Independence logic and tuple existence atoms
... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...
... Definition R relation, ~x , ~y , ~z tuples of attributes. Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ). Huge literature on the topic; If ~x ~y ~z contains all attri ...
Many-Valued Logic
... Some expressions, such as fairly, sort of, and in between, suggest that there are intermediate positions between truth and falsehood, or at any rate that there are degrees of truth and falsehood. ...
... Some expressions, such as fairly, sort of, and in between, suggest that there are intermediate positions between truth and falsehood, or at any rate that there are degrees of truth and falsehood. ...
The Herbrand Manifesto
... There are benefits and disadvantages to doing things this way. On the one hand, with Herbrand semantics, we no longer have many of the nice features of Tarskian semantics compactness, inferential completeness, and semidecidability. On the other hand, there are some real benefits to Herbrand semantic ...
... There are benefits and disadvantages to doing things this way. On the one hand, with Herbrand semantics, we no longer have many of the nice features of Tarskian semantics compactness, inferential completeness, and semidecidability. On the other hand, there are some real benefits to Herbrand semantic ...
Consequence relations and admissible rules
... admissibility one has to describe what it means to extend a theory or logic by a rule. If the theory is given to us via a proof system, this might be quite straightforward, but if the theory is characterized in another way, say via a set of models or algebras, it is less clear what is meant. In thi ...
... admissibility one has to describe what it means to extend a theory or logic by a rule. If the theory is given to us via a proof system, this might be quite straightforward, but if the theory is characterized in another way, say via a set of models or algebras, it is less clear what is meant. In thi ...
Propositional inquisitive logic: a survey
... KP, there is a whole range of intermediate logics which, when extended with classical atoms, yield inquisitive logic: as shown in [9], this range consists exactly of those intermediate logics which include Maksimova’s logic [15] and are included in Medvedev’s logic of finite problems [17], [18]. In ...
... KP, there is a whole range of intermediate logics which, when extended with classical atoms, yield inquisitive logic: as shown in [9], this range consists exactly of those intermediate logics which include Maksimova’s logic [15] and are included in Medvedev’s logic of finite problems [17], [18]. In ...
Predicate logic
... Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for s ...
... Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for s ...
LOGIC MAY BE SIMPLE Logic, Congruence - Jean
... involving an algebraic structure and a structure of order, but it can be conceived as a pure algebraic structure, in particular because the relation of order is definable in terms of functions; however there are some order-algebraic structures which are not reducible to algebras. It is clear that the ...
... involving an algebraic structure and a structure of order, but it can be conceived as a pure algebraic structure, in particular because the relation of order is definable in terms of functions; however there are some order-algebraic structures which are not reducible to algebras. It is clear that the ...
propositional logic extended with a pedagogically useful relevant
... deemed correct by PC, even the slowest students start complaining after a while. Having described the paradoxes, textbooks and logic teachers sometimes try to reason them away. Two types of moves are invoked in this connection. The first move is legitimate but insufficient: one shows that it is corr ...
... deemed correct by PC, even the slowest students start complaining after a while. Having described the paradoxes, textbooks and logic teachers sometimes try to reason them away. Two types of moves are invoked in this connection. The first move is legitimate but insufficient: one shows that it is corr ...
Canonicity and representable relation algebras
... By first-order compactness, if there is a canonical axiomatisation of RRA, then Aσ can be made as representable as we like, by making A sufficiently representable. ...
... By first-order compactness, if there is a canonical axiomatisation of RRA, then Aσ can be made as representable as we like, by making A sufficiently representable. ...
Bilattices and the Semantics of Logic Programming
... traditional one has been classical two-valued ([1], [21]). This is very satisfactory when negations are not allowed in clause bodies. A three-valued semantics has been urged ([7], [8], [17], [18]) as a way of coping with the problems of negation. Also the two valued semantics has been extended via t ...
... traditional one has been classical two-valued ([1], [21]). This is very satisfactory when negations are not allowed in clause bodies. A three-valued semantics has been urged ([7], [8], [17], [18]) as a way of coping with the problems of negation. Also the two valued semantics has been extended via t ...
Aspects of relation algebras
... If S is finite and AtB = S, then b 7→ {x ∈ X : x ≤ b} is an isomorphism : B → A. So we can specify a finite relation algebra by specifying its atom structure. 0.1.6 Examples of relation algebras Example 0.1.10 The smallest non-trivial relation algebra, I, has atoms 1’ and ], both self-converse. The ...
... If S is finite and AtB = S, then b 7→ {x ∈ X : x ≤ b} is an isomorphism : B → A. So we can specify a finite relation algebra by specifying its atom structure. 0.1.6 Examples of relation algebras Example 0.1.10 The smallest non-trivial relation algebra, I, has atoms 1’ and ], both self-converse. The ...
Admissible rules in the implication-- negation fragment of intuitionistic logic
... 1. Introduction Following Lorenzen [17], a rule is said to be admissible for a logic (understood as a finitary structural consequence relation) if it can be added to a proof system for the logic without producing any new theorems. While the admissible rules of classical propositional logic CPC are a ...
... 1. Introduction Following Lorenzen [17], a rule is said to be admissible for a logic (understood as a finitary structural consequence relation) if it can be added to a proof system for the logic without producing any new theorems. While the admissible rules of classical propositional logic CPC are a ...
