x - Agus Aan
... sentence we are trying to prove is TRUE). • If we can infer that FALSE is TRUE we know the knowledgebase is corrupt. • The only thing that might not be TRUE is the negation of the goal that we added, so if must be FALSE. Therefore the goal is true. ...
... sentence we are trying to prove is TRUE). • If we can infer that FALSE is TRUE we know the knowledgebase is corrupt. • The only thing that might not be TRUE is the negation of the goal that we added, so if must be FALSE. Therefore the goal is true. ...
Predicate logic
... Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for s ...
... Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for s ...
Lecture 14 Notes
... It is easy (though tedious) to show that truth sets correspond to valuations in the sense that every first-order truth set is exactly the set of all formulas that are true under a fixed first-order valuation. The definition of first-order valuations can be extended to sentences with parameters as f ...
... It is easy (though tedious) to show that truth sets correspond to valuations in the sense that every first-order truth set is exactly the set of all formulas that are true under a fixed first-order valuation. The definition of first-order valuations can be extended to sentences with parameters as f ...
GLukG logic and its application for non-monotonic reasoning
... Note that the first 8 axioms somewhat constraint the meaning of the →, ∧ and ∨ connectives to match our usual intuition. It is a well known result that in any logic satisfying axioms Pos1 and Pos2, and with modus ponens as its unique inference rule, the deduction theorem holds [9]. Multivalued logic ...
... Note that the first 8 axioms somewhat constraint the meaning of the →, ∧ and ∨ connectives to match our usual intuition. It is a well known result that in any logic satisfying axioms Pos1 and Pos2, and with modus ponens as its unique inference rule, the deduction theorem holds [9]. Multivalued logic ...
Introduction to Theoretical Computer Science, lesson 3
... An argument is valid iff the conclusion is true in every model of the set of the premises. But the set of models can be infinite! And, of course, we cannot examine an infinite number of models; but we can verify the ‘logical form’ of the argument, and check whether the models of premises do satisfy ...
... An argument is valid iff the conclusion is true in every model of the set of the premises. But the set of models can be infinite! And, of course, we cannot examine an infinite number of models; but we can verify the ‘logical form’ of the argument, and check whether the models of premises do satisfy ...
Lecture 7. Model theory. Consistency, independence, completeness
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
Chapter 1: The Foundations: Logic and Proofs
... •QP is the CONVERSE of P Q •¬ Q ¬ P is the CONTRAPOSITIVE of P Q •¬ P ¬ Q is the inverse of P Q •Example: Find the converse of the following statement: R: ‘Raining tomorrow is a sufficient condition for my not going to town.’ ...
... •QP is the CONVERSE of P Q •¬ Q ¬ P is the CONTRAPOSITIVE of P Q •¬ P ¬ Q is the inverse of P Q •Example: Find the converse of the following statement: R: ‘Raining tomorrow is a sufficient condition for my not going to town.’ ...
• Propositional definite clauses ctd • Monotone functions and power
... Completeness of the inference procedure We want to show that if S ` q, then the inference procedure will give us a derivation. The procedure we have described does not explicitly construct derivations, however. So it is not complete in that sense. It is complete in the sense that if S ` q, the proce ...
... Completeness of the inference procedure We want to show that if S ` q, then the inference procedure will give us a derivation. The procedure we have described does not explicitly construct derivations, however. So it is not complete in that sense. It is complete in the sense that if S ` q, the proce ...
A systematic proof theory for several modal logics
... for a more restrictive language can naturally be projected to richer languages by application of substitution. Two logical systems are equivalent if their theories are the same, they are distinct otherwise. If the theory of one logical system strictly contains the theory of another, then we call the ...
... for a more restrictive language can naturally be projected to richer languages by application of substitution. Two logical systems are equivalent if their theories are the same, they are distinct otherwise. If the theory of one logical system strictly contains the theory of another, then we call the ...
Propositional Dynamic Logic of Regular Programs*+
... [lo] can be expressed as “‘~3 [a]+” The fact that a can terminate can be expressed by the assertion “(a) true.” The determinacy of a program a can be expressed by the formula “(a)~ I) [a] p,” where p expresses the condition of determinacy. One goal in developing ...
... [lo] can be expressed as “‘~3 [a]+” The fact that a can terminate can be expressed by the assertion “(a) true.” The determinacy of a program a can be expressed by the formula “(a)~ I) [a] p,” where p expresses the condition of determinacy. One goal in developing ...
Slide 1
... your class has taken a course in Calculus” – “There is a student in your class who has not taken a course in Calculus” ...
... your class has taken a course in Calculus” – “There is a student in your class who has not taken a course in Calculus” ...
A Calculus for Belnap`s Logic in Which Each Proof Consists of Two
... This is the notion of entailment considered in Belnap [5, 6], but not that of Arieli & Avron [1], who use a single-barrelled notion. The two notions of entailment are coextensional on sets of formulas based on classical connectives only, but not on formulas based on a functionally complete set of co ...
... This is the notion of entailment considered in Belnap [5, 6], but not that of Arieli & Avron [1], who use a single-barrelled notion. The two notions of entailment are coextensional on sets of formulas based on classical connectives only, but not on formulas based on a functionally complete set of co ...
