Clausal Connection-Based Theorem Proving in
... during the proof search in an easy way. The transformation of a prefixed matrix to clausal form is done like for classical logic. Note that we apply the substitution σ< to the clausal matrix, i.e. to terms and prefixes of atomic formulae. Definition 11 (Prefixed Clausal Matrix). Let < be a tree orde ...
... during the proof search in an easy way. The transformation of a prefixed matrix to clausal form is done like for classical logic. Note that we apply the substitution σ< to the clausal matrix, i.e. to terms and prefixes of atomic formulae. Definition 11 (Prefixed Clausal Matrix). Let < be a tree orde ...
One-dimensional Fragment of First-order Logic
... in [9], [15]. It was subsequently proved to be NEXPTIME-complete in [16]. Research concerning decidability of variants of two-variable logic has been very active in recent years. Recent articles in the field include for example [3] [5], [11], [17], and several others. The recent research efforts hav ...
... in [9], [15]. It was subsequently proved to be NEXPTIME-complete in [16]. Research concerning decidability of variants of two-variable logic has been very active in recent years. Recent articles in the field include for example [3] [5], [11], [17], and several others. The recent research efforts hav ...
Ultrasheaves
... Grothendieck gave a more general definition of sheaves by replacing the partially ordered collection of open subsets of a topological space by objects in a category C, in which some suitable families of maps Ui → X (for i ∈ I) form “covers” of objects X in C. For such a “Grothendieck topology” a she ...
... Grothendieck gave a more general definition of sheaves by replacing the partially ordered collection of open subsets of a topological space by objects in a category C, in which some suitable families of maps Ui → X (for i ∈ I) form “covers” of objects X in C. For such a “Grothendieck topology” a she ...
Argument construction and reinstatement in logics for
... reasoning, but it has also been applied to the problem of reasoning with legal precedents (Prakken and Sartor 1998). The second system considered here is presented by Kowalski and Toni (1996) as an application of the theory set out in Bandarenko et al. (1997). This system, which will be referred to ...
... reasoning, but it has also been applied to the problem of reasoning with legal precedents (Prakken and Sartor 1998). The second system considered here is presented by Kowalski and Toni (1996) as an application of the theory set out in Bandarenko et al. (1997). This system, which will be referred to ...
Essentials Of Symbolic Logic
... Inference is a thought process in which one proposition is arrived at on the basis of other proposition or propositions. Corresponding to every inference, there is an argument. An argument consists of a group of propositions in which one proposition is claimed to follow from the other propositions p ...
... Inference is a thought process in which one proposition is arrived at on the basis of other proposition or propositions. Corresponding to every inference, there is an argument. An argument consists of a group of propositions in which one proposition is claimed to follow from the other propositions p ...
ND for predicate logic ∀-elimination, first attempt Variable capture
... suppose that Γ |= A and M |= Γ. To see that M |= ∀x.A, we need to show that M [a/x] |= A for all a ∈ U . Because M |= Γ and x does not occur freely in Γ, we have M [a/x] |= Γ. Because Γ |= A, we get M [a/x] |= A. ...
... suppose that Γ |= A and M |= Γ. To see that M |= ∀x.A, we need to show that M [a/x] |= A for all a ∈ U . Because M |= Γ and x does not occur freely in Γ, we have M [a/x] |= Γ. Because Γ |= A, we get M [a/x] |= A. ...
Completeness and Decidability of a Fragment of Duration Calculus
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
... Duration Calculus (DC) was introduced by Zhou, Hoare and Ravn in 1991 as a logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a re ...
A Concise Introduction to Mathematical Logic
... The first was the axiomatization of set theory in various ways. The most important approaches are those of Zermelo (improved by Fraenkel and von Neumann) and the theory of types by Whitehead and Russell. The latter was to become the sole remnant of Frege’s attempt to reduce mathematics to logic. Ins ...
... The first was the axiomatization of set theory in various ways. The most important approaches are those of Zermelo (improved by Fraenkel and von Neumann) and the theory of types by Whitehead and Russell. The latter was to become the sole remnant of Frege’s attempt to reduce mathematics to logic. Ins ...
LINEAR LOGIC AS A FRAMEWORK FOR SPECIFYING SEQUENT
... Such meta logics or logical frameworks have generally been based on intuitionistic logic in which quantification at (non-predicate) higher-order types is available. Identifying a framework that allows the specification of a wide range of logics has proved to be most practical since a single implemen ...
