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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 ...
Canad. Math. Bull. Vol. 24 (2), 1981 INDEPENDENT SETS OF
... §0. Introduction. A set of sentences T is called independent if for every
... §0. Introduction. A set of sentences T is called independent if for every
Adding the Everywhere Operator to Propositional Logic (pdf file)
... Modal logic2 S5 includes 2P among its formulas. As is well known, S5 is not complete with respect to model C, which consists of all states (total functions from the set of all propositional variables to {t, f } , with the conventional definition of evaluation), where every state is accessible from ev ...
... Modal logic2 S5 includes 2P among its formulas. As is well known, S5 is not complete with respect to model C, which consists of all states (total functions from the set of all propositional variables to {t, f } , with the conventional definition of evaluation), where every state is accessible from ev ...
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... Instead, we obtain a sound axiomatization of C that has a nite number of axioms as follows. First, extend language C to a language C . The formulas of C will include those of C the original formulas of C will be called concrete formulas. Then, we give an axiomatization of C |using a nite number o ...
... Instead, we obtain a sound axiomatization of C that has a nite number of axioms as follows. First, extend language C to a language C . The formulas of C will include those of C the original formulas of C will be called concrete formulas. Then, we give an axiomatization of C |using a nite number o ...
completeness theorem for a first order linear
... the until operator U were introduced and the resulting logics were shown to be more expressive than the logics without them. Even in the case of propositional temporal logics, one's cosmology has important consequences: one can assume that time is linear or branching, with or without last moment, di ...
... the until operator U were introduced and the resulting logics were shown to be more expressive than the logics without them. Even in the case of propositional temporal logics, one's cosmology has important consequences: one can assume that time is linear or branching, with or without last moment, di ...
Methods of Proofs Recall we discussed the following methods of
... integer n is odd if there exists an integer k such that n = 2k + 1. Example: Use the definition of odd to explain why 9 is odd, but why 8 is not odd. Axiom (Closure of addition over the integers): If a and b are integers, then a + b is an integer. Axiom (Closure of multiplication over the integers): ...
... integer n is odd if there exists an integer k such that n = 2k + 1. Example: Use the definition of odd to explain why 9 is odd, but why 8 is not odd. Axiom (Closure of addition over the integers): If a and b are integers, then a + b is an integer. Axiom (Closure of multiplication over the integers): ...
Truth, Conservativeness and Provability
... The argument for (P) is as follows: we note that any person accepting PA should also accept (D). The reason is that (D) expresses simply the fact that the person in question accepts PA; and the claim would be that the data on which (D) is based, whether introspective or empirical, are in principle e ...
... The argument for (P) is as follows: we note that any person accepting PA should also accept (D). The reason is that (D) expresses simply the fact that the person in question accepts PA; and the claim would be that the data on which (D) is based, whether introspective or empirical, are in principle e ...
Aristotle`s particularisation
... We begin by noting that, in a first order language, the formula ‘[(∃x)P (x)]’ is an abbreviation for the formula ‘[¬(∀x)¬P (x)]’9 . The commonly accepted interpretation of this formula appeals—generally tacitly, but sometimes explicitly10 —to Aristotle’s particularisation. This is a fundamental tene ...
... We begin by noting that, in a first order language, the formula ‘[(∃x)P (x)]’ is an abbreviation for the formula ‘[¬(∀x)¬P (x)]’9 . The commonly accepted interpretation of this formula appeals—generally tacitly, but sometimes explicitly10 —to Aristotle’s particularisation. This is a fundamental tene ...
On the paradoxes of set theory
... of the greatest ordinal number some definitions are necessary. These include: Definition 1.—— A set M is called ordered if there exists a rule which tells us that for each two distinct elements in M which one precedes the other. Definition 2,~.— A~ ordered set is said to be well-ordered if every non ...
