Slide 1
... • Proofs: valid arguments that establish the truth of mathematical statements – Argument: a sequence of statements that end with a conclusion – Valid: the conclusion must follow from the preceding statements (premises) of the argument ...
... • Proofs: valid arguments that establish the truth of mathematical statements – Argument: a sequence of statements that end with a conclusion – Valid: the conclusion must follow from the preceding statements (premises) of the argument ...
Natural deduction for predicate logic
... In the next module, we will describe the semantics of predicate logic, and discuss soundness and completeness without proof. Still to come are other proof systems for predicate logic, and a discussion of how to ensure that specific mathematical situations (such as number theory or set theory) are pr ...
... In the next module, we will describe the semantics of predicate logic, and discuss soundness and completeness without proof. Still to come are other proof systems for predicate logic, and a discussion of how to ensure that specific mathematical situations (such as number theory or set theory) are pr ...
Formal logic
... logical arguments in which, starting from two premises, a conclusion is reached. For example, given that There are horses in Spain. All horses are mammals. it can be inferred that There are mammals in Spain. Of course, if instead of the second premise we had the weaker one Some horses are mammals. w ...
... logical arguments in which, starting from two premises, a conclusion is reached. For example, given that There are horses in Spain. All horses are mammals. it can be inferred that There are mammals in Spain. Of course, if instead of the second premise we had the weaker one Some horses are mammals. w ...
Adding the Everywhere Operator to Propositional Logic (pdf file)
... Comparison with earlier complete axiomatizations As mentioned in Sec. 1, a number of complete axiomatizations of C have been given [13, 2, 1, 11, 9]. All of them are similar in nature to the following one, which we take from [9]. Begin with Schematic S5 (see Table 2). Instead of adding inference ru ...
... Comparison with earlier complete axiomatizations As mentioned in Sec. 1, a number of complete axiomatizations of C have been given [13, 2, 1, 11, 9]. All of them are similar in nature to the following one, which we take from [9]. Begin with Schematic S5 (see Table 2). Instead of adding inference ru ...
CHAPTER 5 SOME EXTENSIONAL SEMANTICS
... Many valued logics in general and 3-valued logics in particular is an old object of study which had its beginning in the work of Lukasiewicz (1920). He was the first to define a 3- valued semantics for a language L¬,∩,∪,⇒ of classical logic, and called it a three valued logic for short. He left the ...
... Many valued logics in general and 3-valued logics in particular is an old object of study which had its beginning in the work of Lukasiewicz (1920). He was the first to define a 3- valued semantics for a language L¬,∩,∪,⇒ of classical logic, and called it a three valued logic for short. He left the ...
The semantics of predicate logic
... Readings: Section 2.4, 2.5, 2.6. In this module, we will precisely define the semantic interpretation of formulas in our predicate logic. In propositional logic, every formula had a fixed, finite number of models (interpretations); this is not the case in predicate logic. As a consequence, we must t ...
... Readings: Section 2.4, 2.5, 2.6. In this module, we will precisely define the semantic interpretation of formulas in our predicate logic. In propositional logic, every formula had a fixed, finite number of models (interpretations); this is not the case in predicate logic. As a consequence, we must t ...
Rewriting Predicate Logic Statements
... New Proof Strategy ‘Antecedent Assumption’” of the next slide set, you should be able for each proof strategy below to: (1) identify the form of statement the strategy can prove and (2) sketch the structure of a proof that uses the strategy. Strategies: constructive/non-constructive proofs of existe ...
... New Proof Strategy ‘Antecedent Assumption’” of the next slide set, you should be able for each proof strategy below to: (1) identify the form of statement the strategy can prove and (2) sketch the structure of a proof that uses the strategy. Strategies: constructive/non-constructive proofs of existe ...
The Surprise Examination Paradox and the Second Incompleteness
... theory, there exists a (large enough) integer L (depending on the theory and on the programming language that is used to define Kolmogorov complexity) such that, for any integer x, the statement “K(x) > L” cannot be proved within the theory. The proof given by Chaitin is as follows. Let L be a large ...
... theory, there exists a (large enough) integer L (depending on the theory and on the programming language that is used to define Kolmogorov complexity) such that, for any integer x, the statement “K(x) > L” cannot be proved within the theory. The proof given by Chaitin is as follows. Let L be a large ...
GLukG logic and its application for non-monotonic reasoning
... might spoil the whole program. Most logics (at least classical logic and all constructive intermediate logics) share the theorem (a ∧ ¬a) → b, meaning that in the presence of an inconsistency (a ∧ ¬a) then one prove anything (such as the ...
... might spoil the whole program. Most logics (at least classical logic and all constructive intermediate logics) share the theorem (a ∧ ¬a) → b, meaning that in the presence of an inconsistency (a ∧ ¬a) then one prove anything (such as the ...
Logical Prior Probability - Institute for Creative Technologies
... from its use here, rather than set ahead of time. If the same symbol is used with multiple arities, the uses are independent (so f2 would notate distinct functions in f2 (v1 ) versus f2 (v1 , v2 )). – An expression is a composition of function symbols and variable symbols, for example f1 (f1 (v1 )). ...
... from its use here, rather than set ahead of time. If the same symbol is used with multiple arities, the uses are independent (so f2 would notate distinct functions in f2 (v1 ) versus f2 (v1 , v2 )). – An expression is a composition of function symbols and variable symbols, for example f1 (f1 (v1 )). ...
Infinite Games - International Mathematical Union
... Each sequence (z 0 , ..., z„) must belong to T. I wins & play of G just in case the sequence {zt: i^co)£A. The notions of strategy for I (or II) for G and winning strategy for I (or II) for G are defined in the obvious way. G is determined if either I or II has a winning strategy for G. Gale and Ste ...
... Each sequence (z 0 , ..., z„) must belong to T. I wins & play of G just in case the sequence {zt: i^co)£A. The notions of strategy for I (or II) for G and winning strategy for I (or II) for G are defined in the obvious way. G is determined if either I or II has a winning strategy for G. Gale and Ste ...
Formal Logic, Models, Reality
... This gives the meaning of ''. The 'if-then' on the right-hand side is the usual nonformal conditional. The meaning of 'A B' is defined in the metalanguage of the formal language. This is unavoidable because, by Tarski's theorem on truth definitions, the truth predicate cannot be represented in a ...
... This gives the meaning of ''. The 'if-then' on the right-hand side is the usual nonformal conditional. The meaning of 'A B' is defined in the metalanguage of the formal language. This is unavoidable because, by Tarski's theorem on truth definitions, the truth predicate cannot be represented in a ...
Bilattices In Logic Programming
... least fixed point supplies a denotational meaning for the program. We showed in [9] that these ideas carry over in a straightforward way to logic programming languages with an interlaced bilattice as the space of truth values. The most natural ‘direction’ in which to evaluate a least fixed point is ...
... least fixed point supplies a denotational meaning for the program. We showed in [9] that these ideas carry over in a straightforward way to logic programming languages with an interlaced bilattice as the space of truth values. The most natural ‘direction’ in which to evaluate a least fixed point is ...
Discrete Structure
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...
... Applications of Predicate Logic • It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in module 2) for any branch of mathematics. • Predicate logic with function symbols, the “=” operator, and a few proof-buil ...