The Development of Categorical Logic
... space, or a general category). The usual category of sets (itself, of course, a topos) could then be thought of as being composed of “constant” sets. Lawvere and Tierney’s ideas were taken up with enthusiasm and quickly developed further by several mathematicians, notably P. Freyd, J. Bénabou and hi ...
... space, or a general category). The usual category of sets (itself, of course, a topos) could then be thought of as being composed of “constant” sets. Lawvere and Tierney’s ideas were taken up with enthusiasm and quickly developed further by several mathematicians, notably P. Freyd, J. Bénabou and hi ...
Introduction to Logic
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
Ground Nonmonotonic Modal Logics - Dipartimento di Informatica e
... The goal of our work1 is to study the family of ground logics, from the semantical, computational and epistemological viewpoint. With respect to the first issue, we present an appropriate semantic characterization for ground logics, that has been advocated in [31]. In particular, our proposal is an ...
... The goal of our work1 is to study the family of ground logics, from the semantical, computational and epistemological viewpoint. With respect to the first issue, we present an appropriate semantic characterization for ground logics, that has been advocated in [31]. In particular, our proposal is an ...
Chapter 2 Propositional Logic
... wff. That’s why we use the metalinguistic variables “φ” and “ψ”.2 The practice of using variables to express generality is familiar; we can say, for example, “for any integer n, if n is even, then n + 2 is even as well”. Just as “n” here is a variable for numbers, metalinguistic variables are variab ...
... wff. That’s why we use the metalinguistic variables “φ” and “ψ”.2 The practice of using variables to express generality is familiar; we can say, for example, “for any integer n, if n is even, then n + 2 is even as well”. Just as “n” here is a variable for numbers, metalinguistic variables are variab ...
Deep Sequent Systems for Modal Logic
... approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to use, which have good structural properties like contractionadmissibility and invertibility of all rules, and which give rise to decision p ...
... approaches. It allows to capture a wide class of modal logics and does so systematically. In many important cases it yields systems which are natural and easy to use, which have good structural properties like contractionadmissibility and invertibility of all rules, and which give rise to decision p ...
Continuous first order logic and local stability
... can be carried out in one if and only if it can be carried out in the other; but it may still happen that notions which arise naturally from one of the presentations are more useful, and render clear and obvious what was obscure with the other one. This indeed seems to be the case with continuous fir ...
... can be carried out in one if and only if it can be carried out in the other; but it may still happen that notions which arise naturally from one of the presentations are more useful, and render clear and obvious what was obscure with the other one. This indeed seems to be the case with continuous fir ...
Propositional Discourse Logic
... Paul is wrong so Frank must be wrong too. In a similar way all possibilities end up undermining themselves and from this we are forced to conclude, by logic alone, that the discourse has malfunctioned. We are concerned here with diagnosing the disease, the main symptom of which is that the discourse ...
... Paul is wrong so Frank must be wrong too. In a similar way all possibilities end up undermining themselves and from this we are forced to conclude, by logic alone, that the discourse has malfunctioned. We are concerned here with diagnosing the disease, the main symptom of which is that the discourse ...
Expressiveness of Logic Programs under the General Stable Model
... answer set programming, armed with powerful tools from classical logic. The main goal of this work is to identify the expressiveness of logic programs, which is one of the central topics in Knowledge Representation and Reasoning. We will focus on two important classes of logic programs – normal logi ...
... answer set programming, armed with powerful tools from classical logic. The main goal of this work is to identify the expressiveness of logic programs, which is one of the central topics in Knowledge Representation and Reasoning. We will focus on two important classes of logic programs – normal logi ...
The Foundations
... Two syntactically (i.e., textually) different compound propositions may be semantically identical (i.e., have the same meaning). We call them equivalent. Learn: Various equivalence rules or laws. How to prove equivalences using symbolic derivations. Analogy: x * (5 + y) and xy + 5x are alway ...
... Two syntactically (i.e., textually) different compound propositions may be semantically identical (i.e., have the same meaning). We call them equivalent. Learn: Various equivalence rules or laws. How to prove equivalences using symbolic derivations. Analogy: x * (5 + y) and xy + 5x are alway ...
