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Logic and Proof - Collaboratory for Advanced Computing and
... Methods of Proving Theorems Proving implications p → q: Direct proof: Assume p is T, and use rules of inference to prove that q is T Indirect proof: Prove its contrapositive; assume ¬q, and prove ¬p Proof by cases: Prove (p1 ∨ p2) → q by proving (p1 → q) and (p1 → q) • Based on [(p1 ∨ p2) → q ...
... Methods of Proving Theorems Proving implications p → q: Direct proof: Assume p is T, and use rules of inference to prove that q is T Indirect proof: Prove its contrapositive; assume ¬q, and prove ¬p Proof by cases: Prove (p1 ∨ p2) → q by proving (p1 → q) and (p1 → q) • Based on [(p1 ∨ p2) → q ...
Slide 1
... • The checking algorithm A would then verify that the tour really does visit all of the cities and really does have total length K. without seeking all possible K solutions through each of the vertices. Polynomial. • The TSP, therefore, also belongs to NP. • How could a problem fail to belong to NP? ...
... • The checking algorithm A would then verify that the tour really does visit all of the cities and really does have total length K. without seeking all possible K solutions through each of the vertices. Polynomial. • The TSP, therefore, also belongs to NP. • How could a problem fail to belong to NP? ...
Constructive Mathematics, in Theory and Programming Practice
... importance than the mathematical activity from which they were abstracted. From the 1940s there also grew, in the former Soviet Union, a substantial group of analysts, led by A.A. Markov, who practised what was essentially recursive mathematics using intuitionistic logic. Although this group accompl ...
... importance than the mathematical activity from which they were abstracted. From the 1940s there also grew, in the former Soviet Union, a substantial group of analysts, led by A.A. Markov, who practised what was essentially recursive mathematics using intuitionistic logic. Although this group accompl ...
The Future of Post-Human Mathematical Logic
... (and Other Mental States) ............................................................67 Table 1.11. The Theoretical Levels of Consciousness (and Other Mental States) ............................................................68 Table 1.12. The Thematic Issues of Consciousness (and Other Mental Sta ...
... (and Other Mental States) ............................................................67 Table 1.11. The Theoretical Levels of Consciousness (and Other Mental States) ............................................................68 Table 1.12. The Thematic Issues of Consciousness (and Other Mental Sta ...
RR-01-02
... plicability of FEC is proven to be the K Features and Fluents taxonomy. The proof is given with respect to the original definition of this preference logic, where no adjustments of the language or reasoning method formally captures all of the were necessary. As K above characteristics, this assessme ...
... plicability of FEC is proven to be the K Features and Fluents taxonomy. The proof is given with respect to the original definition of this preference logic, where no adjustments of the language or reasoning method formally captures all of the were necessary. As K above characteristics, this assessme ...
Computational foundations of basic recursive function theory
... such concepts in this framework. We represent in the theory those mathematical objects that algorithms compute, numbers andfunctions; iffand g compute functions then asfunctions they are equal precisely iff(a) = g(u) for all a in the domain offand g. This is thus an extensional notion of equality. F ...
... such concepts in this framework. We represent in the theory those mathematical objects that algorithms compute, numbers andfunctions; iffand g compute functions then asfunctions they are equal precisely iff(a) = g(u) for all a in the domain offand g. This is thus an extensional notion of equality. F ...
CPS130, Lecture 1: Introduction to Algorithms
... Much of the material to be presented this week will be a review, but it is so important that I feel this is justified. For example, when I taught cps 196, Discrete Mathematics, Fall 2001, some students who had already taken cps130 did not feel entirely comfortable with this material. To make the pre ...
... Much of the material to be presented this week will be a review, but it is so important that I feel this is justified. For example, when I taught cps 196, Discrete Mathematics, Fall 2001, some students who had already taken cps130 did not feel entirely comfortable with this material. To make the pre ...
The Formulae-as-Classes Interpretation of Constructive Set Theory
... classical Zermelo-Fraenkel Set Theory with the axiom of choice relates to classical Cantorian mathematics. CST provides a standard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop [8]. One of the hallmarks of constructive set theory is that it p ...
... classical Zermelo-Fraenkel Set Theory with the axiom of choice relates to classical Cantorian mathematics. CST provides a standard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop [8]. One of the hallmarks of constructive set theory is that it p ...
(draft)
... that method cannot be used in the Constructive logic system. So the following classical logic proof rules are not found in Constructive logic: ¬¬φ φ ...
... that method cannot be used in the Constructive logic system. So the following classical logic proof rules are not found in Constructive logic: ¬¬φ φ ...
Chapter 4, Mathematics
... That opened the way for a rigorous re-examination of the status of Mathematics. Even in the eighteenth century Leibnitz had dreamt of a sort of arithmetic of logic, by means of which the truth value of any proposition could be determined by some sort of calculation. He hoped that we might get to a p ...
... That opened the way for a rigorous re-examination of the status of Mathematics. Even in the eighteenth century Leibnitz had dreamt of a sort of arithmetic of logic, by means of which the truth value of any proposition could be determined by some sort of calculation. He hoped that we might get to a p ...
