Lecture01 - Mathematics
... between objects). This is not a definition so much as a feeling, and the line between discrete and continuum mathematics is not sharp. Customarily the term discrete mathematics includes the following branches. i) Formal Logic: This includes symbolic logic, propositional logic, and predicate logic. S ...
... between objects). This is not a definition so much as a feeling, and the line between discrete and continuum mathematics is not sharp. Customarily the term discrete mathematics includes the following branches. i) Formal Logic: This includes symbolic logic, propositional logic, and predicate logic. S ...
Propositional Logic
... describes, as in “It’s raining and it's not raining.” • P entails Q, written P |= Q, means that whenever P is True, so is Q. In other words, all models of P are also models of Q. ...
... describes, as in “It’s raining and it's not raining.” • P entails Q, written P |= Q, means that whenever P is True, so is Q. In other words, all models of P are also models of Q. ...
Incompleteness - the UNC Department of Computer Science
... that human mathematical understanding cannot be encapsulated in any (knowably sound) computational procedure. This has the implication that there is something involved in human understanding that lies beyond the actions of any computer. Understanding is a particular manifestation of human consciousn ...
... that human mathematical understanding cannot be encapsulated in any (knowably sound) computational procedure. This has the implication that there is something involved in human understanding that lies beyond the actions of any computer. Understanding is a particular manifestation of human consciousn ...
On Sets of Premises - Matematički Institut SANU
... guage of propositional logic or a first-order language. Instead of the turnstile ⊢ Gentzen writes → (which is more commonly used nowadays for the binary connective of implication; we use it below, as usual, for separating the sources and targets of arrows in categories), for A and B he uses Gothic l ...
... guage of propositional logic or a first-order language. Instead of the turnstile ⊢ Gentzen writes → (which is more commonly used nowadays for the binary connective of implication; we use it below, as usual, for separating the sources and targets of arrows in categories), for A and B he uses Gothic l ...
1992-Ideal Introspective Belief
... atoms (of the form ~Lc$) are not soundly derivable from the premises alone. For example, consider the premise set {lLp > q,p V q}. We would like since there is no reasonable way of to conclude ‘Lp, coming to believe p. But an inference rule that would allow us to conclude 1Lp would have to take into ...
... atoms (of the form ~Lc$) are not soundly derivable from the premises alone. For example, consider the premise set {lLp > q,p V q}. We would like since there is no reasonable way of to conclude ‘Lp, coming to believe p. But an inference rule that would allow us to conclude 1Lp would have to take into ...
Distributed Knowledge
... some examples of semantics that can with some plausibility be called variations on the possible worlds view those that introduce `impossible possible worlds', the so-called `awareness logics,' and perhaps situation theory can be classi ed under this heading as well. However, the operators Ka , even ...
... some examples of semantics that can with some plausibility be called variations on the possible worlds view those that introduce `impossible possible worlds', the so-called `awareness logics,' and perhaps situation theory can be classi ed under this heading as well. However, the operators Ka , even ...
Linear Contextual Modal Type Theory
... Inspired by the double negation translation and [MS07], we make the relation between ! and →, and [Γ; ∆] and [Γ; ∆]→ precise. We show that both formulations of the rules prove the same theorems via the following translation and its inverse. The injection ·i translates propositions that use modalitie ...
... Inspired by the double negation translation and [MS07], we make the relation between ! and →, and [Γ; ∆] and [Γ; ∆]→ precise. We show that both formulations of the rules prove the same theorems via the following translation and its inverse. The injection ·i translates propositions that use modalitie ...
Recall... Venn Diagrams Disjunctive normal form Disjunctive normal
... of a logic table (equivalently: given an interpretation for all possible assignments). This begs the question: ...
... of a logic table (equivalently: given an interpretation for all possible assignments). This begs the question: ...
Implicative Formulae in the Vroofs as Computations” Analogy
... 8. A formula is either an atomic proposition or the product A8B of two formulae. An intuitionistic sequent has the syntactic structure l? I- B where r is a finite (possibly ...
... 8. A formula is either an atomic proposition or the product A8B of two formulae. An intuitionistic sequent has the syntactic structure l? I- B where r is a finite (possibly ...
An admissible second order frame rule in region logic
... antecedent {ϕ} C {ϕ0 } bounds the effects of C ; loosely speaking, its footprint is within the confines of ϕ and ϕ0 . The second order frame rule lifts this reasoning to the level of assumptions, distilling the essence of Hoare’s mismatch. In separation logic, the second order frame rule has nontriv ...
