Ultrasheaves
... First in this section we give a background in categorical logic and general topos theory. Then follows a background directly related to ultrasheaves. 1.1. Background in categorical logic. The study of sheaf theory was pioneered by Grothendieck. He was motivated by examples of sheaves in algebraic ge ...
... First in this section we give a background in categorical logic and general topos theory. Then follows a background directly related to ultrasheaves. 1.1. Background in categorical logic. The study of sheaf theory was pioneered by Grothendieck. He was motivated by examples of sheaves in algebraic ge ...
Clausal Logic and Logic Programming in Algebraic Domains*
... In this paper we show how to represent the Smyth powerdomain of a coherent algebraic dcpo using an elementary logic built over such a domain. This is a clausal logic, different from the modal logic introduced by Winskel [Win83] for the Smyth powerdomain. We obtain the logic by regarding finite sets ...
... In this paper we show how to represent the Smyth powerdomain of a coherent algebraic dcpo using an elementary logic built over such a domain. This is a clausal logic, different from the modal logic introduced by Winskel [Win83] for the Smyth powerdomain. We obtain the logic by regarding finite sets ...
Suszko`s Thesis, Inferential Many-Valuedness, and the
... from two to three, Malinowski did a step towards logical many-valuedness, but in a way that gives up the idea of entailment as preservation of a logical value (from the premises to the conclusion, or vice versa) and thereby at the price of violating reflexivity. In the present paper, it is suggested ...
... from two to three, Malinowski did a step towards logical many-valuedness, but in a way that gives up the idea of entailment as preservation of a logical value (from the premises to the conclusion, or vice versa) and thereby at the price of violating reflexivity. In the present paper, it is suggested ...
Propositional Calculus
... Some Cuyahoga River water is not pure. (the horizontal line is short for therefore.) The argument is valid. A valid (correct, sound) argument is one in which it would be contradictory for the premises to be true but the conclusion false. Propositional Calculus ...
... Some Cuyahoga River water is not pure. (the horizontal line is short for therefore.) The argument is valid. A valid (correct, sound) argument is one in which it would be contradictory for the premises to be true but the conclusion false. Propositional Calculus ...
A(x)
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
pdf [local copy]
... of ambiguous descriptions. There are clearly historical issues concerning questions like: How did Russell & Whitehead understand their elimination procedure? Did the language of Principia allow vacuous quantification? Did it allow 0-ary predicates? Etcetera. We do not intend to take a standpoint on ...
... of ambiguous descriptions. There are clearly historical issues concerning questions like: How did Russell & Whitehead understand their elimination procedure? Did the language of Principia allow vacuous quantification? Did it allow 0-ary predicates? Etcetera. We do not intend to take a standpoint on ...
An Abridged Report - Association for the Advancement of Artificial
... of belief, but is a new metalogical property of certain sets of sentences. Because of this, the derivation from only knowing (1) and (2) to knowing (3) must be carried out completely outside the logic, as in McDermott and Doyle’s logic or in Reiter’s (in their case with appropriate metalogical argum ...
... of belief, but is a new metalogical property of certain sets of sentences. Because of this, the derivation from only knowing (1) and (2) to knowing (3) must be carried out completely outside the logic, as in McDermott and Doyle’s logic or in Reiter’s (in their case with appropriate metalogical argum ...
what are we to accept, and what are we to reject
... distinct properties may have logically equivalent possession conditions. Regardless, we can introduce a coarser account of properties, by bundling together all logically coextensive properties. If from a is P is it logically follows that a is Q and vice versa, we will say that the properties P and Q ...
... distinct properties may have logically equivalent possession conditions. Regardless, we can introduce a coarser account of properties, by bundling together all logically coextensive properties. If from a is P is it logically follows that a is Q and vice versa, we will say that the properties P and Q ...
Outline of Lecture 2 First Order Logic and Second Order Logic Basic
... • For H any simple graph, let F orbind(H) class of finite graphs which have no induced copy of H. • Cographs were first defined inductively: The class of cographs is the smallest class of graphs which contains the single vertex graph E1 and is closed under disjoint unions and (loopfree) graph comple ...
... • For H any simple graph, let F orbind(H) class of finite graphs which have no induced copy of H. • Cographs were first defined inductively: The class of cographs is the smallest class of graphs which contains the single vertex graph E1 and is closed under disjoint unions and (loopfree) graph comple ...
Remarks on Second-Order Consequence
... axioms is a definite matter. This is not an idle concern, since the definition of logical consequence involves talk of all models, and it is far from clear that it is determinate what all models are2. The situation is rather different for second-order languages. In second-order languages we deal wit ...
... axioms is a definite matter. This is not an idle concern, since the definition of logical consequence involves talk of all models, and it is far from clear that it is determinate what all models are2. The situation is rather different for second-order languages. In second-order languages we deal wit ...
Ordered Groups: A Case Study In Reverse Mathematics 1 Introduction
... Mathematics ∗ Reed Solomon August 28, 2003 ...
... Mathematics ∗ Reed Solomon August 28, 2003 ...
Concept Hierarchies from a Logical Point of View
... logic is commonly seen as the most basic sort of logic – witness any textbook on logic. Conceptually, however, it seems rather awkward to regard attributes as propositions. If attributes are formalized within a logical language at all then the most natural way to do so is to represent them as monadi ...
... logic is commonly seen as the most basic sort of logic – witness any textbook on logic. Conceptually, however, it seems rather awkward to regard attributes as propositions. If attributes are formalized within a logical language at all then the most natural way to do so is to represent them as monadi ...
