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Modal Logic and Model Theory
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... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://dv1litvip.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an e ...
ordinal logics and the characterization of informal concepts of proof
... numerical terms and function symbols, but also proof predicates. This is independently justified by the accepted sense of finitist according to which finitist proofs can themselves be the subject-matter of finitist reasoning; also it is in accordance with Heyting's and Gödel's view of the place of p ...
... numerical terms and function symbols, but also proof predicates. This is independently justified by the accepted sense of finitist according to which finitist proofs can themselves be the subject-matter of finitist reasoning; also it is in accordance with Heyting's and Gödel's view of the place of p ...
(A B) |– A
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
Constructions of the real numbers
... In this section, the set of the rational numbers, denoted by Q, will be constructed from Z. The ideas and methods used will exhibit many similarities to the ones for the construction of Z from N. We will therefore only give the relevant definitions and prove the most important properties of the newl ...
... In this section, the set of the rational numbers, denoted by Q, will be constructed from Z. The ideas and methods used will exhibit many similarities to the ones for the construction of Z from N. We will therefore only give the relevant definitions and prove the most important properties of the newl ...
(A B) |– A
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
... Notes: 1. A, B are not formulas, but meta-symbols denoting any formula. Each axiom schema denotes an infinite class of formulas of a given form. If axioms were specified by concrete formulas, like 1. p (q p) 2. (p (q r)) ((p q) (p r)) 3. (q p) (p q) we would have to extend th ...
UNIVERSITY OF LONDON BA EXAMINATION PHILOSOPHY
... (ii) What is it for a set to be well-ordered by a relation? Give an example of a set B and a relation R on B such that B is totally ordered by R but not well-ordered by R, and explain why it is not well-ordered by R. (iii) What is a transitive set? What is an ordinal? Prove that every member of an o ...
... (ii) What is it for a set to be well-ordered by a relation? Give an example of a set B and a relation R on B such that B is totally ordered by R but not well-ordered by R, and explain why it is not well-ordered by R. (iii) What is a transitive set? What is an ordinal? Prove that every member of an o ...
Real Analysis - University of Illinois at Chicago
... These notes are all about the Real Numbers and Calculus. We start from scratch with definitions and a set of nine axioms. Then, using basic notions of sets and logical reasoning, we derive what we need to know about real numbers in order to advance through a rigorous development of the theorems of C ...
... These notes are all about the Real Numbers and Calculus. We start from scratch with definitions and a set of nine axioms. Then, using basic notions of sets and logical reasoning, we derive what we need to know about real numbers in order to advance through a rigorous development of the theorems of C ...
pdf format
... Theorem 2 (Integer Induction) Let ϕ(x, y1, . . . , yk ) be a first-order formula. Let y1 , . . . , yk be fixed sets. Suppose that ϕ(0, ~y) is true and that (∀x ∈ ω)(ϕ(x, ~y) → ϕ(S(x), ~y)). Then ϕ(x, ~y) is true for all x ∈ ω . Definition A set x is transitive if every member of x is a subset of x. ...
... Theorem 2 (Integer Induction) Let ϕ(x, y1, . . . , yk ) be a first-order formula. Let y1 , . . . , yk be fixed sets. Suppose that ϕ(0, ~y) is true and that (∀x ∈ ω)(ϕ(x, ~y) → ϕ(S(x), ~y)). Then ϕ(x, ~y) is true for all x ∈ ω . Definition A set x is transitive if every member of x is a subset of x. ...
Lecture Notes
... Problem: Given a structure M , a world w of M and a formula φ, decide if M, w |= φ. Theorem: For finite M , and φ ∈ L{K1 ,...,Kn ,CG } there exists an algorithm that solves the problem in time linear in |M | · |φ|, where |M | and |φ| are the amount of space needed to write down M and φ, ...
... Problem: Given a structure M , a world w of M and a formula φ, decide if M, w |= φ. Theorem: For finite M , and φ ∈ L{K1 ,...,Kn ,CG } there exists an algorithm that solves the problem in time linear in |M | · |φ|, where |M | and |φ| are the amount of space needed to write down M and φ, ...
Tactical and Strategic Challenges to Logic (KAIST
... that these inconsistencies aren’t expungable without serious damage to their practical utility? Even granting that these assumptions are common knowledge in various precincts of informatica, couldn’t we have some supporting evidence? These are fair and necessary questions, for which we’ll have no ti ...
... that these inconsistencies aren’t expungable without serious damage to their practical utility? Even granting that these assumptions are common knowledge in various precincts of informatica, couldn’t we have some supporting evidence? These are fair and necessary questions, for which we’ll have no ti ...
Class Notes
... Hilbert says nothing about what the “things” are. Axioms. An axiom is a proposition about the objects in question which we do not attempt to prove but rather which we accept as given. One of Euclid’s axioms, for example, was “It shall be possible to draw a straight line joining any two points.” Aris ...
... Hilbert says nothing about what the “things” are. Axioms. An axiom is a proposition about the objects in question which we do not attempt to prove but rather which we accept as given. One of Euclid’s axioms, for example, was “It shall be possible to draw a straight line joining any two points.” Aris ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... semantics in certain statements, or the fact that equality is a two-valued predicate in both theories. There are two major differences inbetween the set of axioms of lattice-valued set theory and that of FST. The first difference consists in connectives used in the formulations of axioms: LZFZ uses ...
