The Taming of the (X)OR
... satz [Li99] degraded as soon as formulae containing exclusive-or appeared in the original formulation. Thus, solving real crypto-problems with CNF-provers looks unlikely. A similar situation is found in circuit verification where the usage of successful BDD-packages [BRB90] has proven to be utterly ...
... satz [Li99] degraded as soon as formulae containing exclusive-or appeared in the original formulation. Thus, solving real crypto-problems with CNF-provers looks unlikely. A similar situation is found in circuit verification where the usage of successful BDD-packages [BRB90] has proven to be utterly ...
Modal Logic for Artificial Intelligence
... Some rules are very simple: if you can prove ϕ and you can prove ψ, then you can also prove their conjunction ϕ ∧ ψ. Other rules are more complicated. For example, the only way to ‘eliminate’ the disjunction ϕ ∨ ψ is by proving, first that ϕ ∨ ψ, and second, that some conclusion χ can be proven both ...
... Some rules are very simple: if you can prove ϕ and you can prove ψ, then you can also prove their conjunction ϕ ∧ ψ. Other rules are more complicated. For example, the only way to ‘eliminate’ the disjunction ϕ ∨ ψ is by proving, first that ϕ ∨ ψ, and second, that some conclusion χ can be proven both ...
Factoring Out the Impossibility of Logical Aggregation
... individual whatever the pro…le - if and only if it is nonconstant and satis…es independence of irrelevant alternatives (IIA). In the logical context, the latter condition says that if a formula enters the judgment sets of exactly the same individuals in two pro…les, it is a member of either both or ...
... individual whatever the pro…le - if and only if it is nonconstant and satis…es independence of irrelevant alternatives (IIA). In the logical context, the latter condition says that if a formula enters the judgment sets of exactly the same individuals in two pro…les, it is a member of either both or ...
A Note on the Relation between Inflationary Fixpoints and Least
... formulas. It turns out that combining first-order logic with the ability to nest and complement fixpoint operators is powerful enough so that every formula of inflationary fixpoint logic is equivalent to a formula using least fixpoints of formulas positive in their fixpoint variable. This was first ...
... formulas. It turns out that combining first-order logic with the ability to nest and complement fixpoint operators is powerful enough so that every formula of inflationary fixpoint logic is equivalent to a formula using least fixpoints of formulas positive in their fixpoint variable. This was first ...
Frege`s Other Program
... that, as far as is currently known, it is too weak as it does not entail Peano Arithmetic, but only weaker systems, in particular Robinson’s Q. Alternatively, unlike the approach just described, we can break the bond of logicism and extensionalism, rejecting one while maintaining the other. The most ...
... that, as far as is currently known, it is too weak as it does not entail Peano Arithmetic, but only weaker systems, in particular Robinson’s Q. Alternatively, unlike the approach just described, we can break the bond of logicism and extensionalism, rejecting one while maintaining the other. The most ...
HPL-2008 - HP Labs
... represents a principal E in the role F , toegether with resources R. Note that the role F is itself represented by a process and that it is intended to have fewer abilities than E. Along with this construct comes a logical modality, given by the following forcing definition: R, G |= {E}φ iff ∃ F s.t ...
... represents a principal E in the role F , toegether with resources R. Note that the role F is itself represented by a process and that it is intended to have fewer abilities than E. Along with this construct comes a logical modality, given by the following forcing definition: R, G |= {E}φ iff ∃ F s.t ...
Relevant deduction
... rest on certain irrelevant deductions, which are, although mathematically valid, nonsensical and often enough harmful in applied arguments. Moreover the reason why these deductions are inappropriate as applied arguments is itself a logical one and can be defined within the framework of the underlyin ...
... rest on certain irrelevant deductions, which are, although mathematically valid, nonsensical and often enough harmful in applied arguments. Moreover the reason why these deductions are inappropriate as applied arguments is itself a logical one and can be defined within the framework of the underlyin ...
CS 399: Constructive Logic Final Exam (Sample Solution) Name Instructions
... once. If we relax this restriction and say that each hypothesis must be used at most once, we obtain what is called affine logic. The hypothesis rules become: Γ; ∆, A ` A ...