Tableau-based decision procedure for the full
... development of efficient decision procedures for them. In the recent precursor [5] to the present paper, we set out to fill in the above-mentioned gap by developing a practically efficient (within the theoretical complexity bounds) tableau-based decision procedure for the coalitional multiagent temp ...
... development of efficient decision procedures for them. In the recent precursor [5] to the present paper, we set out to fill in the above-mentioned gap by developing a practically efficient (within the theoretical complexity bounds) tableau-based decision procedure for the coalitional multiagent temp ...
LOGICAL CONSEQUENCE AS TRUTH-PRESERVATION STEPHEN READ Abstract
... We are left then with Lewis’ claim that relevance is a necessary condition on inference and proof. Lewis, following MacColl, thought that relevance could be reduced to modality: rejecting the truth-functional analysis as permitting irrelevance, they both sought to capture the essence of implication ...
... We are left then with Lewis’ claim that relevance is a necessary condition on inference and proof. Lewis, following MacColl, thought that relevance could be reduced to modality: rejecting the truth-functional analysis as permitting irrelevance, they both sought to capture the essence of implication ...
Relevant Logic A Philosophical Examination of Inference Stephen Read February 21, 2012
... analysis. For the rules of the formal theory of deduction state the conditions under which formulae may be asserted—it connects the formulae with their use. I present a general framework of proof-theoretic analysis, and in particular show how the theory of negation in chapter 4 can be fitted to this ...
... analysis. For the rules of the formal theory of deduction state the conditions under which formulae may be asserted—it connects the formulae with their use. I present a general framework of proof-theoretic analysis, and in particular show how the theory of negation in chapter 4 can be fitted to this ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order fr ...
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order fr ...
Ways Things Can`t Be
... In what follows I will construct a notion of “truth according to a corpus” which respects all five conditions. One obvious way to do this is to start with a different logic. A paraconsistent logic is one in which the inference A, ∼ A B fails. If we choose such a logic, we can construct “impossible ...
... In what follows I will construct a notion of “truth according to a corpus” which respects all five conditions. One obvious way to do this is to start with a different logic. A paraconsistent logic is one in which the inference A, ∼ A B fails. If we choose such a logic, we can construct “impossible ...
Decision procedures in Algebra and Logic
... • Algebra over a field: An algebra over a ring except that R is a field instead of a commutative ring. • Jordan algebra: a Jordan ring except that R is a field. • Lie algebra: an algebra over a field respecting the Jacobi identity, whose vector multiplication, the Lie bracket denoted [u,v], anticomm ...
... • Algebra over a field: An algebra over a ring except that R is a field instead of a commutative ring. • Jordan algebra: a Jordan ring except that R is a field. • Lie algebra: an algebra over a field respecting the Jacobi identity, whose vector multiplication, the Lie bracket denoted [u,v], anticomm ...
On the Construction of Analytic Sequent Calculi for Sub
... Γ and ∆ are finite sets of formulas. We employ the standard sequent notations, e.g. when writing expressions like Γ, ψ ⇒ ∆ or ⇒ ψ. The union of sequents is defined by (Γ1 ⇒ ∆1 ) ∪ (Γ2 ⇒ ∆2 ) = Γ1 ∪ Γ2 ⇒ ∆1 ∪ ∆2 . For a sequent Γ ⇒ ∆, frm(Γ ⇒ ∆) = Γ ∪ ∆. This notation is naturally extended to sets of ...
... Γ and ∆ are finite sets of formulas. We employ the standard sequent notations, e.g. when writing expressions like Γ, ψ ⇒ ∆ or ⇒ ψ. The union of sequents is defined by (Γ1 ⇒ ∆1 ) ∪ (Γ2 ⇒ ∆2 ) = Γ1 ∪ Γ2 ⇒ ∆1 ∪ ∆2 . For a sequent Γ ⇒ ∆, frm(Γ ⇒ ∆) = Γ ∪ ∆. This notation is naturally extended to sets of ...