Linearity and nonlinearity
... second can be described as “evaluation of f distributes over addition.” A function that is not linear is called nonlinear. Proof by contradiction A proof technique that is generally useful throughout mathematics is proof by contradiction. If you want to prove X is true, assume that X is not true and ...
... second can be described as “evaluation of f distributes over addition.” A function that is not linear is called nonlinear. Proof by contradiction A proof technique that is generally useful throughout mathematics is proof by contradiction. If you want to prove X is true, assume that X is not true and ...
Exam-Computational_Logic-Subjects_2016
... The theorem of deduction and its reverse. 7. Definitions: tautology, theorem, logical consequence, syntactic consequence, logical equivalence, consistent/contingent/valid/inconsistent formula, interpretation, model, anti-model. The axiomatic system of propositional logic. The axiomatic system of pro ...
... The theorem of deduction and its reverse. 7. Definitions: tautology, theorem, logical consequence, syntactic consequence, logical equivalence, consistent/contingent/valid/inconsistent formula, interpretation, model, anti-model. The axiomatic system of propositional logic. The axiomatic system of pro ...
Propositional Logic
... The validity of an argument do not depend on the truth of the premises but with the fact that if someone accepts the truth of the premises he/she must accept the conclusion. If someone does not accept the premises, he/she wont accept the conclusions but this does not invalidate the argument. ...
... The validity of an argument do not depend on the truth of the premises but with the fact that if someone accepts the truth of the premises he/she must accept the conclusion. If someone does not accept the premises, he/she wont accept the conclusions but this does not invalidate the argument. ...
Tautologies Arguments Logical Implication
... A derivation (or proof ) in an axiom system AX is a sequence of formulas C1 , . . . , C N ; each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX: ...
... A derivation (or proof ) in an axiom system AX is a sequence of formulas C1 , . . . , C N ; each formula Ck is either an axiom in AX or follows from previous formulas using an inference rule in AX: ...
Gödel on Conceptual Realism and Mathematical Intuition
... set theory describes a transfinite iteration of the setforming operations of the simple theory of types. Mathematical propositions are true by virtue of the meaning of the terms occurring in them. Parsons remarks that Gödel uses mathematical intuition in both object-relational and propositional atti ...
... set theory describes a transfinite iteration of the setforming operations of the simple theory of types. Mathematical propositions are true by virtue of the meaning of the terms occurring in them. Parsons remarks that Gödel uses mathematical intuition in both object-relational and propositional atti ...
We showed on Tuesday that Every relation in the arithmetical
... A (first-order) proof system is a set of rules which allows certain formulas to be derived from other formulas. Proposition The usual proof system (for arithmetic) is computable. For those who worry about the deductive power of the “usual proof system”: Gödel’s Completeness Theorem The usual proof ...
... A (first-order) proof system is a set of rules which allows certain formulas to be derived from other formulas. Proposition The usual proof system (for arithmetic) is computable. For those who worry about the deductive power of the “usual proof system”: Gödel’s Completeness Theorem The usual proof ...
2/TRUTH-FUNCTIONS
... 2 refers to the number of truth values and n refers to the number of distinct statement variables. s9. Truth-Functions: Any expression whose truth value is defined in all cases by its logical operator. 9a. Negation: Not (-) curl -> -p Expressions: it is not true that/ it is false that/ it is not the ...
... 2 refers to the number of truth values and n refers to the number of distinct statement variables. s9. Truth-Functions: Any expression whose truth value is defined in all cases by its logical operator. 9a. Negation: Not (-) curl -> -p Expressions: it is not true that/ it is false that/ it is not the ...
Document
... This is not true. To disprove it, it is enough to find one integer (counter-example) 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 other number?) case 1: x2 is 0. Thus y2 is 3. But from previou ...
... This is not true. To disprove it, it is enough to find one integer (counter-example) 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 other number?) case 1: x2 is 0. Thus y2 is 3. But from previou ...
Mathematical Logic Deciding logical consequence Complexity of
... For reasoning to be correct, this process should generally preserve truth. That is, the arguments should be valid. How can we be sure our arguments are valid? Reasoning takes place in many different ways in everyday life: Word of Authority: we derive conclusions from a source that we trust; e.g. rel ...
... For reasoning to be correct, this process should generally preserve truth. That is, the arguments should be valid. How can we be sure our arguments are valid? Reasoning takes place in many different ways in everyday life: Word of Authority: we derive conclusions from a source that we trust; e.g. rel ...
Propositional Logic, Predicates, and Equivalence
... more critical and creative thinking than the other material. Examples 1215 involve conjecturing a formula or rule for generating the terms of a sequence when only the first few terms are known. Encourage to try the On-Line Encyclopedia of Integer Sequences, mentioned in this section. The second part ...
... more critical and creative thinking than the other material. Examples 1215 involve conjecturing a formula or rule for generating the terms of a sequence when only the first few terms are known. Encourage to try the On-Line Encyclopedia of Integer Sequences, mentioned in this section. The second part ...