Seventy-five problems for testing automatic
... ATPers in mind that the following list is offered. None of these problems will be the sort whose solution is, of itself, of any mathematical or logical interest. Such ‘open problems’ are regularly published in the Newsletter of the Association for Automated Reasoning. Most (but not all) of my proble ...
... ATPers in mind that the following list is offered. None of these problems will be the sort whose solution is, of itself, of any mathematical or logical interest. Such ‘open problems’ are regularly published in the Newsletter of the Association for Automated Reasoning. Most (but not all) of my proble ...
A Well-Founded Semantics for Logic Programs with Abstract
... sets, well-founded models (Van Gelder, Ross, and Schlipf 1991) have been found to be very useful as well. First, computing the well-founded model of a normal logic program is tractable. This compares to the NP-completeness of computing an answer set. Secondly, the well-founded model of a normal logi ...
... sets, well-founded models (Van Gelder, Ross, and Schlipf 1991) have been found to be very useful as well. First, computing the well-founded model of a normal logic program is tractable. This compares to the NP-completeness of computing an answer set. Secondly, the well-founded model of a normal logi ...
Truth Value Solver: A Software for Calculating Truth Value with
... Classical logic is used to deal with certain information, but on the other hand multi-valued logic is used to deal with uncertain information. Multi-valued logic was proposed by Lukasiewicz and extended by many researchers such as Adams [1], Shafer [9], Zadeh [10], etc. Multi-valued logic has been w ...
... Classical logic is used to deal with certain information, but on the other hand multi-valued logic is used to deal with uncertain information. Multi-valued logic was proposed by Lukasiewicz and extended by many researchers such as Adams [1], Shafer [9], Zadeh [10], etc. Multi-valued logic has been w ...
Truth, Conservativeness and Provability
... That is, provided that we take (F) for granted. Indeed, one could still wonder about the exact sense, in which the reflective axioms express some of the content of (D) —what does ‘express’ mean here? It is an intricate question, which I am not going to discuss in this paper—I will just concentrate o ...
... That is, provided that we take (F) for granted. Indeed, one could still wonder about the exact sense, in which the reflective axioms express some of the content of (D) —what does ‘express’ mean here? It is an intricate question, which I am not going to discuss in this paper—I will just concentrate o ...
An Introduction to Prolog Programming
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
Intuitionistic modal logic made explicit
... Justification logics are explicit modal logics in the sense that they unfold the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics f ...
... Justification logics are explicit modal logics in the sense that they unfold the -modality in families of so-called justification terms. Instead of formulas A, meaning that A is known, justification logics include formulas t : A, meaning that A is known for reason t. Artemov’s original semantics f ...
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 ...
1992-Ideal Introspective Belief
... atoms (of the form ~Lc$) are not soundly derivable from the premises alone. For example, consider the premise set {lLp > q,p V q}. We would like since there is no reasonable way of to conclude ‘Lp, coming to believe p. But an inference rule that would allow us to conclude 1Lp would have to take into ...
... atoms (of the form ~Lc$) are not soundly derivable from the premises alone. For example, consider the premise set {lLp > q,p V q}. We would like since there is no reasonable way of to conclude ‘Lp, coming to believe p. But an inference rule that would allow us to conclude 1Lp would have to take into ...
article in press - School of Computer Science
... We generalise the result of [H. Ganzinger, C. Meyer, M. Veanes, The two-variable guarded fragment with transitive relations, in: Proc. 14th IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 24–34] on decidability of the two variable monadic guarded fragment of first ...
... We generalise the result of [H. Ganzinger, C. Meyer, M. Veanes, The two-variable guarded fragment with transitive relations, in: Proc. 14th IEEE Symposium on Logic in Computer Science, IEEE Computer Society Press, 1999, pp. 24–34] on decidability of the two variable monadic guarded fragment of first ...
Chapter 1 - National Taiwan University
... Please read Section §1.7 of your textbook. It takes effort to know how to write correct proofs. When you read the text, please try to understand how the statements are proved, instead of what the statements are proving. Sometimes, we may make a statement without knowing whether it is true or not. Suc ...
... Please read Section §1.7 of your textbook. It takes effort to know how to write correct proofs. When you read the text, please try to understand how the statements are proved, instead of what the statements are proving. Sometimes, we may make a statement without knowing whether it is true or not. Suc ...
Intuitionistic Logic
... Negation is also defined by means of proofs: p : ¬A says that each proof a of A can be converted by the construction p into a proof of an absurdity, say 0 = 1. A proof of ¬A thus tells us that A has no proof! The most interesting propositional connective is the implication. The classical solution, i ...
... Negation is also defined by means of proofs: p : ¬A says that each proof a of A can be converted by the construction p into a proof of an absurdity, say 0 = 1. A proof of ¬A thus tells us that A has no proof! The most interesting propositional connective is the implication. The classical solution, i ...
INTERPLAYS OF KNOWLEDGE AND NON
... complete with respect to some given class of Kripke frames, in the same way normal modal logics are (see [14]). Since the work developed in [7], a lot of important results have appeared in the domain of epistemic logics. Many authors have studied these systems especially concerning applications in C ...
... complete with respect to some given class of Kripke frames, in the same way normal modal logics are (see [14]). Since the work developed in [7], a lot of important results have appeared in the domain of epistemic logics. Many authors have studied these systems especially concerning applications in C ...