A joint logic of problems and propositions, a modified BHK
... depending on variables that all range over the same domain of discourse) and the constant ⊥ Formulas are of two types: problems (denoted by Greek letters) and propositions (denoted by Roman letters) Classical connectives: propositions → propositions Intuitionistic connectives: problems → problems Tw ...
... depending on variables that all range over the same domain of discourse) and the constant ⊥ Formulas are of two types: problems (denoted by Greek letters) and propositions (denoted by Roman letters) Classical connectives: propositions → propositions Intuitionistic connectives: problems → problems Tw ...
vmcai - of Philipp Ruemmer
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order ...
... In program verification, an interpolating theorem prover often interacts tightly with various decision procedures. It is therefore advantageous for the interpolants computed by the prover to be expressible in simple logic fragments. Unfortunately, interpolation procedures for expressive first-order ...
Reductio ad Absurdum Argumentation in Normal Logic
... Scenarios here defined. Normal Logic Programs are approached as assumption-based argumentation systems. We generalize this setting by allowing both negative and positive assumptions. Negative assumptions are made maximal, consistent with existence of a semantics, and positive assumptions are adopted ...
... Scenarios here defined. Normal Logic Programs are approached as assumption-based argumentation systems. We generalize this setting by allowing both negative and positive assumptions. Negative assumptions are made maximal, consistent with existence of a semantics, and positive assumptions are adopted ...
Symbolic Logic I: The Propositional Calculus
... Much of mathematical reasoning is made up of conditional statements, called implications. These are statements that a certain condition P produces a certain consequence Q. It is important to realize that we may often not know whether P is true or not, but we do know that if it is true, then Q follow ...
... Much of mathematical reasoning is made up of conditional statements, called implications. These are statements that a certain condition P produces a certain consequence Q. It is important to realize that we may often not know whether P is true or not, but we do know that if it is true, then Q follow ...
Multiverse Set Theory and Absolutely Undecidable Propositions
... Von Neumann refers to Skolem and Löwenheim [18] as sources of the noncategoricity of his, or any other set theory. It is worth noting that von Neumann puts so much weight on categoricity. Indeed, if set theory had a categorical axiomatization, the categoricity proof itself, carried out in set theor ...
... Von Neumann refers to Skolem and Löwenheim [18] as sources of the noncategoricity of his, or any other set theory. It is worth noting that von Neumann puts so much weight on categoricity. Indeed, if set theory had a categorical axiomatization, the categoricity proof itself, carried out in set theor ...
Introduction to Logic
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
Simple multiplicative proof nets with units
... Here is a passage from Girard’s Proof Nets: the Parallel Syntax for Proof Theory [Gir96, §A.2]1 : There are two multiplicative neutrals, 1 and ⊥, and two rules, the axiom ⊢ 1 and the weakening rule: from ⊢ Γ, deduce ⊢ Γ, ⊥. Both rules are handled by means of links with one conclusion and no premise; ...
... Here is a passage from Girard’s Proof Nets: the Parallel Syntax for Proof Theory [Gir96, §A.2]1 : There are two multiplicative neutrals, 1 and ⊥, and two rules, the axiom ⊢ 1 and the weakening rule: from ⊢ Γ, deduce ⊢ Γ, ⊥. Both rules are handled by means of links with one conclusion and no premise; ...
071 Embeddings
... For technical explanation of this term see Monk [1976] chapters 13 to 16, and Tarski, Mostowski and Robinson ...
... For technical explanation of this term see Monk [1976] chapters 13 to 16, and Tarski, Mostowski and Robinson ...
Default Logic (Reiter) - Department of Computing
... Given E, the reduct DE is a (possibly empty) set of ordinary, non-default, monotonic rules. So we have available all the properties of (monotonic, non-default) closures. For example: E is an extension of (D, W ) when E is the smallest set of formulas S such that: S = Th(W ∪ TDE (S) ) And we have var ...
... Given E, the reduct DE is a (possibly empty) set of ordinary, non-default, monotonic rules. So we have available all the properties of (monotonic, non-default) closures. For example: E is an extension of (D, W ) when E is the smallest set of formulas S such that: S = Th(W ∪ TDE (S) ) And we have var ...