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 ...
A joint logic of problems and propositions, a modified BHK
... Surprisingly, the calculus of problems coincides in form with Brouwer’s intuitionistic logic, as recently formalized by Heyting. [In fact, we shall argue] that [intuitionistic logic] should be replaced with the calculus of problems, since its objects are in reality not theoretical propositions but r ...
... Surprisingly, the calculus of problems coincides in form with Brouwer’s intuitionistic logic, as recently formalized by Heyting. [In fact, we shall argue] that [intuitionistic logic] should be replaced with the calculus of problems, since its objects are in reality not theoretical propositions but r ...
relevance logic - Consequently.org
... The title of this piece is not ‘A Survey of Relevance Logic’. Such a project was impossible in the mid 1980s when the first version of this article was published, due to the development of the field and even the space limitations of the Handbook. The situation is if anything, more difficult now. For ...
... The title of this piece is not ‘A Survey of Relevance Logic’. Such a project was impossible in the mid 1980s when the first version of this article was published, due to the development of the field and even the space limitations of the Handbook. The situation is if anything, more difficult now. For ...
Relevant and Substructural Logics
... Logics tend to be viewed of in one of two ways — with an eye to proofs, or with an eye to models.2 Relevant and substructural logics are no different: you can focus on notions of proof, inference rules and structural features of deduction in these logics, or you can focus on interpretations of the l ...
... Logics tend to be viewed of in one of two ways — with an eye to proofs, or with an eye to models.2 Relevant and substructural logics are no different: you can focus on notions of proof, inference rules and structural features of deduction in these logics, or you can focus on interpretations of the l ...
Everything Else Being Equal: A Modal Logic for Ceteris Paribus
... von Wright’s original work in preference logic, in order to motivate some of the notions we develop later, but also as a foundational standard against which we can evaluate our own results. In Section 3, we present a basic modal logic of weak and strict preference interpreted in ordered models of po ...
... von Wright’s original work in preference logic, in order to motivate some of the notions we develop later, but also as a foundational standard against which we can evaluate our own results. In Section 3, we present a basic modal logic of weak and strict preference interpreted in ordered models of po ...
Higher Order Logic - Theory and Logic Group
... important aspects of higher order logic receive here a scant treatment, and others are omitted altogether, notably most uses of higher order constructs in mathematical practice, in recursion theory, and in computer science. Such choices of topics can not be independent of an author's interests and b ...
... important aspects of higher order logic receive here a scant treatment, and others are omitted altogether, notably most uses of higher order constructs in mathematical practice, in recursion theory, and in computer science. Such choices of topics can not be independent of an author's interests and b ...
Higher Order Logic - Indiana University
... important aspects of higher order logic receive here a scant treatment, and others are omitted altogether, notably most uses of higher order constructs in mathematical practice, in recursion theory, and in computer science. Such choices of topics can not be independent of an author's interests and b ...
... important aspects of higher order logic receive here a scant treatment, and others are omitted altogether, notably most uses of higher order constructs in mathematical practice, in recursion theory, and in computer science. Such choices of topics can not be independent of an author's interests and b ...
Classical Propositional Logic
... Completeness A Henkin-style Completeness Proof for Natural Deduction Computability ...
... Completeness A Henkin-style Completeness Proof for Natural Deduction Computability ...
The Herbrand Manifesto
... In much of the literature, Herbrand semantics is treated (somewhat understandably) as a special case of Tarskian semantics - the case where we look at so-called Herbrand interpretations. Although the two are similar in many ways, they are not the same. First of all, in Tarskian semantics, there are ...
... In much of the literature, Herbrand semantics is treated (somewhat understandably) as a special case of Tarskian semantics - the case where we look at so-called Herbrand interpretations. Although the two are similar in many ways, they are not the same. First of all, in Tarskian semantics, there are ...
x - Homepages | The University of Aberdeen
... Applications of Predicate Logic It is one of the most-used formal notations for writing mathematical definitions, axioms, and theorems. For example, in linear algebra, a partial order is introduced saying that a relation R is reflexive and transitive – and these notions are defined using predicate l ...
... Applications of Predicate Logic It is one of the most-used formal notations for writing mathematical definitions, axioms, and theorems. For example, in linear algebra, a partial order is introduced saying that a relation R is reflexive and transitive – and these notions are defined using predicate l ...
Many-Valued Logic
... Every possible answer to it presupposes something—that Jones has been beating his wife. ...
... Every possible answer to it presupposes something—that Jones has been beating his wife. ...
Modal Consequence Relations
... rules: h{p, q}, p ∧ qi and both h{p ∧ q}, pi and h{p ∧ q}, qi. In addition, if ` = `M for some logical matrix we say that ` has all constants if for each s ∈ T there exists a nullary term function s such that for all valuations v v(s) = s. (Note that since var(s) = ∅ the value of s does not depend a ...