... antecedent {ϕ} C {ϕ0 } bounds the effects of C ; loosely speaking, its footprint is within the confines of ϕ and ϕ0 . The second order frame rule lifts this reasoning to the level of assumptions, distilling the essence of Hoare’s mismatch. In separation logic, the second order frame rule has nontriv ...
Kripke completeness revisited
... definitions and the standard notions of what is nowadays regarded as “Kripke semantics.” ...
... definitions and the standard notions of what is nowadays regarded as “Kripke semantics.” ...
On Natural Deduction in Classical First-Order Logic: Curry
... as valid as ever, because it provides a theoretical justification for an important quest: the search for the constructive content of classical proofs. Herbrand’s Theorem tells us what is the immediate computational content of classical first-order logic: the list of witnesses contained in any Herbra ...
... as valid as ever, because it provides a theoretical justification for an important quest: the search for the constructive content of classical proofs. Herbrand’s Theorem tells us what is the immediate computational content of classical first-order logic: the list of witnesses contained in any Herbra ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... negation in the context of residuated logic programming is provided in terms of the notion of coherence as a generalization in the fuzzy framework of the concept of consistence. Then, fuzzy answer sets for general residuated logic programs are defined as a suitable generalization of the Gelfond-Lifs ...
... negation in the context of residuated logic programming is provided in terms of the notion of coherence as a generalization in the fuzzy framework of the concept of consistence. Then, fuzzy answer sets for general residuated logic programs are defined as a suitable generalization of the Gelfond-Lifs ...
S2 - CALCULEMUS.ORG
... result, until the questions and problems concerning researches on the foundations of computer science arose, most of the mathematicians did not pay too much attention to semantics restricted to finite models. The most extreme example of this is the monograph [Chang – Keisler 1990] reassuming some im ...
... result, until the questions and problems concerning researches on the foundations of computer science arose, most of the mathematicians did not pay too much attention to semantics restricted to finite models. The most extreme example of this is the monograph [Chang – Keisler 1990] reassuming some im ...
Geometry Notes 2.2 Logic Determining Truths Values
... q: A triangle has two acute angles 1. p ∧ q 2. ~p∧ q ...
... q: A triangle has two acute angles 1. p ∧ q 2. ~p∧ q ...
Predicate_calculus
... In order to formulate the predicate calculus one must first fix an exact logico-mathematical language Ω . In the most common case of single-sorted first-order languages, such a language contains predicate variables x,y,z,…, function symbols f,g,h,… with a varying number of argument places, and predi ...
... In order to formulate the predicate calculus one must first fix an exact logico-mathematical language Ω . In the most common case of single-sorted first-order languages, such a language contains predicate variables x,y,z,…, function symbols f,g,h,… with a varying number of argument places, and predi ...
CS243, Logic and Computation Propositional Logic 1 Propositions
... truth assignment to a set P of propositional symbols. The truth value with respect to t of any propositional statement over P is then defined as follows. 1. (Basis) The truth value of each basic proposition is as given directly by t. 2. (Recursion) If p and q are propositions over P with truth value ...
... truth assignment to a set P of propositional symbols. The truth value with respect to t of any propositional statement over P is then defined as follows. 1. (Basis) The truth value of each basic proposition is as given directly by t. 2. (Recursion) If p and q are propositions over P with truth value ...
Birkhoff`s variety theorem in many sorts
... X is a finite S-sorted set and t, t are elements of FΣ X of the same sort. A Σ-algebra A satisfies the equation provided that for every S-sorted function f : X → A, the homomorphism f¯: FΣ X → A extending f merges t and t . An equational class is a full subcategory of Σ-Alg specified by a set of equa ...
... X is a finite S-sorted set and t, t are elements of FΣ X of the same sort. A Σ-algebra A satisfies the equation provided that for every S-sorted function f : X → A, the homomorphism f¯: FΣ X → A extending f merges t and t . An equational class is a full subcategory of Σ-Alg specified by a set of equa ...
Section 1
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is ...
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is ...
Chapter 0 - Ravikumar - Sonoma State University
... • Assertions: Mathematical statement expresses some property of a set of defined objects. Assertions may or may not be true. ...
... • Assertions: Mathematical statement expresses some property of a set of defined objects. Assertions may or may not be true. ...