Nonmonotonic Reasoning - Computer Science Department
... structures (in case of predicate logic) or Kripke structures (for intuitionistic and modal logics). Semantics always generates a semantic consequence relation defined by means of semantic entailment. Let us see how it works in the case of propositional logic. We say that a formula ϕ is a semantic con ...
... structures (in case of predicate logic) or Kripke structures (for intuitionistic and modal logics). Semantics always generates a semantic consequence relation defined by means of semantic entailment. Let us see how it works in the case of propositional logic. We say that a formula ϕ is a semantic con ...
Lesson 12
... Propositional Logic and Predicate Logic each with Modus Ponens as their inference produce are sound but not complete. We shall see that we need further (sound) rules of inference to achieve completeness. In fact we shall see that we shall even restrict the language in order to achieve an effective i ...
... Propositional Logic and Predicate Logic each with Modus Ponens as their inference produce are sound but not complete. We shall see that we need further (sound) rules of inference to achieve completeness. In fact we shall see that we shall even restrict the language in order to achieve an effective i ...
Second-Order Logic and Fagin`s Theorem
... Corollary 7.10 Every polynomial-time query is expressible as a second-order, existential Horn formula: P ⊆ SO∃-Horn. The proof of Theorem 7.8 shows that nondeterministic time nk is contained in (SO∃, arity 2k). Lynch improved this to arity k. His proof uses the numeric predicate PLUS. Fagin’s theore ...
... Corollary 7.10 Every polynomial-time query is expressible as a second-order, existential Horn formula: P ⊆ SO∃-Horn. The proof of Theorem 7.8 shows that nondeterministic time nk is contained in (SO∃, arity 2k). Lynch improved this to arity k. His proof uses the numeric predicate PLUS. Fagin’s theore ...
INTERPLAYS OF KNOWLEDGE AND NON
... translated into classical normal modal logics and they are sound and complete with respect to some given class of Kripke frames, in the same way normal modal logics are (see [14]). Since the work developed in [7], a lot of important results have appeared in the domain of epistemic logics. Many autho ...
... translated into classical normal modal logics and they are sound and complete with respect to some given class of Kripke frames, in the same way normal modal logics are (see [14]). Since the work developed in [7], a lot of important results have appeared in the domain of epistemic logics. Many autho ...
Informal proofs
... • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often help us to decide the soundness o ...
... • The steps of the proofs are not expressed in any formal language as e.g. propositional logic • Steps are argued less formally using English, mathematical formulas and so on • One must always watch the consistency of the argument made, logic and its rules can often help us to decide the soundness o ...
Robot Morality and Review of classical logic.
... Analytic philosophy (like proving God’ Existence, free will, the problem of evil, etc) Many other… At this point I should ask all students to give another examples of similar problems that they want to solve ...
... Analytic philosophy (like proving God’ Existence, free will, the problem of evil, etc) Many other… At this point I should ask all students to give another examples of similar problems that they want to solve ...
Chapter 0 - Ravikumar - Sonoma State University
... Assertion: Every integer is a sum of squares of two integers. This is not true. To disprove it, it is enough to find one integer (counterexample) that can’t be written as sum of two squares. Consider 3. Suppose x2 + y2 = 3 for some integers x and y. This means, x2 is either 0, 1 or 2. (Why not any ...
... Assertion: Every integer is a sum of squares of two integers. This is not true. To disprove it, it is enough to find one integer (counterexample) that can’t be written as sum of two squares. Consider 3. Suppose x2 + y2 = 3 for some integers x and y. This means, x2 is either 0, 1 or 2. (Why not any ...
Lecture01 - Mathematics
... homeschooling my eldest son, I took him through half a college text in symbolic logic during ninth grade when he was also learning Algebra I. He had great success with that material despite his limited mathematics background.) ...
... homeschooling my eldest son, I took him through half a college text in symbolic logic during ninth grade when he was also learning Algebra I. He had great success with that material despite his limited mathematics background.) ...