... once. If we relax this restriction and say that each hypothesis must be used at most once, we obtain what is called affine logic. The hypothesis rules become: Γ; ∆, A ` A ...
many-valued logics - University of Sydney
... 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} |= p ∨ q). Classical logic is then the language just introduced together with ei ...
... 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} |= p ∨ q). Classical logic is then the language just introduced together with ei ...
Subintuitionistic Logics with Kripke Semantics
... In 1981, A. Visser [7] had already introduced Basic logic (BPC), an extension of F with truth preservation, in the natural deduction form, and proved completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BP ...
... In 1981, A. Visser [7] had already introduced Basic logic (BPC), an extension of F with truth preservation, in the natural deduction form, and proved completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BP ...
Intuitionistic and Modal Logic
... with logicism, Frege’s idea that mathematics is no more than logic, since mathematics can be reduced to it, a view supported by Russell (not a Platonist) at the time. • Formalism. Most famous modern representative: Hilbert. View that there are no mathematical objects, no mathematical truths, just fo ...
... with logicism, Frege’s idea that mathematics is no more than logic, since mathematics can be reduced to it, a view supported by Russell (not a Platonist) at the time. • Formalism. Most famous modern representative: Hilbert. View that there are no mathematical objects, no mathematical truths, just fo ...
Chapter 2 Propositional Logic
... is studied by philosophers, mathematicians and computer scientists. Logic appears in different areas of computer science, such as programming, circuits, artificial intelligence and databases. It is useful to represent knowledge precisely and to help extract information. This last sentence may not be ...
... is studied by philosophers, mathematicians and computer scientists. Logic appears in different areas of computer science, such as programming, circuits, artificial intelligence and databases. It is useful to represent knowledge precisely and to help extract information. This last sentence may not be ...
Lecture 2
... The value of a Boolean expression is its state which can be either true or false. State is a list of variables with associated values. pq is satisfiable because it is satisfied if (p, true) is in state S1. It is not valid in the state S2: (p,false), (q,false). pp p is valid in every state. ...
... The value of a Boolean expression is its state which can be either true or false. State is a list of variables with associated values. pq is satisfiable because it is satisfied if (p, true) is in state S1. It is not valid in the state S2: (p,false), (q,false). pp p is valid in every state. ...
From p
... Propositional Logic Propositional logic is a good vehicle to introduce basic properties of logic. It does not provide means to determine the validity (truth or false) of atomic statements. Instead, it allows you to evaluate the validity of compound statements given the validity of its atomic compone ...
... Propositional Logic Propositional logic is a good vehicle to introduce basic properties of logic. It does not provide means to determine the validity (truth or false) of atomic statements. Instead, it allows you to evaluate the validity of compound statements given the validity of its atomic compone ...
Teach Yourself Logic 2017: A Study Guide
... courses, or are none at all? It seems then that many beginning graduate students in philosophy – if they are not to be quite dismally uneducated in logic and therefore cut off from working in some of the most exciting areas of their discipline – will need to teach themselves from books, either solo ...
... courses, or are none at all? It seems then that many beginning graduate students in philosophy – if they are not to be quite dismally uneducated in logic and therefore cut off from working in some of the most exciting areas of their discipline – will need to teach themselves from books, either solo ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... generalization of the Gelfond-Lifschitz reduct. As our interest in this work is on the notion of coherence, our natural environment is that of extended residuated logic programs, that is, those which do not contain default negation. Note that, as our interpretations are defined on the set of literal ...
... generalization of the Gelfond-Lifschitz reduct. As our interest in this work is on the notion of coherence, our natural environment is that of extended residuated logic programs, that is, those which do not contain default negation. Note that, as our interpretations are defined on the set of literal ...
Teach Yourself Logic 2016: A Study Guide
... courses, or are none at all? It seems then that many beginning graduate students in philosophy – if they are not to be quite dismally uneducated in logic and therefore cut off from working in some of the most exciting areas of their discipline – will need to teach themselves from books, either solo ...
... courses, or are none at all? It seems then that many beginning graduate students in philosophy – if they are not to be quite dismally uneducated in logic and therefore cut off from working in some of the most exciting areas of their discipline – will need to teach themselves from books, either solo ...
