logical axiom
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
... ponens”, which states that from formulas A and A → B, one my deduce B. It is easy to see that this rule preserves logical validity. The axioms, together with modus ponens, form a sound deductive system for the classical propositional logic. In addition, it is also complete. Note that in the above se ...
22.1 Representability of Functions in a Formal Theory
... 1930’s. A variety of formal models has been developed, such as the λ-calculus (functional programming), Turing machines (imperative programming), recursive functions, and a huge number of programming languages. All these models have turned out to be equivalent. None is stronger than the others, so w ...
... 1930’s. A variety of formal models has been developed, such as the λ-calculus (functional programming), Turing machines (imperative programming), recursive functions, and a huge number of programming languages. All these models have turned out to be equivalent. None is stronger than the others, so w ...
Slides for Rosen, 5th edition
... • Mathematics is much more than that: Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts. ...
... • Mathematics is much more than that: Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts. ...
powerpoint - IDA.LiU.se
... Vocabulary for a logic formula: set of symbols containing all those that occur in the formula (and maybe some more) Interpretation for a logic formula: a mapping from a vocabulary for it, to truth-values T or F Model for a logic formula: an interpretation where the value of the formula is T Joint vo ...
... Vocabulary for a logic formula: set of symbols containing all those that occur in the formula (and maybe some more) Interpretation for a logic formula: a mapping from a vocabulary for it, to truth-values T or F Model for a logic formula: an interpretation where the value of the formula is T Joint vo ...
Proof
... • So, a2 and b2 would both have an even number of 2s in their factorizations. • So, 2b2 would have an odd number of 2s. • Since 2b2 has an odd number of 2s in its factorization, and a2 has an even number of 2s, by the UFT they can’t be equal! Contradiction. ...
... • So, a2 and b2 would both have an even number of 2s in their factorizations. • So, 2b2 would have an odd number of 2s. • Since 2b2 has an odd number of 2s in its factorization, and a2 has an even number of 2s, by the UFT they can’t be equal! Contradiction. ...
Jacques Herbrand (1908 - 1931) Principal writings in logic
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
... ES(A,p) that makes the expansion true and assigns the numerical value q to the constant c. œxœy∑zı(x,y,z) expresses the existence, for any p and q, of interpretations that make ES(A,p) true, and give the constant c the value q. If the theory is a true theory of arithmetic, then œxœy∑zı(x,y,z) is tru ...
Course discipline/number/title: MATH 1050: Foundations of
... 7. Convert Hindu-Arabic numbers to their equivalents in Egyptian, Mayan, and Roman numeration systems. 8. Use multi-base blocks to convert numbers between base ten and other bases. 9. Apply set theory and Venn diagrams to solve problems. 10. Demonstrate an understanding of the properties of addition ...
... 7. Convert Hindu-Arabic numbers to their equivalents in Egyptian, Mayan, and Roman numeration systems. 8. Use multi-base blocks to convert numbers between base ten and other bases. 9. Apply set theory and Venn diagrams to solve problems. 10. Demonstrate an understanding of the properties of addition ...
PDF
... It turns out that there is there is a deep connection between the type systems we have been exploring for the lambda calculus, and proof systems for a variety of logic known as intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics, which takes a more conservative view o ...
... It turns out that there is there is a deep connection between the type systems we have been exploring for the lambda calculus, and proof systems for a variety of logic known as intuitionistic logic. Intuitionistic logic is the basis of constructive mathematics, which takes a more conservative view o ...
Book Review: Lorenz J. Halbeisen: “Combinatorial Set Theory.”
... ℵ0 < a ≤ 2ℵ0 . An exciting area of set theory studies the possible values of cardinal characteristics and their relations to each other, cf. e.g. [1]. Part II of the book introduces the key method of this area: the technique of forcing, which originally was developped by P. Cohen to prove the indepe ...
... ℵ0 < a ≤ 2ℵ0 . An exciting area of set theory studies the possible values of cardinal characteristics and their relations to each other, cf. e.g. [1]. Part II of the book introduces the key method of this area: the technique of forcing, which originally was developped by P. Cohen to prove the indepe ...