Propositions as Types - Informatics Homepages Server
... consideration at one moment must be bounded (“We cannot tell at a glance whether 9999999999999999 and 999999999999999 are the same”). Later, Gandy [18] would point out that Turing’s argument amounts to a theorem asserting that any computation a human with paper and pencil can perform can also be per ...
... consideration at one moment must be bounded (“We cannot tell at a glance whether 9999999999999999 and 999999999999999 are the same”). Later, Gandy [18] would point out that Turing’s argument amounts to a theorem asserting that any computation a human with paper and pencil can perform can also be per ...
Standardization of Formulæ
... An existential quantifier can be removed by replacing the variable it bounds by a Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r ( ...
... An existential quantifier can be removed by replacing the variable it bounds by a Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r ( ...
MMConceptualComputationalRemainder
... The conceptual proof given above provides a geometric visualization of the situation required by the hypothesis of the theorem, and this visualization makes the truth of the theorem obvious. But there is a sense of "conceptual", related to the idea of conceptual definition given under elementary, th ...
... The conceptual proof given above provides a geometric visualization of the situation required by the hypothesis of the theorem, and this visualization makes the truth of the theorem obvious. But there is a sense of "conceptual", related to the idea of conceptual definition given under elementary, th ...
A Proof Theory for Generic Judgments: An extended abstract
... γ to denote that t is a simply typed λ-term of type γ in which there may appear the (fixed) logical and non-logical constants as well as those eigenvariables in Σ. We shall also say t is a Σ-term (of type γ), and, if γ is o, t is a Σ-formula. In the displayed sequent above, n ≥ 0 and B0 , B1 , . . . ...
... γ to denote that t is a simply typed λ-term of type γ in which there may appear the (fixed) logical and non-logical constants as well as those eigenvariables in Σ. We shall also say t is a Σ-term (of type γ), and, if γ is o, t is a Σ-formula. In the displayed sequent above, n ≥ 0 and B0 , B1 , . . . ...
An Invitation to Proofs Without Words
... help the reader see why a particular mathematical statement may be true, and also to see how one might begin to go about proving it true. As Yuri Ivanovich Manin said, “A good proof is one that makes us wiser,” a sentiment echoed by Andrew Gleason: “Proofs really aren’t there to convince you that so ...
... help the reader see why a particular mathematical statement may be true, and also to see how one might begin to go about proving it true. As Yuri Ivanovich Manin said, “A good proof is one that makes us wiser,” a sentiment echoed by Andrew Gleason: “Proofs really aren’t there to convince you that so ...
2.1-2.3: Reasoning in Geometry
... “Anything that has wheels and moves people from place to place.” Not all definitions may be precise, so when creating or following one, read carefully! ...
... “Anything that has wheels and moves people from place to place.” Not all definitions may be precise, so when creating or following one, read carefully! ...
Midterm Exam 1 Solutions, Comments, and Feedback
... • Implications involving variables: In statements such as “f (x) < f (y) whenever x < y” or “if n is odd, then n is prime” the variables (x, y, and n) are understood to be arbitrary elements of the underlying universe, i.e., in the sense of a “for all” quantifier. When negating such statements, this ...
... • Implications involving variables: In statements such as “f (x) < f (y) whenever x < y” or “if n is odd, then n is prime” the variables (x, y, and n) are understood to be arbitrary elements of the underlying universe, i.e., in the sense of a “for all” quantifier. When negating such statements, this ...
Knowledge Representation: Logic
... Logic gives processable form to all the information that can be precisely expressed in any other language. Logic allows to express all the information that can be stored in computer memory. If some knowledge is not logic-conveyable it cannot be processed by computers no matter the notation. ...
... Logic gives processable form to all the information that can be precisely expressed in any other language. Logic allows to express all the information that can be stored in computer memory. If some knowledge is not logic-conveyable it cannot be processed by computers no matter the notation. ...