... semantics in certain statements, or the fact that equality is a two-valued predicate in both theories. There are two major differences inbetween the set of axioms of lattice-valued set theory and that of FST. The first difference consists in connectives used in the formulations of axioms: LZFZ uses ...
Gödel`s ontological argument: a reply to Oppy
... gödel’s ontological argument: a reply to oppy 311 some non-positive essential property. Take, for example, P1, which by Definition 1* is a positive property. It follows then from Axiom 7 that being necessarily P1 is positive. Now look at Axiom 1. It states that if a property is positive, then its n ...
... gödel’s ontological argument: a reply to oppy 311 some non-positive essential property. Take, for example, P1, which by Definition 1* is a positive property. It follows then from Axiom 7 that being necessarily P1 is positive. Now look at Axiom 1. It states that if a property is positive, then its n ...
Prime numbers
... • Assume that P (j) holds for all j ≤ n (this is a strong induction): Assume a natural number j can be written as a product of primes for all j ≤ n. • We wish to show P (n + 1) holds: We wish to show that n + 1 can be written as a product of primes. • Consider n + 1. There are two cases: – n + 1 is ...
... • Assume that P (j) holds for all j ≤ n (this is a strong induction): Assume a natural number j can be written as a product of primes for all j ≤ n. • We wish to show P (n + 1) holds: We wish to show that n + 1 can be written as a product of primes. • Consider n + 1. There are two cases: – n + 1 is ...
Nonmonotonic Logic II: Nonmonotonic Modal Theories
... The first two of these follow by predicate calculus. The third follows because ~CANFLY(FRED) is not a member of the fixed point. In other words, M CANFLY(FRED) is in Astheory(fixed-point). So by the first proper axiom, CANFLY(FRED) is in the fixed point as well. Of course, I have not proven that thi ...
... The first two of these follow by predicate calculus. The third follows because ~CANFLY(FRED) is not a member of the fixed point. In other words, M CANFLY(FRED) is in Astheory(fixed-point). So by the first proper axiom, CANFLY(FRED) is in the fixed point as well. Of course, I have not proven that thi ...
January 12
... Frege developed modern logic, later employed as the basis of analytic philosophy of language, in order to prove a view of arithmetic. Ultimately, Frege didn’t realize what a great accomplishment he had made, and spent the last 20 years of his life depressed, believing that he was a failure. Psycholo ...
... Frege developed modern logic, later employed as the basis of analytic philosophy of language, in order to prove a view of arithmetic. Ultimately, Frege didn’t realize what a great accomplishment he had made, and spent the last 20 years of his life depressed, believing that he was a failure. Psycholo ...
Welcome to CS 39 - Dartmouth Computer Science
... To prove a theorem by induction: • Prove the theorem for a general case by assuming the same theorem to be true (“induction hypothesis”) for all smaller cases. • Separately prove the theorem, without making any assumptions, for all “base” cases, i.e., those cases for which there is nothing smaller. ...
... To prove a theorem by induction: • Prove the theorem for a general case by assuming the same theorem to be true (“induction hypothesis”) for all smaller cases. • Separately prove the theorem, without making any assumptions, for all “base” cases, i.e., those cases for which there is nothing smaller. ...
A(x)
... question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, th ...
... question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, th ...
A(x)
... question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, th ...
... question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, th ...
A(x)
... The first equivalence is obtained by applying the Deduction Theorem m-times, the second is valid due to the soundness and completeness, the third one is the semantic equivalence. ...
... The first equivalence is obtained by applying the Deduction Theorem m-times, the second is valid due to the soundness and completeness, the third one is the semantic equivalence. ...
A(x)
... The first equivalence is obtained by applying the Deduction Theorem m-times, the second is valid due to the soundness and completeness, the third one is the semantic equivalence. ...
... The first equivalence is obtained by applying the Deduction Theorem m-times, the second is valid due to the soundness and completeness, the third one is the semantic equivalence. ...
Transfinite progressions: A second look at completeness.
... Autonomy plays a large role in Franzén [2004], which is concerned with the peculiar role of reflection principles in demonstrating the apparent inexhaustibility of our mathematical knowledge. The topic of the present paper is the completeness theorem for progressions, which was only touched on briefl ...
... Autonomy plays a large role in Franzén [2004], which is concerned with the peculiar role of reflection principles in demonstrating the apparent inexhaustibility of our mathematical knowledge. The topic of the present paper is the completeness theorem for progressions, which was only touched on briefl ...
(A B) |– A
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
... if T, A |– B and |– A, then T |– B. It is not necessary to state theorems in the assumptions. if A |– B, then T, A |– B. (Monotonicity of proving) if T |– A and T, A |– B, then T |– B. if T |– A and A |– B, then T |– B. if T |– A; T |– B; A, B |– C then T |– C. if T |– A and T |– B, then T |– A B. ...
Lecture 8: Back-and-forth - to go back my main page.
... s’s at the end. The inductive condition ensures that g preserves the interpretation of all symbols in LA , and fixes I pointwise. Clearly, the inductive condition is initially satisfied. During the construction, we need to make sure g is total, surjective, and moves arbitrarily small points above I. ...
... s’s at the end. The inductive condition ensures that g preserves the interpretation of all symbols in LA , and fixes I pointwise. Clearly, the inductive condition is initially satisfied. During the construction, we need to make sure g is total, surjective, and moves arbitrarily small points above I. ...
Peano axioms
![](https://commons.wikimedia.org/wiki/Special:FilePath/Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg?width=300)
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set ""number"". The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the ""underlying logic"". The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.