Introduction to logic
... features is reasoning. Reasoning can be viewed as the process of having a knowledge base (KB) and manipulating it to create new knowledge. Reasoning can be seen as comprised of 3 processes: 1. Perceiving stimuli from the environment; 2. Translating the stimuli in components of the KB; 3. Working on ...
... features is reasoning. Reasoning can be viewed as the process of having a knowledge base (KB) and manipulating it to create new knowledge. Reasoning can be seen as comprised of 3 processes: 1. Perceiving stimuli from the environment; 2. Translating the stimuli in components of the KB; 3. Working on ...
CS 40: Foundations of Computer Science
... to follow from our assumptions, so let's find a case in which the assumptions hold but this conditional statement does not. This conditional statement fails in the case in which s is true and e is false. If we take d to be true as well, then both of our assumptions are true. There fore this conclusi ...
... to follow from our assumptions, so let's find a case in which the assumptions hold but this conditional statement does not. This conditional statement fails in the case in which s is true and e is false. If we take d to be true as well, then both of our assumptions are true. There fore this conclusi ...
Proof theory for modal logic
... An axiom system for modal logic can be an extension of intuitionistic or classical propositional logic. In the latter, the notions of necessity and possibility are interdefinable by the equivalence 2A ⊃⊂ ¬3¬A. It is seen that necessity and possibility behave analogously to the quantifiers: In one in ...
... An axiom system for modal logic can be an extension of intuitionistic or classical propositional logic. In the latter, the notions of necessity and possibility are interdefinable by the equivalence 2A ⊃⊂ ¬3¬A. It is seen that necessity and possibility behave analogously to the quantifiers: In one in ...
A Mathematical Introduction to Modal Logic
... Exercise 2.10. Prove that the axiom scheme (ϕ → ψ) → (ϕ → ψ) is sound. The second axiom is called Kripke axiom or normality axiom. The logics which possess the normality axiom are surprisingly called normal modal logics. We have two proof rules. The first one, modus ponens, is a familiar one: If ...
... Exercise 2.10. Prove that the axiom scheme (ϕ → ψ) → (ϕ → ψ) is sound. The second axiom is called Kripke axiom or normality axiom. The logics which possess the normality axiom are surprisingly called normal modal logics. We have two proof rules. The first one, modus ponens, is a familiar one: If ...
on Computability
... axiomatizations of mathematics • A logic usually refers to a set of rules about constructing valid sentences. Here are a few logics. Propositional logic concerns sentences such as (p ∨¬q) ∧ (¬p ∧ r) where p, q, r are boolean variables. Recall that the SAT problem consists of determining the satisfia ...
... axiomatizations of mathematics • A logic usually refers to a set of rules about constructing valid sentences. Here are a few logics. Propositional logic concerns sentences such as (p ∨¬q) ∧ (¬p ∧ r) where p, q, r are boolean variables. Recall that the SAT problem consists of determining the satisfia ...
Plural Quantifiers
... The lesson to be drawn from the foregoing reflections on plurals and secondorder logic is that neither the use of plurals nor the employment of secondorder logic commits us to the existence of extra items beyond those to which we are already committed. We need not construe second-order quantifiers a ...
... The lesson to be drawn from the foregoing reflections on plurals and secondorder logic is that neither the use of plurals nor the employment of secondorder logic commits us to the existence of extra items beyond those to which we are already committed. We need not construe second-order quantifiers a ...
Discrete Structures & Algorithms Propositional Logic
... In day-to-day speech, sometimes we use “or” as an “exclusive or”. ...
... In day-to-day speech, sometimes we use “or” as an “exclusive or”. ...
Strong Completeness and Limited Canonicity for PDL
... is true in a world (i.e. a maximal consistent set) of the canonical model iff it is an element of that world. In other words: at the formula level, there is agreement between the semantics and the proof theoretical aspects of the canonical model. Therefore we call this property formula harmony, and ...
... is true in a world (i.e. a maximal consistent set) of the canonical model iff it is an element of that world. In other words: at the formula level, there is agreement between the semantics and the proof theoretical aspects of the canonical model. Therefore we call this property formula harmony, and ...