... Such meta logics or logical frameworks have generally been based on intuitionistic logic in which quantification at (non-predicate) higher-order types is available. Identifying a framework that allows the specification of a wide range of logics has proved to be most practical since a single implemen ...
Primitive Recursive Arithmetic and its Role in the Foundations of
... higher-order equality that I advocated above in connection with the theory T and say something about its remedy. The relation s =A,B t or for simplicity s = t of extensional equality between objects respectively of type A and type B (unlike that of intensional equality) is definable in the theory it ...
... higher-order equality that I advocated above in connection with the theory T and say something about its remedy. The relation s =A,B t or for simplicity s = t of extensional equality between objects respectively of type A and type B (unlike that of intensional equality) is definable in the theory it ...
Reasoning about Action and Change
... was just this ‘global nature’ that originally made nonmonotonic approaches so appealing. This is best captured in the so-called persistence assumption which states that all facts usually persist to hold after the performance of all actions, if not stated otherwise. To the best of our knowledge Georg ...
... was just this ‘global nature’ that originally made nonmonotonic approaches so appealing. This is best captured in the so-called persistence assumption which states that all facts usually persist to hold after the performance of all actions, if not stated otherwise. To the best of our knowledge Georg ...
Modalities in the Realm of Questions: Axiomatizing Inquisitive
... (vi) If ϕ ∈ L! ∪ L? and a ∈ A, then Ka ϕ ∈ L! (vii) If ϕ ∈ L! ∪ L? and a ∈ A, then Ea ϕ ∈ L! Importantly, conjunction, implication, and the modalities are allowed to apply to interrogatives as well as declaratives. Also, notice how the two syntactic categories are intertwined: from a sequence of dec ...
... (vi) If ϕ ∈ L! ∪ L? and a ∈ A, then Ka ϕ ∈ L! (vii) If ϕ ∈ L! ∪ L? and a ∈ A, then Ea ϕ ∈ L! Importantly, conjunction, implication, and the modalities are allowed to apply to interrogatives as well as declaratives. Also, notice how the two syntactic categories are intertwined: from a sequence of dec ...
The Foundations: Logic and Proofs
... A lemma is a ‘helping theorem’ or a result which is needed to prove a theorem. A corollary is a result which follows directly from a theorem. Less important theorems are sometimes called propositions. A conjecture is a statement that is being proposed to be true. Once a proof of a ...
... A lemma is a ‘helping theorem’ or a result which is needed to prove a theorem. A corollary is a result which follows directly from a theorem. Less important theorems are sometimes called propositions. A conjecture is a statement that is being proposed to be true. Once a proof of a ...
Glivenko sequent classes in the light of structural proof theory
... in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been ...
... in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been ...
Reasoning without Contradiction
... view of negation as cancellation that is, in my view, attractive and natural. The rule &-E is valid except where the conjunction contains a wff and its negation. Reductio ad Absurdum is a healthy rule of inference unaffected by the claim that contradictions say nothing. Informally, the rule is: if y ...
... view of negation as cancellation that is, in my view, attractive and natural. The rule &-E is valid except where the conjunction contains a wff and its negation. Reductio ad Absurdum is a healthy rule of inference unaffected by the claim that contradictions say nothing. Informally, the rule is: if y ...
On the use of fuzzy stable models for inconsistent classical logic
... The existence of stable models can be guaranteed by simply imposing conditions on the underlying residuated lattice: Theorem 3. Let L ≡ ([0, 1], ≤, ∗, ←, ¬) be a residuated lattice with negation. If ∗ and ¬ are continuous operators, then every finite normal program P defined over L has at least a st ...
... The existence of stable models can be guaranteed by simply imposing conditions on the underlying residuated lattice: Theorem 3. Let L ≡ ([0, 1], ≤, ∗, ←, ¬) be a residuated lattice with negation. If ∗ and ¬ are continuous operators, then every finite normal program P defined over L has at least a st ...
Chapter 15 Logic Name Date Objective: Students will use
... Chapter 15 Logic Name _______________________________________________ Date __________________ Objective: Students will use propositions to create truth tables and logical equivalences in order to draw logical conclusions Proposition Propositions are statements that may be true or false. Propositions ...