... of the greatest ordinal number some definitions are necessary. These include: Definition 1.—— A set M is called ordered if there exists a rule which tells us that for each two distinct elements in M which one precedes the other. Definition 2,~.— A~ ordered set is said to be well-ordered if every non ...
eprint_4_1049_36.doc
... then 3 = 2 + 1, then 4 = 3 + 1, and so on. The principle makes precise the vague phrase “and so on.” Principle of Mathematical Induction: Let S be a set of positive integers with the following two properties: (i) 1 belongs to S. (ii) If k belongs to S, then k + 1 belongs to S. Then S is the set of a ...
... then 3 = 2 + 1, then 4 = 3 + 1, and so on. The principle makes precise the vague phrase “and so on.” Principle of Mathematical Induction: Let S be a set of positive integers with the following two properties: (i) 1 belongs to S. (ii) If k belongs to S, then k + 1 belongs to S. Then S is the set of a ...
FORMALIZATION OF HILBERT`S GEOMETRY OF INCIDENCE AND
... try to use Axiom I3 which says that on any line there are at least two points, say, B and C for line a. Then, if we succeed in proving that point A is distinct from B , we can construct the line AB : Assume A and B are equal. Since B is incident with a and A is outside a, we have a contradiction. Th ...
... try to use Axiom I3 which says that on any line there are at least two points, say, B and C for line a. Then, if we succeed in proving that point A is distinct from B , we can construct the line AB : Assume A and B are equal. Since B is incident with a and A is outside a, we have a contradiction. Th ...
pdf
... [9] showed that the validity (and satisfiability) problem for every modal logic between K and S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiability problem for S5 is NP-complete. What causes the gap between NP and PSPACE here? We show tha ...
... [9] showed that the validity (and satisfiability) problem for every modal logic between K and S4 is PSPACE-hard; and is PSPACE-complete for the modal logics K, T, and S4. He also showed that the satisfiability problem for S5 is NP-complete. What causes the gap between NP and PSPACE here? We show tha ...
Practice Problem Set 1
... • These problems will not be graded. • Mutual discussion and discussion with the instructor/TA is strongly encouraged. 1. [From HW1, Autumn 2011] Use the proof system of first order logic studied in class to prove each of the following sequents. You must indicate which proof rule you are applying at ...
... • These problems will not be graded. • Mutual discussion and discussion with the instructor/TA is strongly encouraged. 1. [From HW1, Autumn 2011] Use the proof system of first order logic studied in class to prove each of the following sequents. You must indicate which proof rule you are applying at ...
On the Question of Absolute Undecidability
... in accepting the system obtained by expanding the language to include the truth predicate and expanding the axioms by adding the elementary axioms governing the truth predicate and allowing the truth predicate to figure in the induction scheme. The statement Con(PA) is provable in the resulting sys ...
... in accepting the system obtained by expanding the language to include the truth predicate and expanding the axioms by adding the elementary axioms governing the truth predicate and allowing the truth predicate to figure in the induction scheme. The statement Con(PA) is provable in the resulting sys ...
A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part
... Formulae (well-formed modal formulae–wfmf) are built, via the standard inductive definitions, from atomic formulae–which may involve predi000 cate symbols P, Q, R, P54 , . . .–and the primary logical symbols. The latter are the Boolean variables p, q, p0 , p00 , q13 , . . ., and the connectives: ¬, ...
... Formulae (well-formed modal formulae–wfmf) are built, via the standard inductive definitions, from atomic formulae–which may involve predi000 cate symbols P, Q, R, P54 , . . .–and the primary logical symbols. The latter are the Boolean variables p, q, p0 , p00 , q13 , . . ., and the connectives: ¬, ...
3463: Mathematical Logic
... Figure 1: Turing machine configuration. The current symbol is 0 and the current state is q2 . For mathematical logic, they have another, more basic, but important use: they allow us to define what we mean by a natural number (nonnegative integer). In fact, we shall give three alternative definitions ...
... Figure 1: Turing machine configuration. The current symbol is 0 and the current state is q2 . For mathematical logic, they have another, more basic, but important use: they allow us to define what we mean by a natural number (nonnegative integer). In fact, we shall give three alternative definitions ...
