Philosophy 240: Symbolic Logic
... P He has provided a formal construction in an artificial language. P Does it capture our ordinary notion? P “It seems to me obvious that the only rational approach to [questions about the correct notion of truth] would be the following: We should reconcile ourselves with the fact that we are confron ...
... P He has provided a formal construction in an artificial language. P Does it capture our ordinary notion? P “It seems to me obvious that the only rational approach to [questions about the correct notion of truth] would be the following: We should reconcile ourselves with the fact that we are confron ...
Intuitionistic and Modal Logic
... 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 modern representatives: Frege, Gödel. View that mathematical objects have independent existence outside ...
... 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 modern representatives: Frege, Gödel. View that mathematical objects have independent existence outside ...
LINEAR LOGIC AS A FRAMEWORK FOR SPECIFYING SEQUENT
... where ∆ is the multiset union of ∆1 and ∆2 , and A is the multiset union of A1 and A2 . In other words, those subgoals immediately to the left of an ⇒ are attempted with empty bounded contexts: the bounded contexts, here ∆ and A, are divided up to be used to prove those goals immediately to the left ...
... where ∆ is the multiset union of ∆1 and ∆2 , and A is the multiset union of A1 and A2 . In other words, those subgoals immediately to the left of an ⇒ are attempted with empty bounded contexts: the bounded contexts, here ∆ and A, are divided up to be used to prove those goals immediately to the left ...
Existential Definability of Modal Frame Classes
... All of the frame constructions used in the theorem – bounded morphisms, disjoint unions, generated subframes and ultrafilter extensions – are presented in detail in [1] (the same notation is used in this paper). Just to be clear, we say that a class K reflects a construction if its complement Kc , th ...
... All of the frame constructions used in the theorem – bounded morphisms, disjoint unions, generated subframes and ultrafilter extensions – are presented in detail in [1] (the same notation is used in this paper). Just to be clear, we say that a class K reflects a construction if its complement Kc , th ...
Subintuitionistic Logics with Kripke Semantics
... satisfying some specific properties can be treated in much the same way as F with the same proofs, and a form of strong completeness for F due to Restall [4] is shown. We then apply the results to logics stronger than F. Thus we will prove a strong completeness theorem for BPC with a Hilbert style p ...
... satisfying some specific properties can be treated in much the same way as F with the same proofs, and a form of strong completeness for F due to Restall [4] is shown. We then apply the results to logics stronger than F. Thus we will prove a strong completeness theorem for BPC with a Hilbert style p ...
Certamen 1 de Representación del Conocimiento
... Solution: Let LP (ϕ) and RP (ϕ) denote the number of left and right parenthesis of ϕ Base case: for proposition p we have LP (p) = RP (p) = 0. Inductive cases: assume we have formulas φ1 and φ2 such that LP (φ1 ) = RP (φ1 ) = c1 and LP (φ2 ) = RP (φ2 ) = c2 : 1. Given a formula φ such that LP (φ) = ...
... Solution: Let LP (ϕ) and RP (ϕ) denote the number of left and right parenthesis of ϕ Base case: for proposition p we have LP (p) = RP (p) = 0. Inductive cases: assume we have formulas φ1 and φ2 such that LP (φ1 ) = RP (φ1 ) = c1 and LP (φ2 ) = RP (φ2 ) = c2 : 1. Given a formula φ such that LP (φ) = ...
logica and critical thinking
... Think it twice: Don’t take things for granted so easily. Always ask the why-question: Try to find out the reason (the premises) why certain claim (the conclusion) can be supported. Examine and evaluate the relationship between the reasons and the claim. ...
... Think it twice: Don’t take things for granted so easily. Always ask the why-question: Try to find out the reason (the premises) why certain claim (the conclusion) can be supported. Examine and evaluate the relationship between the reasons and the claim. ...
Knowledge Representation and Reasoning
... symbol ‘ → ’ is called a propositional connective. Many systems of propositional logic have been developed. In this lecture we are studying classical — i.e. the best established — propositional logic. In classical propositional logic it is taken as a principle that: ...
... symbol ‘ → ’ is called a propositional connective. Many systems of propositional logic have been developed. In this lecture we are studying classical — i.e. the best established — propositional logic. In classical propositional logic it is taken as a principle that: ...
Logic, Human Logic, and Propositional Logic Human Logic
... The truth table method and the proof method succeed in exactly the same cases. On large problems, the proof method often takes fewer steps than the truth table method. However, in the worst case, the proof method may take just as many or more steps to find an answer as the truth table method. Usuall ...
... The truth table method and the proof method succeed in exactly the same cases. On large problems, the proof method often takes fewer steps than the truth table method. However, in the worst case, the proof method may take just as many or more steps to find an answer as the truth table method. Usuall ...
Modal logic and the approximation induction principle
... guishes the top states of the two LTSs above, by means of any formula hai( n∈N hain T) with N infinite. Namely, such a formula holds for the top state at the right, but not for the top state at the left. However, OFIN does not distinguish these states; all formulas in OFIN hold for both states. Gold ...
... guishes the top states of the two LTSs above, by means of any formula hai( n∈N hain T) with N infinite. Namely, such a formula holds for the top state at the right, but not for the top state at the left. However, OFIN does not distinguish these states; all formulas in OFIN hold for both states. Gold ...
Paper - Department of Computer Science and Information Systems
... additional equations corresponding the axioms of L. A closely related algorithmic problem for L is the admissibility problem for inference rules: given an inference rule ϕ1 , . . . , ϕn /ϕ, decide whether it is admissible in L, that is, for every substitution s, we have L ` s(ϕ) whenever L ` s(ϕi ), ...