Modal Consequence Relations
... Logic is generally defined as the science of reasoning. Mathematical logic is mainly concerned with forms of reasoning that lead from true premises to true conclusions. Thus we say that the argument from σ0 ; σ1 ; · · · ; σn−1 to δ is logically correct if whenever σi is true for all i < n, then so i ...
... Logic is generally defined as the science of reasoning. Mathematical logic is mainly concerned with forms of reasoning that lead from true premises to true conclusions. Thus we say that the argument from σ0 ; σ1 ; · · · ; σn−1 to δ is logically correct if whenever σi is true for all i < n, then so i ...
PDF
... theoremhood of wff’s is preserved by the inference rule (that is, if ` A and ` A → B, then ` B), all we need to show is that every axiom of PLi is a theorem of PLc . 1. A → (B → A). This is just an axiom schema for PLc . 2. A → (B → A ∧ B). A, B, A → ¬B, ¬B, ⊥ leads to A, B, A → ¬B `⊥. Applying the ...
... theoremhood of wff’s is preserved by the inference rule (that is, if ` A and ` A → B, then ` B), all we need to show is that every axiom of PLi is a theorem of PLc . 1. A → (B → A). This is just an axiom schema for PLc . 2. A → (B → A ∧ B). A, B, A → ¬B, ¬B, ⊥ leads to A, B, A → ¬B `⊥. Applying the ...
On Sets of Premises - Matematički Institut SANU
... ⊢ Gentzen writes → (which is more commonly used nowadays for the binary connective of implication; we use it below, as usual, for separating the sources and targets of arrows in categories), for A and B he uses Gothic letters, and for n and m Greek letters (see [6], Section I.2.3). The natural numbe ...
... ⊢ Gentzen writes → (which is more commonly used nowadays for the binary connective of implication; we use it below, as usual, for separating the sources and targets of arrows in categories), for A and B he uses Gothic letters, and for n and m Greek letters (see [6], Section I.2.3). The natural numbe ...
Default Reasoning in a Terminological Logic
... logic that combines the tools for describing taxonomic organizations of complex objects which are typical of TLs, the ability to describe default information which is typical of general nonmonotonic formalisms, and the incrementality in KB construction which is typical of MINEs. Such an endeavour co ...
... logic that combines the tools for describing taxonomic organizations of complex objects which are typical of TLs, the ability to describe default information which is typical of general nonmonotonic formalisms, and the incrementality in KB construction which is typical of MINEs. Such an endeavour co ...
Modal Reasoning
... Basic modal logic satisfies the finite model property or FMP: Theorem 5.2. Every satisfiable modal formula has a finite model. The finite model property doesn’t itself give decidability; we may still have to check all finite models– all infinitely many of them. A strengthened version of the finite ...
... Basic modal logic satisfies the finite model property or FMP: Theorem 5.2. Every satisfiable modal formula has a finite model. The finite model property doesn’t itself give decidability; we may still have to check all finite models– all infinitely many of them. A strengthened version of the finite ...
Geometric Modal Logic
... claims incomparably more than saying that this proposition is simply necessary. Speaking of something as ‘possibly possible’, we implicitly let the variation system itself vary, we shift from a given system of possibility into a frame inside which this system is only one among others, and we say tha ...
... claims incomparably more than saying that this proposition is simply necessary. Speaking of something as ‘possibly possible’, we implicitly let the variation system itself vary, we shift from a given system of possibility into a frame inside which this system is only one among others, and we say tha ...
Kripke completeness revisited
... excessive use of “intuitive” arguments on the geometry of tableau proofs. Kaplan suggested a different, more “mathematical” and more elegant approach based on an adaptation of Henkin’s completeness proof for classical logic. Indeed, a Henkin-style completeness proof for S5 had already been published ...
... excessive use of “intuitive” arguments on the geometry of tableau proofs. Kaplan suggested a different, more “mathematical” and more elegant approach based on an adaptation of Henkin’s completeness proof for classical logic. Indeed, a Henkin-style completeness proof for S5 had already been published ...