A General Proof Method for ... without the Barcan Formula.*
... necessity and possibility, but they can also provide a basis for reasoning about knowledge, belief, time and change, e.g. [Halpern & Moses, 19851. Automated reasoning in modal logics is made difficult, however, by (i) the absence of a normal form for expressions containing modal operators, and (ii) ...
... necessity and possibility, but they can also provide a basis for reasoning about knowledge, belief, time and change, e.g. [Halpern & Moses, 19851. Automated reasoning in modal logics is made difficult, however, by (i) the absence of a normal form for expressions containing modal operators, and (ii) ...
many-valued logics - University of Sydney
... case, a tautology is a proposition which gets the value 1 on every model (e.g. p ∨ ¬p, p → p), and a proposition α is a logical consequence of the set of propositions Γ (written Γ |= α) if, on every model on which every proposition in Γ has the value 1, α has the value 1 (e.g. {p, p → q} |= q, {p} | ...
... case, a tautology is a proposition which gets the value 1 on every model (e.g. p ∨ ¬p, p → p), and a proposition α is a logical consequence of the set of propositions Γ (written Γ |= α) if, on every model on which every proposition in Γ has the value 1, α has the value 1 (e.g. {p, p → q} |= q, {p} | ...
Propositional Logic - University of San Francisco
... A knowledge base plus a model allow us to perform inference. For a given set of sentences, plus some assignment of values to variables, what can we conclude? Entailment tells us that a sentence can be derived. Inference tells us how it is derived. An algorithm that only derives entailed sentences is ...
... A knowledge base plus a model allow us to perform inference. For a given set of sentences, plus some assignment of values to variables, what can we conclude? Entailment tells us that a sentence can be derived. Inference tells us how it is derived. An algorithm that only derives entailed sentences is ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... In this paper we develop a modal formalism called cylindric modal logic we investigate its basic semantics and axiomatics. The motivation for introducing this formalism is twofold: first, it forms an interesting bridge over the gap between propositional formalisms and first-order logic. And second, ...
... In this paper we develop a modal formalism called cylindric modal logic we investigate its basic semantics and axiomatics. The motivation for introducing this formalism is twofold: first, it forms an interesting bridge over the gap between propositional formalisms and first-order logic. And second, ...
Temporal Here and There - Computational Cognition Lab
... In [10], Michael Gelfond and Vladimir Lifschitz introduced the so-called 0 semantics that subsumed many of the existing Logic Programming alternatives but without the syntactic restrictions made by previous approaches. The modelbased orientation of this semantics led to a paradigm suitable for const ...
... In [10], Michael Gelfond and Vladimir Lifschitz introduced the so-called 0 semantics that subsumed many of the existing Logic Programming alternatives but without the syntactic restrictions made by previous approaches. The modelbased orientation of this semantics led to a paradigm suitable for const ...
p q
... Example: For all real numbers d, d1, d2, and x, if d=min{d1,d2} and xd, then xd1 and xd2. Proof. From the definition of min, it follows that dd1 and dd2. From xd and dd1, we may derive xd1 by the transitive property of . From xd and dd2, we may derive xd2 by the same ...
... Example: For all real numbers d, d1, d2, and x, if d=min{d1,d2} and xd, then xd1 and xd2. Proof. From the definition of min, it follows that dd1 and dd2. From xd and dd1, we may derive xd1 by the transitive property of . From xd and dd2, we may derive xd2 by the same ...
1 Preliminaries 2 Basic logical and mathematical definitions
... When considering logic programs there exists a particular class of interpretations which are relevant, namely Herbrand interpretations. Let assume that the first order language L is defined on a signature Σ which contains at least one 0-ary function symbol. The set τ (Σ) of the ground terms is calle ...
... When considering logic programs there exists a particular class of interpretations which are relevant, namely Herbrand interpretations. Let assume that the first order language L is defined on a signature Σ which contains at least one 0-ary function symbol. The set τ (Σ) of the ground terms is calle ...