... additional equations corresponding the axioms of L. A closely related algorithmic problem for L is the admissibility problem for inference rules: given an inference rule ϕ1 , . . . , ϕn /ϕ, decide whether it is admissible in L, that is, for every substitution s, we have L ` s(ϕ) whenever L ` s(ϕi ), ...
Tactics for Separation Logic Abstract Andrew W. Appel INRIA Rocquencourt & Princeton University
... linear logic, in that in general P ∗ Q does not entail P ∗ P ∗ Q or vice versa. When doing machine-checked proofs of imperative programs, one faces a choice: one could implement Hoare logic (or separation logic) directly (or directly in a logical framework such as Twelf or Isabelle); or one could de ...
... linear logic, in that in general P ∗ Q does not entail P ∗ P ∗ Q or vice versa. When doing machine-checked proofs of imperative programs, one faces a choice: one could implement Hoare logic (or separation logic) directly (or directly in a logical framework such as Twelf or Isabelle); or one could de ...
A Well-Founded Semantics for Logic Programs with Abstract
... model semantics was first proposed for normal logic programs by Gelfond and Lifschitz in 1988, various extensions have been put forward for theoretical and/or practical reasons. These include disjunctive logic programs (Gelfond and Lifschitz 1991), nested logic programs (Lifschitz, Tang, and Turner ...
... model semantics was first proposed for normal logic programs by Gelfond and Lifschitz in 1988, various extensions have been put forward for theoretical and/or practical reasons. These include disjunctive logic programs (Gelfond and Lifschitz 1991), nested logic programs (Lifschitz, Tang, and Turner ...
Propositional Logic - Department of Computer Science
... Many important functions F which have as domain the set of all propositional formulas are defined by specifying the values • F (pi ), for all propositional variables pi , • F (P ∧ Q), given the values F (P ) and F (Q), • F (P ∨ Q), given the values F (P ) and F (Q), • F (¬P ), given the value F (P ) ...
... Many important functions F which have as domain the set of all propositional formulas are defined by specifying the values • F (pi ), for all propositional variables pi , • F (P ∧ Q), given the values F (P ) and F (Q), • F (P ∨ Q), given the values F (P ) and F (Q), • F (¬P ), given the value F (P ) ...
Implication
... Note that we must do something, otherwise p ⇒ q would not be a well defined statement, since it would not be defined as either true or false on all the possible inputs. We make the convention that p ⇒ q is always true if p is false. The major reason for defining things this way is the following obse ...
... Note that we must do something, otherwise p ⇒ q would not be a well defined statement, since it would not be defined as either true or false on all the possible inputs. We make the convention that p ⇒ q is always true if p is false. The major reason for defining things this way is the following obse ...
Higher-Order Modal Logic—A Sketch
... In first-order logic, relation symbols have an arity. In higher-order logic this gets replaced by a typing mechanism. There are several ways this can be done: logical connectives can be considered primitive, or as constants of the language; a boolean type can be introduced, or not. We adopt a straig ...
... In first-order logic, relation symbols have an arity. In higher-order logic this gets replaced by a typing mechanism. There are several ways this can be done: logical connectives can be considered primitive, or as constants of the language; a boolean type can be introduced, or not. We adopt a straig ...
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 ...
Discrete Mathematics
... The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, written P ∨ Q, is a proposition. The conditional statement (implies), written P −→ Q, is a proposition. The Boolean va ...
... The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, written P ∨ Q, is a proposition. The conditional statement (implies), written P −→ Q, is a proposition. The Boolean va ...
A mathematical sentence is a sentence that states a fact or contains
... V. Law of Disjunctive Inference: In the first four laws of inference presented, one or more of the premises was in the form of a conditional statement p → q. In this next inference law, there are no conditional statements in the given premises. As we watch a mystery show on television, we believe t ...
... V. Law of Disjunctive Inference: In the first four laws of inference presented, one or more of the premises was in the form of a conditional statement p → q. In this next inference law, there are no conditional statements in the given premises. As we watch a mystery show on television, we believe t ...
Logic and Proofs1 1 Overview. 2 Sentential Connectives.
... I deduce that β is true. You may view this as a definition of what “deduce” means. The idea behind modus ponens is that words like “deduce” belong to the language of ordinary mathematics, not to the special language of logic. Thus, modus ponens provides a bridge between a formalism and the formalism ...
... I deduce that β is true. You may view this as a definition of what “deduce” means. The idea behind modus ponens is that words like “deduce” belong to the language of ordinary mathematics, not to the special language of logic. Thus, modus ponens provides a bridge between a formalism and the formalism ...
Concept Hierarchies from a Logical Point of View
... a formal context hU, Σ, i uniquely corresponds to an interpretation M of Σ, and vice versa: simply define M (p) = p⊳ = {x ∈ U | x p}. The notion of an interpretation gives us the notion of truth and model as well: a statement ∀φ is true with respect to the interpretation M if M (φ) = U (with M ex ...
... a formal context hU, Σ, i uniquely corresponds to an interpretation M of Σ, and vice versa: simply define M (p) = p⊳ = {x ∈ U | x p}. The notion of an interpretation gives us the notion of truth and model as well: a statement ∀φ is true with respect to the interpretation M if M (φ) = U (with M ex ...
THE ABUNDANCE OF THE FUTURE A Paraconsistent Approach to
... Abundance has at least some intuitive grounding in our linguistic use: most of the times, when we say “tomorrow it is going to rain” we do not claim that this is certain, only that there are reasons to assert it, but since there may also be reasons for asserting its negation, both seem to be tenable ...
... Abundance has at least some intuitive grounding in our linguistic use: most of the times, when we say “tomorrow it is going to rain” we do not claim that this is certain, only that there are reasons to assert it, but since there may also be reasons for asserting its negation, both seem to be tenable ...