... rules: h{p, q}, p ∧ qi and both h{p ∧ q}, pi and h{p ∧ q}, qi. In addition, if ` = `M for some logical matrix we say that ` has all constants if for each s ∈ T there exists a nullary term function s such that for all valuations v v(s) = s. (Note that since var(s) = ∅ the value of s does not depend a ...
Intuitionistic and Modal Logic
... Intuitionism, Platonism, Formalism • Intuitionism is as one of the three basic points of view opposed to Platonism and formalism. View that mathematics and mathematical truths are creations of the human mind: true = provable. N.B! provable in the informal, not formal sense. • Platonism. Most famous ...
... Intuitionism, Platonism, Formalism • Intuitionism is as one of the three basic points of view opposed to Platonism and formalism. View that mathematics and mathematical truths are creations of the human mind: true = provable. N.B! provable in the informal, not formal sense. • Platonism. Most famous ...
Logic Part II: Intuitionistic Logic and Natural Deduction
... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...
... 2. This proof contains of a proof of a. 3. It also contains a proof of b . 4. So if we take the proof of b and put it together with the proof of a, we obtain a proof of b ...
Essentials Of Symbolic Logic
... 2000 years. Both the Greek successors of Aristotle and the medieval scholastics made several important logical discoveries. But the importance of these discoveries was not realized at the time they were made. The general belief that all the important logical discoveries have been made by Aristotle n ...
... 2000 years. Both the Greek successors of Aristotle and the medieval scholastics made several important logical discoveries. But the importance of these discoveries was not realized at the time they were made. The general belief that all the important logical discoveries have been made by Aristotle n ...
LINEAR LOGIC AS A FRAMEWORK FOR SPECIFYING SEQUENT
... §1. Introduction. Logics and type systems have been exploited in recent years as frameworks for the specification of deduction in a number of logics. Such meta logics or logical frameworks have generally been based on intuitionistic logic in which quantification at (non-predicate) higher-order types ...
... §1. Introduction. Logics and type systems have been exploited in recent years as frameworks for the specification of deduction in a number of logics. Such meta logics or logical frameworks have generally been based on intuitionistic logic in which quantification at (non-predicate) higher-order types ...
relevant reasoning as the logical basis of
... conditions. Indeed, a major work of almost all scientists is to discover some conditional and/or causal relationships between various phenomena, data, and laws in their research fields. In logic, the notion abstracted from various conditionals is called “entailment.” In general, an entailment, for i ...
... conditions. Indeed, a major work of almost all scientists is to discover some conditional and/or causal relationships between various phenomena, data, and laws in their research fields. In logic, the notion abstracted from various conditionals is called “entailment.” In general, an entailment, for i ...
Let me begin by reminding you of a number of passages ranging
... maximally general science. As Frege says in the opening paragraph of “Der Gedanke”, The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean prescriptions which ought to be obeyed but with which actual occurrences are not always in conformity. Laws of nature [on the other h ...
... maximally general science. As Frege says in the opening paragraph of “Der Gedanke”, The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean prescriptions which ought to be obeyed but with which actual occurrences are not always in conformity. Laws of nature [on the other h ...
Formal deduction in propositional logic
... Finiteness of premise set Theorem. If Σ ` A, then there is some finite Σ0 ⊆ Σ such that Σ0 ` A. Proof. By induction on the complexity of Σ ` A. Basis: The premise A of A ` A generated by (Ref) is itself finite. Induction Step: We distinguish ten cases. For each case, assume that the premises have t ...
... Finiteness of premise set Theorem. If Σ ` A, then there is some finite Σ0 ⊆ Σ such that Σ0 ` A. Proof. By induction on the complexity of Σ ` A. Basis: The premise A of A ` A generated by (Ref) is itself finite. Induction Step: We distinguish ten cases. For each case, assume that the premises have t ...
Introduction to first order logic for knowledge representation
... Language, real world and mathematical structure Language In describing a phenomena or a portion of the world, we adopt a language. The phrases of this language are used to describe objects of the real worlds, their properties, and facts that holds. This language can be informal (natural language, g ...
... Language, real world and mathematical structure Language In describing a phenomena or a portion of the world, we adopt a language. The phrases of this language are used to describe objects of the real worlds, their properties, and facts that holds. This language can be informal (natural language, g ...
Proofs in Propositional Logic
... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...
... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...
Proofs in Propositional Logic
... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...
... This tactic can be useful for avoiding proof duplication inside some interactive proof. Notice that the scope of the declaration H :B is limited to the second subgoal. If a proof of B is needed elsewhere, it would be better to prove a lemma stating B. Remark : Sometimes the overuse of assert may lea ...
ICS 353: Design and Analysis of Algorithms
... • Question 6 page 58: Let C(x,y) mean that x is enrolled in y, where the universe of discourse for x is the set of all students in your school and the universe of discourse for y is the set of classes being given at your school. Express each of the following statements by a simple English ...
... • Question 6 page 58: Let C(x,y) mean that x is enrolled in y, where the universe of discourse for x is the set of all students in your school and the universe of discourse for y is the set of classes being given at your school. Express each of the following statements by a simple English ...