-Distribution: ...