HOARE`S LOGIC AND PEANO`S ARITHMETIC
... As far as the proof theory ~)f a data type axioma.tisation T is concerned, the semantics of the specification is ModiTL Before looking at Peano arithmetic and the special problems at hand, consider the algebraic specification methods for data types where one invariably has a particzdor semantic mode ...
... As far as the proof theory ~)f a data type axioma.tisation T is concerned, the semantics of the specification is ModiTL Before looking at Peano arithmetic and the special problems at hand, consider the algebraic specification methods for data types where one invariably has a particzdor semantic mode ...
A Proof Theory for Generic Judgments: An extended abstract
... need to discover invariants. Another more intensional approach, however, involves introducing a new, generic variable, say, c : γ, that has not been introduced before in the proof, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are calle ...
... need to discover invariants. Another more intensional approach, however, involves introducing a new, generic variable, say, c : γ, that has not been introduced before in the proof, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are calle ...
The disjunction introduction rule: Syntactic and semantics
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
Taming method in modal logic and mosaic method in temporal logic
... Using previous lemma (1) we can prove: ...
... Using previous lemma (1) we can prove: ...
PPT
... Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19th and early 20th century. George Boole (Boolean algebra) introduced Mathematica ...
... Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19th and early 20th century. George Boole (Boolean algebra) introduced Mathematica ...
Basic Logic and Fregean Set Theory - MSCS
... S4 (and BQC with K4). The theories FQC and IQC are relatively inconsistent. In Section 4 we introduce Fregean set theories over BQC. Over F the traditional proof of the Russell Paradox turns into a derivation of the additional axiom schema of FQC. This result illustrates that logic sometimes doesn’t ...
... S4 (and BQC with K4). The theories FQC and IQC are relatively inconsistent. In Section 4 we introduce Fregean set theories over BQC. Over F the traditional proof of the Russell Paradox turns into a derivation of the additional axiom schema of FQC. This result illustrates that logic sometimes doesn’t ...
Natural deduction for predicate logic
... This suggests that to prove a formula of the form ∀xφ, we can prove φ with some arbitrary but fresh variable x0 substituted for x. That is, we want to prove the formula φ[x0 /x]. On the previous slide, we used n as a fresh variable, but in our formal proofs, we adopt the convention of using subscri ...
... This suggests that to prove a formula of the form ∀xφ, we can prove φ with some arbitrary but fresh variable x0 substituted for x. That is, we want to prove the formula φ[x0 /x]. On the previous slide, we used n as a fresh variable, but in our formal proofs, we adopt the convention of using subscri ...
THE MODAL LOGIC OF INNER MODELS §1. Introduction. In [10, 11
... finite sequence (z0 , ..., zn ) such that z0 = x, zn = y and for any i < n, we have either zi ≤ zi+1 or zi+1 ≤ zi ; we say that a cluster C ⊆ P is maximal if for v ∈ C and w ≥ v, we have w ∈ C; and we say that a cluster C ⊆ P is the top cluster if for any v ∈ P and w ∈ C, we have that v ≤ w. Clearly ...
... finite sequence (z0 , ..., zn ) such that z0 = x, zn = y and for any i < n, we have either zi ≤ zi+1 or zi+1 ≤ zi ; we say that a cluster C ⊆ P is maximal if for v ∈ C and w ≥ v, we have w ∈ C; and we say that a cluster C ⊆ P is the top cluster if for any v ∈ P and w ∈ C, we have that v ≤ w. Clearly ...
Handling Exceptions in nonmonotonic reasoning
... of this very simple defeasible axiomatic basis. Some argue that only g2 should be derived and others argue for splitting the expansions: one applying g1 and other applying g2 . Translating the example in Reiter’s default logic (according with Def. 2.4), it applies only the second rule, and this has ...
... of this very simple defeasible axiomatic basis. Some argue that only g2 should be derived and others argue for splitting the expansions: one applying g1 and other applying g2 . Translating the example in Reiter’s default logic (according with Def. 2.4), it applies only the second rule, and this has ...
Is the Liar Sentence Both True and False? - NYU Philosophy
... with acceptance. To a Þrst approximation anyway, accepting A is having a high degree of belief in it; say a degree of belief over a certain threshold T , which may depend on context but must be greater than 12 . (Degrees of belief are assumed to be real numbers in the interval [0, 1].) To the same ...
... with acceptance. To a Þrst approximation anyway, accepting A is having a high degree of belief in it; say a degree of belief over a certain threshold T , which may depend on context but must be greater than 12 . (Degrees of belief are assumed to be real numbers in the interval [0, 1].) To the same ...
Properties of Independently Axiomatizable Bimodal Logics
... L := L ∩ L are normal mono-modal logics. Conversely, given two mono-modal logics M, N we can form the fusion M ⊗ N which is the least bimodal logic containing both M and N where the modal operator of M is translated as 2 and the operator of N by . If L = L2 ⊗ L we call L independently axiomatiza ...
... L := L ∩ L are normal mono-modal logics. Conversely, given two mono-modal logics M, N we can form the fusion M ⊗ N which is the least bimodal logic containing both M and N where the modal operator of M is translated as 2 and the operator of N by . If L = L2 ⊗ L we call L independently axiomatiza ...