The Foundations: Logic and Proofs - UTH e
... The Connective Or in English In English “or” has two distinct meanings. “Inclusive Or” - In the sentence “Students who have taken CS202 or Math120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction ...
... The Connective Or in English In English “or” has two distinct meanings. “Inclusive Or” - In the sentence “Students who have taken CS202 or Math120 may take this class,” we assume that students need to have taken one of the prerequisites, but may have taken both. This is the meaning of disjunction ...
P Q
... Truth symbols true and false (reserved) Constant symbols are symbol expressions having the first character lowercase Variable symbols are symbol expressions beginning with an uppercase character Function symbols are symbol expressions having the first character lowercase ...
... Truth symbols true and false (reserved) Constant symbols are symbol expressions having the first character lowercase Variable symbols are symbol expressions beginning with an uppercase character Function symbols are symbol expressions having the first character lowercase ...
On the Finite Model Property in Order-Sorted Logic
... model-finder that deduces sort information for unsorted problems and, under certain conditions, can bound the size of domains for certain sorts and improve the performance of the instantiation procedure. Order-sorting is not used, and there are restrictions on the use of equality. Momtahan [23] defi ...
... model-finder that deduces sort information for unsorted problems and, under certain conditions, can bound the size of domains for certain sorts and improve the performance of the instantiation procedure. Order-sorting is not used, and there are restrictions on the use of equality. Momtahan [23] defi ...
On Equivalent Transformations of Infinitary Formulas under the
... Our goal here is to develop methods for proving that pairs F , G of infinitary formulas have the same stable models. From the results of Pearce [7] and Ferraris [1] we know that in the case of grounded logic programs in the sense of Gelfond and Lifschitz [2] and, more generally, finite propositiona ...
... Our goal here is to develop methods for proving that pairs F , G of infinitary formulas have the same stable models. From the results of Pearce [7] and Ferraris [1] we know that in the case of grounded logic programs in the sense of Gelfond and Lifschitz [2] and, more generally, finite propositiona ...
Document
... quantifiers, predicates and logical connectives. A valid argument for predicate logic need not be a tautology. The meaning and the structure of the quantifiers and predicates determines the interpretation and the validity of the arguments Basic approach to prove arguments: ...
... quantifiers, predicates and logical connectives. A valid argument for predicate logic need not be a tautology. The meaning and the structure of the quantifiers and predicates determines the interpretation and the validity of the arguments Basic approach to prove arguments: ...
Robot Morality and Review of classical logic.
... Suppose your waiter tells you that you can have either rice pilaf or baked potato with your dinner. In such circumstances, he plainly does not mean either rice pilaf or baked potato or both. You have to choose. So this use of “or” doesn’t fit the definition of disjunction given above. ...
... Suppose your waiter tells you that you can have either rice pilaf or baked potato with your dinner. In such circumstances, he plainly does not mean either rice pilaf or baked potato or both. You have to choose. So this use of “or” doesn’t fit the definition of disjunction given above. ...
First-Order Queries over One Unary Function
... acyclic conjunctive queries [Yan81,PY99], or for full first-order queries on relational structures of bounded degree [See96,DG06,Lin06] or on tree-decomposable structures [FG01] (see also [FFG02]). A quasi-unary signature consists of one unary function and any number of monadic predicates. First-or ...
... acyclic conjunctive queries [Yan81,PY99], or for full first-order queries on relational structures of bounded degree [See96,DG06,Lin06] or on tree-decomposable structures [FG01] (see also [FFG02]). A quasi-unary signature consists of one unary function and any number of monadic predicates. First-or ...
Knowledge representation 1
... variable and in the other case a constant, substitute the constant for the variable, everywhere that that constant appears in the clause. ...
... variable and in the other case a constant, substitute the constant for the variable, everywhere that that constant appears in the clause. ...
Definability properties and the congruence closure
... in the vocabulary zw {E, V}. Add a new binary relation symbol R and consider the following sentence X(V) of type zu{E, R, V}, which belongs to L by regularity: Vx [( Vx -~ o-~,lRx,~) ^ ( Vx -~ o-2~ JRxr~)]. ...