(pdf)
... Definition 1.9. A model or structure M for some language L is an ordered triple M = (A, I, β), where A is a nonempty set, β is a variable assignment function and I is an interpretation function with domain the set of all constants, relations and function symbols in L such that: (1) For every consta ...
... Definition 1.9. A model or structure M for some language L is an ordered triple M = (A, I, β), where A is a nonempty set, β is a variable assignment function and I is an interpretation function with domain the set of all constants, relations and function symbols in L such that: (1) For every consta ...
Binary aggregation with integrity constraints Grandi, U. - UvA-DARE
... Sections 5.2.2 and 5.3. Recall that a binary relation is a linear order if it is irreflexive, transitive and complete. The term aPi b stands for “individual i strictly prefers alternative a to alternative b”. The choice of a linear order Pi for each individual constitutes a preference profile P = (P ...
... Sections 5.2.2 and 5.3. Recall that a binary relation is a linear order if it is irreflexive, transitive and complete. The term aPi b stands for “individual i strictly prefers alternative a to alternative b”. The choice of a linear order Pi for each individual constitutes a preference profile P = (P ...
INTERPLAYS OF KNOWLEDGE AND NON
... In addition, note that given the dual ϕ → 3ϕ of the famous axiom (T) in alethic shape formulated as ϕ → ϕ here called in epistemic version as (E) it is easy to see that knowledge is a sufficient condition for possible knowledge: (K ϕ → 3 K ϕ). However, it is not immediate to check what are suff ...
... In addition, note that given the dual ϕ → 3ϕ of the famous axiom (T) in alethic shape formulated as ϕ → ϕ here called in epistemic version as (E) it is easy to see that knowledge is a sufficient condition for possible knowledge: (K ϕ → 3 K ϕ). However, it is not immediate to check what are suff ...
Quine`s Conjecture on Many-Sorted Logic
... they entail precisely the same sentences. It is therefore easy to see that logical equivalence is too strict to capture any sense in which Quine’s conjecture is true. Theories can only be logically equivalent if they are formulated in the same signature, so no many-sorted theory is logically equival ...
... they entail precisely the same sentences. It is therefore easy to see that logical equivalence is too strict to capture any sense in which Quine’s conjecture is true. Theories can only be logically equivalent if they are formulated in the same signature, so no many-sorted theory is logically equival ...
A(x)
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
... A1,…,Am |– iff A1,…,Am |= . Proof. If the Theorem of Deduction holds, then A1,…,Am |– iff |– (A1 (A2 …(Am )…)). |– (A1 (A2 …(Am )…)) iff |– (A1 … Am) . If the calculus is sound and complete, then |– (A1 … Am) iff |= (A1 … Am) . |= (A1 … Am) iff A1,…,Am |= ...
Fraïssé`s conjecture in Pi^1_1-comprehension
... showed that WKL0 can prove block-bqos and barrier-bqos are the same thing. The equivalence between block-bqos and continuous-bqos is immediate from the translation between bad continuous functions and bad arrays. The notion of Borel-bqos was introduced by Simpson [Sim85]. His proof that they are the ...
... showed that WKL0 can prove block-bqos and barrier-bqos are the same thing. The equivalence between block-bqos and continuous-bqos is immediate from the translation between bad continuous functions and bad arrays. The notion of Borel-bqos was introduced by Simpson [Sim85]. His proof that they are the ...
CPSC 2105 Lecture 6 - Edward Bosworth, Ph.D.
... Algebraically, this function is denoted f(X) = X’ or f(X) = X . The notation X’ is done for typesetting convenience only; the notation The evaluation of the function is simple: ...
... Algebraically, this function is denoted f(X) = X’ or f(X) = X . The notation X’ is done for typesetting convenience only; the notation The evaluation of the function is simple: ...
Quine`s Conjecture on Many-Sorted Logic∗ - Philsci
... notion of logical consequence. A theory T entails a sentence φ, written T φ, if M φ for every model M of T . We begin with the following preliminary criterion for theoretical equivalence. Definition. Theories T1 and T2 are logically equivalent if they have the same class of models. One can verif ...
... notion of logical consequence. A theory T entails a sentence φ, written T φ, if M φ for every model M of T . We begin with the following preliminary criterion for theoretical equivalence. Definition. Theories T1 and T2 are logically equivalent if they have the same class of models. One can verif ...
SORT LOGIC AND FOUNDATIONS OF MATHEMATICS 1
... exists, namely, Y . The Comprehension Axiom is the traditional (impredicative) axiom schema which gives second order logic, and in our case sort logic, the necessary power to do mathematics [3]. In individual cases less comprehension may be sufficient but this is the general schema. The difference b ...
... exists, namely, Y . The Comprehension Axiom is the traditional (impredicative) axiom schema which gives second order logic, and in our case sort logic, the necessary power to do mathematics [3]. In individual cases less comprehension may be sufficient but this is the general schema. The difference b ...
P Q
... The resolution refutation proof procedure answers a query or deduces a new result by reducing the set of clauses to a contradiction, represented by the null clause () The contradiction is produced by resolving pairs of clauses from the database If a resolution does not produce a contradiction d ...
... The resolution refutation proof procedure answers a query or deduces a new result by reducing the set of clauses to a contradiction, represented by the null clause () The contradiction is produced by resolving pairs of clauses from the database If a resolution does not produce a contradiction d ...