... Chapter 15 Logic Name _______________________________________________ Date __________________ Objective: Students will use propositions to create truth tables and logical equivalences in order to draw logical conclusions Proposition Propositions are statements that may be true or false. Propositions ...
Logic and Computation Lecture notes Jeremy Avigad Assistant Professor, Philosophy
... concepts, is the demonstration that in fact, in the case of first-order predicate logic, they coincide. In other words, for every first order sentence ϕ and set of sentences Γ, Γ proves ϕ if and only if Γ logically implies ϕ. The forwards direction is known as “soundness”: it asserts that our proof ...
... concepts, is the demonstration that in fact, in the case of first-order predicate logic, they coincide. In other words, for every first order sentence ϕ and set of sentences Γ, Γ proves ϕ if and only if Γ logically implies ϕ. The forwards direction is known as “soundness”: it asserts that our proof ...
Notions of locality and their logical characterizations over nite
... Let be an isomorphism type of a structure in the language 1 ( extended with one constant). A point a in a structure A d-realizes , written as d (A; a) = , if NdA (a) is of isomorphism type . By #d [A; ] we denote the number of elements of A which d-realize , that is, the cardinality o ...
... Let be an isomorphism type of a structure in the language 1 ( extended with one constant). A point a in a structure A d-realizes , written as d (A; a) = , if NdA (a) is of isomorphism type . By #d [A; ] we denote the number of elements of A which d-realize , that is, the cardinality o ...
We can only see a short distance ahead, but we can see plenty
... deduction systems, equation calculi, etc. Its first great contribution, of course, was the formalization of the notion of a function being computable by an algorithm and the discovery of many remarkable instances in all branches of mathematics of the dividing line between computability and noncomput ...
... deduction systems, equation calculi, etc. Its first great contribution, of course, was the formalization of the notion of a function being computable by an algorithm and the discovery of many remarkable instances in all branches of mathematics of the dividing line between computability and noncomput ...
duality of quantifiers ¬8xA(x) 9x¬A(x) ¬9xA(x) 8x¬A(x)
... Wolves, foxes, birds, caterpillars, and snails are animals, and there are some of each of them. Also there are some grains, and grains are plants. Every animal either likes to eat all plants or all animals much smaller than itself that like to eat some plants. Caterpillars and snails are much smalle ...
... Wolves, foxes, birds, caterpillars, and snails are animals, and there are some of each of them. Also there are some grains, and grains are plants. Every animal either likes to eat all plants or all animals much smaller than itself that like to eat some plants. Caterpillars and snails are much smalle ...
Dynamic logic of propositional assignments
... indicates the possible execution of an atomic program from one state to another. The modal formula hπiϕ is true at state s if there is an execution of π from s leading to a state satisfying ϕ. The basic dynamic logic is Propositional Dynamic Logic (PDL). Its atomic programs are abstract: they are ju ...
... indicates the possible execution of an atomic program from one state to another. The modal formula hπiϕ is true at state s if there is an execution of π from s leading to a state satisfying ϕ. The basic dynamic logic is Propositional Dynamic Logic (PDL). Its atomic programs are abstract: they are ju ...
Master Thesis - Yoichi Hirai
... 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 characterisation of waitfree computation given by Herlihy, Shavit, Saks and Zaharogl ...
... 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 characterisation of waitfree computation given by Herlihy, Shavit, Saks and Zaharogl ...
Recall... Venn Diagrams Disjunctive normal form Disjunctive normal
... CS304 — Lecture 4: Functional Completeness and Normal Forms ...
... CS304 — Lecture 4: Functional Completeness and Normal Forms ...
Suszko`s Thesis, Inferential Many-Valuedness, and the
... T>+ U V~ is the set of all algebraic values available. The first condition is imposed by Malinowski [24], [25], [26], [27] and Gottwald [22], and the second condition may be used to define systems of paraconsistent logic.4 In order to provide a counterexample to Suszko's Thesis, Malinowski defined t ...
... T>+ U V~ is the set of all algebraic values available. The first condition is imposed by Malinowski [24], [25], [26], [27] and Gottwald [22], and the second condition may be used to define systems of paraconsistent logic.4 In order to provide a counterexample to Suszko's Thesis, Malinowski defined t ...