... in the vocabulary zw {E, V}. Add a new binary relation symbol R and consider the following sentence X(V) of type zu{E, R, V}, which belongs to L by regularity: Vx [( Vx -~ o-~,lRx,~) ^ ( Vx -~ o-2~ JRxr~)]. ...
3.1.3 Subformulas
... Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ∈ S(F ). 3. If (G1 ◦ G2 ) ∈ S(F ) , then G1 , G2 ∈ S(F ). It will be shown in Exercise 3.4 that such a smallest set exis ...
... Definition 3.8 Let F be a propositional formula. The set of subformulas of F is the smallest set S(F ) satisfying the following conditions: 1. F ∈ S(F ). 2. If ¬G ∈ S(F ) , then G ∈ S(F ). 3. If (G1 ◦ G2 ) ∈ S(F ) , then G1 , G2 ∈ S(F ). It will be shown in Exercise 3.4 that such a smallest set exis ...
Logic seminar
... • Customarily, we represent “true” by T and “false” by F. • Furthermore, for convenience, we use an uppercase symbol or a string of uppercase symbols to denote a proposition. • For instance, we may denote the propositions as follows: – P: Snow is white, – Q: Sugar is a hydrocarbon, – R: Smith has a ...
... • Customarily, we represent “true” by T and “false” by F. • Furthermore, for convenience, we use an uppercase symbol or a string of uppercase symbols to denote a proposition. • For instance, we may denote the propositions as follows: – P: Snow is white, – Q: Sugar is a hydrocarbon, – R: Smith has a ...
A Simple and Practical Valuation Tree Calculus for First
... that the reader pays a closer attention to the example given there. ...
... that the reader pays a closer attention to the example given there. ...
Resources - CSE, IIT Bombay
... Version 2 Proof (contd.) Part 2: Suppose S’ is a finite unsatisfiable set of gr. instances of clauses in S - Every interpretation I of S contains an interpretation I’ of S’ - So, if I’ falsifies S’, then I must also falsify S’ - Since S’ is falsified by every interpretation I’, it must also be fals ...
... Version 2 Proof (contd.) Part 2: Suppose S’ is a finite unsatisfiable set of gr. instances of clauses in S - Every interpretation I of S contains an interpretation I’ of S’ - So, if I’ falsifies S’, then I must also falsify S’ - Since S’ is falsified by every interpretation I’, it must also be fals ...
A Uniform Proof Procedure for Classical and Non
... guide the search for a classical matrix-proof in normal-form. The resulting proof procedure is based on ideas originally developed for a generalized connection based proof method for intuitionistic logic [11]. It consists of a connection driven algorithm which checks the complementarity of all paths ...
... guide the search for a classical matrix-proof in normal-form. The resulting proof procedure is based on ideas originally developed for a generalized connection based proof method for intuitionistic logic [11]. It consists of a connection driven algorithm which checks the complementarity of all paths ...
Unification in Propositional Logic
... Unfortunately, however, there are usually infinitely many projective formulas in F (x) implying a given A ∈ F (x). Hence we need a refinement of the argument: bounded bisimulations will help. ...
... Unfortunately, however, there are usually infinitely many projective formulas in F (x) implying a given A ∈ F (x). Hence we need a refinement of the argument: bounded bisimulations will help. ...
Propositional and Predicate Logic - IX
... for each n-ary function symbol f of the language L. (iii) x1 = y1 ∧ · · · ∧ xn = yn → (R(x1 , . . . , xn ) → R(y1 , . . . , yn )) for each n-ary relation symbol R of the language L including =. A tableau proof from a theory T in a language L with equality is a tableau proof from T ∗ where T ∗ denote ...
... for each n-ary function symbol f of the language L. (iii) x1 = y1 ∧ · · · ∧ xn = yn → (R(x1 , . . . , xn ) → R(y1 , . . . , yn )) for each n-ary relation symbol R of the language L including =. A tableau proof from a theory T in a language L with equality is a tableau proof from T ∗ where T ∗ denote ...