![arXiv:1410.5037v2 [cs.LO] 18 Jun 2016](http://s1.studyres.com/store/data/007883898_2-29c425582568c0e0d15c3d815896c9cf-300x300.png)
arXiv:1410.5037v2 [cs.LO] 18 Jun 2016
... We start by establishing a more general translation. We show that for every k ≥ 1 and every Σ11 (FOCk ) definable atom AQ , we have FOk (AQ ) ≤ Σ11 (FOCk ). Note that strictly speaking FOk (AQ ) uses only one atom AQ instead of a finite collection A of atoms, but our proof below generalizes directly ...
... We start by establishing a more general translation. We show that for every k ≥ 1 and every Σ11 (FOCk ) definable atom AQ , we have FOk (AQ ) ≤ Σ11 (FOCk ). Note that strictly speaking FOk (AQ ) uses only one atom AQ instead of a finite collection A of atoms, but our proof below generalizes directly ...
Curry`s Paradox. An Argument for Trivialism
... overview of criticism see Berto 2007, part IV). In this paper we will not discuss the crucial problem concerning the acceptance of a dialetheia. Rather, we will focus on the following claims by Priest: (i) The presence of dialetheiae does not entail trivialism: a) a contradictory theory may not be e ...
... overview of criticism see Berto 2007, part IV). In this paper we will not discuss the crucial problem concerning the acceptance of a dialetheia. Rather, we will focus on the following claims by Priest: (i) The presence of dialetheiae does not entail trivialism: a) a contradictory theory may not be e ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... as least fixpoint of a logic program, it has been due to an excess of information in the program (possibly erroneous information). As a result, rejecting noncoherent interpretations seems convenient as well. An important remark is that coherence can be interpreted with an empirical sense and that th ...
... as least fixpoint of a logic program, it has been due to an excess of information in the program (possibly erroneous information). As a result, rejecting noncoherent interpretations seems convenient as well. An important remark is that coherence can be interpreted with an empirical sense and that th ...
PARADOX AND INTUITION
... mean that we at the same time establish any structure-preserving correspondence between the objects from the initial model and arithmetical objects. Hilary Putnam’s paper Models and Reality (Putnam 1980) has become one of the most frequently quoted works discussing problems connected with determinac ...
... mean that we at the same time establish any structure-preserving correspondence between the objects from the initial model and arithmetical objects. Hilary Putnam’s paper Models and Reality (Putnam 1980) has become one of the most frequently quoted works discussing problems connected with determinac ...
Using linear logic to reason about sequent systems
... Stating and proving that atomically closed sequents are complete. Of the various proposals for non-commutative variants of classical linear logic: it would be interesting to see if these can be used to capture non-commutative object-level logics in a manner done here. To deal with admissibility of i ...
... Stating and proving that atomically closed sequents are complete. Of the various proposals for non-commutative variants of classical linear logic: it would be interesting to see if these can be used to capture non-commutative object-level logics in a manner done here. To deal with admissibility of i ...
A Nonstandard Approach to the. Logical Omniscience Problem
... What about logical omniscience? Notice that notions like "validity" and "logical consequence" (which played a prominent part in our informal description of logical omniscience) are not absolute notions; their formal definitions depend on how truth is defined and on the class of worlds being consider ...
... What about logical omniscience? Notice that notions like "validity" and "logical consequence" (which played a prominent part in our informal description of logical omniscience) are not absolute notions; their formal definitions depend on how truth is defined and on the class of worlds being consider ...
Homework #3 - Jonathan Livengood
... 1. Translate the following argument into our formal language and then use truth tables to determine whether the argument is valid or invalid. If the TV remote isn’t working, then John has to change channels manually. John has to change channels manually. The TV remote isn’t working. 2. Translate the ...
... 1. Translate the following argument into our formal language and then use truth tables to determine whether the argument is valid or invalid. If the TV remote isn’t working, then John has to change channels manually. John has to change channels manually. The TV remote isn’t working. 2. Translate the ...
Propositional Logic What is logic? Propositions Negation
... • This is a contentious question! We will play it safe, and stick to: – “The systematic use of symbolic techniques and mathematical methods to determine the forms of valid deductive argument.” [Shorter Oxford Dictionary]. ...
... • This is a contentious question! We will play it safe, and stick to: – “The systematic use of symbolic techniques and mathematical methods to determine the forms of valid deductive argument.” [Shorter Oxford Dictionary]. ...