Theories.Axioms,Rules of Inference
... and applies Rules of Inference to eventually obtain the formula we need to prove. And hence by definition, the above is a theorem. I show this proof format only to explain the formal definition of a proof, but its sometimes hard to do it in this fashion and in the future we will stick to the format ...
... and applies Rules of Inference to eventually obtain the formula we need to prove. And hence by definition, the above is a theorem. I show this proof format only to explain the formal definition of a proof, but its sometimes hard to do it in this fashion and in the future we will stick to the format ...
A simple proof of Parsons` theorem
... Hilbert’s programmatic ideas didn’t die with Gödel’s theorem. Rather, they were reformulated in the light of Gödel’s results. Beweistheorie, the mathematical discipline that Hilbert invented to carry out finitistic consistency proofs, eventually redirected its aims and broadened its methods (the r ...
... Hilbert’s programmatic ideas didn’t die with Gödel’s theorem. Rather, they were reformulated in the light of Gödel’s results. Beweistheorie, the mathematical discipline that Hilbert invented to carry out finitistic consistency proofs, eventually redirected its aims and broadened its methods (the r ...
THE HISTORY OF LOGIC
... Aristotle may also be credited with the formulation of several metalogical theses, most notably the Law of Noncontradiction, the Principle of the Excluded Middle, and the Law of Bivalence. These are important in his discussion of modal logic and tense logic. Aristotle referred to certain principles ...
... Aristotle may also be credited with the formulation of several metalogical theses, most notably the Law of Noncontradiction, the Principle of the Excluded Middle, and the Law of Bivalence. These are important in his discussion of modal logic and tense logic. Aristotle referred to certain principles ...
Slides
... Instantiating the framework for a specific logic L, requires a deductive system for L that meets several criteria. Linear arithmetic, EUF, arrays etc all have it. ...
... Instantiating the framework for a specific logic L, requires a deductive system for L that meets several criteria. Linear arithmetic, EUF, arrays etc all have it. ...
Exam-Computational_Logic-Subjects_2016
... 6. The theorems of soundness and completeness of the proof methods: The properties of propositional logic: coherence, non-contradiction ,decidability. The theorem of soundness for propositional logic: If | U then | U (a theorem is a tautology). The theorem of completeness for propositional logic ...
... 6. The theorems of soundness and completeness of the proof methods: The properties of propositional logic: coherence, non-contradiction ,decidability. The theorem of soundness for propositional logic: If | U then | U (a theorem is a tautology). The theorem of completeness for propositional logic ...
PDF
... equivalent. None is stronger than the others, so we can be quite sure that they do represent the class of all computable functions completely – if a function is computable, then it can be represented in each of these formalisms (this became known as Church’s thesis). Most theoretical models of compu ...
... equivalent. None is stronger than the others, so we can be quite sure that they do represent the class of all computable functions completely – if a function is computable, then it can be represented in each of these formalisms (this became known as Church’s thesis). Most theoretical models of compu ...
Propositional Logic Predicate Logic
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
... Name of Symbols ∀ (universal quantifier), and ∃ (existential quantifier). Definition. A formula A is valid if A is true no matter how we replace the individual constants in A with concrete individuals and the predicate variables in A with concrete predicates. Note. The set of individuals must be in ...
Frege`s Foundations of Arithmetic
... axioms”. Week 3. Three neat ideas: (1) “The content of a statement of number is an assertion about a concept.” (2) “If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a, even if it is not always in our power to apply this c ...
... axioms”. Week 3. Three neat ideas: (1) “The content of a statement of number is an assertion about a concept.” (2) “If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a, even if it is not always in our power to apply this c ...
A short article for the Encyclopedia of Artificial Intelligence: Second
... then it is η-convertible to λx(M x), provided x is not free in M ). Many standard proof-theoretic results – such as cut-elimination (Girard, 1986), unification (Huet, 1975), resolution (Andrews, 1971), and Skolemization and Herbrand’s Theorem (Miller, 1987) – have been formulated for this fragment. ...
... then it is η-convertible to λx(M x), provided x is not free in M ). Many standard proof-theoretic results – such as cut-elimination (Girard, 1986), unification (Huet, 1975), resolution (Andrews, 1971), and Skolemization and Herbrand’s Theorem (Miller, 1987) – have been formulated for this fragment. ...