Beginning Deductive Logic
... This is rather a deep and tricky question, as of course are relatives like: “What is physics?” and “What is economics?” and “What is art?”. Perhaps one has to be either brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to cr ...
... This is rather a deep and tricky question, as of course are relatives like: “What is physics?” and “What is economics?” and “What is art?”. Perhaps one has to be either brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to cr ...
An Independence Result For Intuitionistic Bounded Arithmetic
... For the definition of Kripke models of intuitionistic bounded arithmetic and basic results about them, see [M2] and [B2]. The general results on intuitionistic logic and arithmetic, and also Kripke models, can be found in [TD]. [MM] contains a study of weak fragments of first-order intuitionistic ar ...
... For the definition of Kripke models of intuitionistic bounded arithmetic and basic results about them, see [M2] and [B2]. The general results on intuitionistic logic and arithmetic, and also Kripke models, can be found in [TD]. [MM] contains a study of weak fragments of first-order intuitionistic ar ...
Lesson 12
... language in order to achieve an effective inference procedure that is sound and complete for a subset of First Order Predicate Logic. The notion of soundness and completeness is more generally applied than in logic. Whenever we create a knowledge based program we use the syntax of the knowledge repr ...
... language in order to achieve an effective inference procedure that is sound and complete for a subset of First Order Predicate Logic. The notion of soundness and completeness is more generally applied than in logic. Whenever we create a knowledge based program we use the syntax of the knowledge repr ...
A Proof Theory for Generic Judgments: An extended abstract
... The generic interpretation of quantifiers generally entails the extensional interpretation: this is a simple consequence of the cut-elimination theorem as follows. Assume that the sequent Γ −→ ∀x.B is proved using the introduction of ∀ on the right from the premise Γ −→ B[c/x], where c is an eigenva ...
... The generic interpretation of quantifiers generally entails the extensional interpretation: this is a simple consequence of the cut-elimination theorem as follows. Assume that the sequent Γ −→ ∀x.B is proved using the introduction of ∀ on the right from the premise Γ −→ B[c/x], where c is an eigenva ...
1 slide/page
... predicate symbols: ◦ for each constant symbol c, I(c) ∈ D ∗ Which domain element is Alice? ◦ for each unary predicate P , I(P ) is a predicate on domain D ∗ formally, I(P )(d) ∈ {true,false} for each d ∈ D ∗ Is Alice Tall? How about Bob? ◦ for each binary predicate Q, I(Q) is a predicate on D × D: ∗ ...
... predicate symbols: ◦ for each constant symbol c, I(c) ∈ D ∗ Which domain element is Alice? ◦ for each unary predicate P , I(P ) is a predicate on domain D ∗ formally, I(P )(d) ∈ {true,false} for each d ∈ D ∗ Is Alice Tall? How about Bob? ◦ for each binary predicate Q, I(Q) is a predicate on D × D: ∗ ...
Binary Decision Diagrams for First Order Predicate Logic
... Appendix B Proof: The transformation operators can be formulated as rewrite rules.l1 and l2 are predicates. l1 > l2 l1 (l2 ( x, y ), z ) l2 (l1 ( x, z ), l1 ( y, z )) ...
... Appendix B Proof: The transformation operators can be formulated as rewrite rules.l1 and l2 are predicates. l1 > l2 l1 (l2 ( x, y ), z ) l2 (l1 ( x, z ), l1 ( y, z )) ...
A course in Mathematical Logic
... Terms and formulas are interpreted in a model. Definition 8. (Definition of a model) Let L be a language. An L-model M is given by a set M of elements (called the universe of the model) and 1. For every function symbol f ∈ L of arity n, a function f M : M n → M ; 2. For every relation symbol R ∈ L o ...
... Terms and formulas are interpreted in a model. Definition 8. (Definition of a model) Let L be a language. An L-model M is given by a set M of elements (called the universe of the model) and 1. For every function symbol f ∈ L of arity n, a function f M : M n → M ; 2. For every relation symbol R ∈ L o ...