S. P. Odintsov “REDUCTIO AD ABSURDUM” AND LUKASIEWICZ`S
... with algebras unit element of which corresponds to the only distinguished value. As usually, for an L-formulae ϕ and algebra A of the language L∪{1}, ϕ is an identity of A if V (ϕ) = 1 for any A-valuation V . A logic of A, LA, is defined as a set of all its identities and a logic of a class K of alge ...
... with algebras unit element of which corresponds to the only distinguished value. As usually, for an L-formulae ϕ and algebra A of the language L∪{1}, ϕ is an identity of A if V (ϕ) = 1 for any A-valuation V . A logic of A, LA, is defined as a set of all its identities and a logic of a class K of alge ...
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 ...
Aristotle`s work on logic.
... reduced: ‘B’ to Barbara, ‘C’ to Celarent, ‘D’ to Darii, and ‘F’ to Ferio. The letter ‘s’ after the ith vowel indicates that the corresponding proposition has to be simply converted, i.e., a use of si . The letter ‘p’ after the ith vowel indicates that the corresponding proposition has to be accident ...
... reduced: ‘B’ to Barbara, ‘C’ to Celarent, ‘D’ to Darii, and ‘F’ to Ferio. The letter ‘s’ after the ith vowel indicates that the corresponding proposition has to be simply converted, i.e., a use of si . The letter ‘p’ after the ith vowel indicates that the corresponding proposition has to be accident ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... generalization of the Gelfond-Lifschitz reduct. ...
... generalization of the Gelfond-Lifschitz reduct. ...
Predicate Logic
... • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain of interest consisting of just two people, john and mary, and we know that tall(mary) and tall(john) are true, we can say that X tall(X) is true. ...
... • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain of interest consisting of just two people, john and mary, and we know that tall(mary) and tall(john) are true, we can say that X tall(X) is true. ...
1 The calculus of “predicates”
... universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for present just on the predicate part of the calculus. A typical formula of the predicate calculus is jus ...
... universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for present just on the predicate part of the calculus. A typical formula of the predicate calculus is jus ...
Written
... b) Group the possible elements of R into 36 groups. For each ordered pair of the form (a,b) with ab, group (a,b) with (b,a). This gives rise to 28 groups. The last 8 groups will each contain one element: either (1,1), (2,2), (3, 3), etc. or (8,8). For each possible symmetric relation, you are eithe ...
... b) Group the possible elements of R into 36 groups. For each ordered pair of the form (a,b) with ab, group (a,b) with (b,a). This gives rise to 28 groups. The last 8 groups will each contain one element: either (1,1), (2,2), (3, 3), etc. or (8,8). For each possible symmetric relation, you are eithe ...
Lecture 11 Artificial Intelligence Predicate Logic
... • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain of interest consisting of just two people, john and mary, and we know that tall(mary) and tall(john) are true, we can say that X tall(X) is true. ...
... • X P(X) means that P(X) must be true for at least one object X in the domain of interest. • So if we have a domain of interest consisting of just two people, john and mary, and we know that tall(mary) and tall(john) are true, we can say that X tall(X) is true. ...
Exam 2 Sample
... means "x is y's parent." For example, if Phil is Sandy's parent, then Phil relates to Sandy, i.e., "Phil R Sandy" is true. a. Is R a transitive relation? ________ Explain: b. Use plain English to describe the inverse relation, R 1 : x R 1 y means... ...
... means "x is y's parent." For example, if Phil is Sandy's parent, then Phil relates to Sandy, i.e., "Phil R Sandy" is true. a. Is R a transitive relation? ________ Explain: b. Use plain English to describe the inverse relation, R 1 : x R 1 y means... ...
Negative translation - Homepages of UvA/FNWI staff
... It is natural to think of classical logic as an extension of intuitionistic logic as it can be obtained from intuitionistic logic by adding an additional axiom (for instance, the Law of Excluded Middle ϕ ∨ ¬ϕ). However, the opposite point of view makes sense as well: one could also think of intuitio ...
... It is natural to think of classical logic as an extension of intuitionistic logic as it can be obtained from intuitionistic logic by adding an additional axiom (for instance, the Law of Excluded Middle ϕ ∨ ¬ϕ). However, the opposite point of view makes sense as well: one could also think of intuitio ...
Identity and Philosophical Problems of Symbolic Logic
... There are philosophical issues concerning the status of sentence connectives in predicate logic. ...
... There are philosophical issues concerning the status of sentence connectives in predicate logic. ...
Science of Logic
Science of Logic (German: Wissenschaft der Logik, first published between 1812 and 1816) is the work in which Georg Wilhelm Friedrich Hegel outlined his vision of logic, which is an ontology that incorporates the traditional Aristotelian syllogism as a sub-component rather than a basis. For Hegel, the most important achievement of German Idealism, starting with Kant and culminating in his own philosophy, was the demonstration that reality is shaped through and through by mind and, when properly understood, is mind. Thus ultimately the structures of thought and reality, subject and object, are identical. And since for Hegel the underlying structure of all of reality is ultimately rational, logic is not merely about reasoning or argument but rather is also the rational, structural core of all of reality and every dimension of it. Thus Hegel's Science of Logic includes among other things analyses of being, nothingness, becoming, existence, reality, essence, reflection, concept, and method. As developed, it included the fullest description of his dialectic. Hegel considered it one of his major works and therefore kept it up to date through revision. Science of Logic is sometimes referred to as the ""Greater Logic"" to distinguish it from the ""Lesser Logic"", the moniker given to the condensed version Hegel presented as the ""Logic"" section of his Encyclopedia of the Philosophical Sciences.