The Future of Post-Human Mathematical Logic
... In this newest tome, Dr. Baofu tackles yet another set of sacrosanct beliefs which few thinkers would dare to question—the foundations of mathematics and logic. He examines the reasoning of forebears, points out specific shortcomings, and offers another perspective to fulfill those shortcomings. The ...
... In this newest tome, Dr. Baofu tackles yet another set of sacrosanct beliefs which few thinkers would dare to question—the foundations of mathematics and logic. He examines the reasoning of forebears, points out specific shortcomings, and offers another perspective to fulfill those shortcomings. The ...
Some Principles of Logic
... • Only man has a sense of humour • No man is a woman • No women have a sense of humour ...
... • Only man has a sense of humour • No man is a woman • No women have a sense of humour ...
Completeness Theorem for Continuous Functions and Product
... in [11]. After that paper, the technique has been applied for proving completeness theorems for many infinitary logics with generalized quantifiers [4–7, 12]. Roughly speaking, adding new quantifiers to infinitary logics is one of the most frequent and high acceptable ways to incorporate into the realm ...
... in [11]. After that paper, the technique has been applied for proving completeness theorems for many infinitary logics with generalized quantifiers [4–7, 12]. Roughly speaking, adding new quantifiers to infinitary logics is one of the most frequent and high acceptable ways to incorporate into the realm ...
CHAPTER 5 SOME EXTENSIONAL SEMANTICS
... We call the H logic a Heyting logic because its connectives are defined as operations on the set {F, ⊥, T } in such a way that they form a 3-element Heyting algebra, called also a 3-element pseudo-boolean algebra. Pseudo-boolean, or Heyting algebras provide algebraic models for the intuitionistic lo ...
... We call the H logic a Heyting logic because its connectives are defined as operations on the set {F, ⊥, T } in such a way that they form a 3-element Heyting algebra, called also a 3-element pseudo-boolean algebra. Pseudo-boolean, or Heyting algebras provide algebraic models for the intuitionistic lo ...
A Proof of Nominalism. An Exercise in Successful
... for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical assumption. Systematically speaking, and even more importantly, the version of the axiom ...
... for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical assumption. Systematically speaking, and even more importantly, the version of the axiom ...
proceedings version
... A here-and-there model (HT model) is made up of two sets of propositional variables H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the materia ...
... A here-and-there model (HT model) is made up of two sets of propositional variables H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the materia ...
Chapter 2 - Princeton University Press
... (b) for each set x there exists a set y with the property that the members of y are the same as the members of members of x. The issue is not whether empty sets and unions “really exist,” but rather, what consequences can be proved about some abstract objects from an abstract system of axioms, consi ...
... (b) for each set x there exists a set y with the property that the members of y are the same as the members of members of x. The issue is not whether empty sets and unions “really exist,” but rather, what consequences can be proved about some abstract objects from an abstract system of axioms, consi ...
Decidable fragments of first-order logic Decidable fragments of first
... Here fragments of first-order logic are distinguish according to That the set of valid L-sentences is not decidable means that there is no effective procedure that on any input eventually terminates and correctly decides whether the input is valid or not. A sentence is valid if and only if its negat ...
... Here fragments of first-order logic are distinguish according to That the set of valid L-sentences is not decidable means that there is no effective procedure that on any input eventually terminates and correctly decides whether the input is valid or not. A sentence is valid if and only if its negat ...
Identity in modal logic theorem proving
... construct proofs within one of these proof theories - - by which I mean both that the result generated would be recognized as a proof in [say] Whitehead Russell's axiom system and also that the "machine internal" strategies and methods are applications of what it is legal to do within the proof theo ...
... construct proofs within one of these proof theories - - by which I mean both that the result generated would be recognized as a proof in [say] Whitehead Russell's axiom system and also that the "machine internal" strategies and methods are applications of what it is legal to do within the proof theo ...
Scoring Rubric for Assignment 1
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
A(x)
... a Greek philosopher, a student of Plato and teacher of Alexander the Great. He wrote on diverse subjects, including physics, metaphysics, poetry (including theater), biology and zoology, logic, rhetoric, politics, government, and ethics. Along with Socrates and Plato, Aristotle was one of the most i ...
... a Greek philosopher, a student of Plato and teacher of Alexander the Great. He wrote on diverse subjects, including physics, metaphysics, poetry (including theater), biology and zoology, logic, rhetoric, politics, government, and ethics. Along with Socrates and Plato, Aristotle was one of the most i ...
Chapter 12 Reasoning, Logic, and Fallacies
... then show that the sample size is too small. Note: a formal proof would require a mathematical calculation. This is the subject of probability theory. For now, you must rely on common sense. • References: Barker: 189, Cedarblom and Paulsen: 372, Davis: 103 26 May 1995 ...
... then show that the sample size is too small. Note: a formal proof would require a mathematical calculation. This is the subject of probability theory. For now, you must rely on common sense. • References: Barker: 189, Cedarblom and Paulsen: 372, Davis: 103 26 May 1995 ...
Logical Argument
... we should even expect an argument to be rigorous, that is, to adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to be a formal proof. In ordinary language, people refer to the logic of an argument or use terminol ...
... we should even expect an argument to be rigorous, that is, to adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to be a formal proof. In ordinary language, people refer to the logic of an argument or use terminol ...