Aristotle, Boole, and Categories
... Proof. Let S be such a set having u as its only universal sentence if any. Form a model of S by taking its universe E to consist of the existential sentences. For each member e of E, set to true at e the literals of e and, if u exists, the literal of u whose predicate symbol does not appear in e. Se ...
... Proof. Let S be such a set having u as its only universal sentence if any. Form a model of S by taking its universe E to consist of the existential sentences. For each member e of E, set to true at e the literals of e and, if u exists, the literal of u whose predicate symbol does not appear in e. Se ...
Hilbert`s investigations on the foundations of arithmetic (1935) Paul
... the view identified above [oben gekennzeichneten Auffassung ]: we do not have to think the totality of all possible laws according to which the elements of a fundamental sequence can proceed as the set of real numbers, but rather — as has just been explained [? dargelegt] — a system of things whose ...
... the view identified above [oben gekennzeichneten Auffassung ]: we do not have to think the totality of all possible laws according to which the elements of a fundamental sequence can proceed as the set of real numbers, but rather — as has just been explained [? dargelegt] — a system of things whose ...
Epsilon Substitution for Transfinite Induction
... T RU E. On the other hand, if |xφ(x)|S then, since obviously 0 t ,→S T RU E then we have φ(0) ,→S T RU E. In either case, CrI ,→S T RU E, so I 6= I(S). In the second case, if vIS = 0 then |s|S = S0, and therefore, by correctness, |xs = Sx|S = 0, so s = S(xs = Sx) ,→S T RU E. Again, CrI ,→S T RU ...
... T RU E. On the other hand, if |xφ(x)|S then, since obviously 0 t ,→S T RU E then we have φ(0) ,→S T RU E. In either case, CrI ,→S T RU E, so I 6= I(S). In the second case, if vIS = 0 then |s|S = S0, and therefore, by correctness, |xs = Sx|S = 0, so s = S(xs = Sx) ,→S T RU E. Again, CrI ,→S T RU ...
A constructive approach to nonstandard analysis*
... The content of the paper is outlined as follows. In Section 2 we give some metamathematical results on nonarchimedean extensions, e.g. Martin-Lof’s interpretation of infinity symbols. We also indicate how such theories might be used. Unfortunately, they have no useful external notions, such as being ...
... The content of the paper is outlined as follows. In Section 2 we give some metamathematical results on nonarchimedean extensions, e.g. Martin-Lof’s interpretation of infinity symbols. We also indicate how such theories might be used. Unfortunately, they have no useful external notions, such as being ...
Classical and Intuitionistic Models of Arithmetic
... Theorem 4.1 Let K be a Kripke model of HA such that, whenever α ≤ β in K, Aβ is an end extension of Aα ( Aα ⊆e Aβ ; such K are called end-extension models). Then, for each α in K , we have Aα |= B1 . Proof: We have to show that for each α ∈ K, every 1 -formula ϕ(x, y, v) and all a, a ∈ Aα , α |= ∀ ...
... Theorem 4.1 Let K be a Kripke model of HA such that, whenever α ≤ β in K, Aβ is an end extension of Aα ( Aα ⊆e Aβ ; such K are called end-extension models). Then, for each α in K , we have Aα |= B1 . Proof: We have to show that for each α ∈ K, every 1 -formula ϕ(x, y, v) and all a, a ∈ Aα , α |= ∀ ...
The Ring of Integers
... Elementary number theory is largely about the ring of integers, denoted by the symbol Z. The integers are an example of an algebraic structure called an integral domain. This means that Z satisfies the following axioms: (a) Z has operations + (addition) and · (multiplication). It is closed under the ...
... Elementary number theory is largely about the ring of integers, denoted by the symbol Z. The integers are an example of an algebraic structure called an integral domain. This means that Z satisfies the following axioms: (a) Z has operations + (addition) and · (multiplication). It is closed under the ...
A Recursively Axiomatizable Subsystem of Levesque`s Logic of Only
... The model of is constructed in several steps. First, we de ne a tree structure that serves as the skeleton for our model. Next, we associate to every node of the tree a maximal consistent set of sentences of a suitable language. Then we associate to every node of the tree a dierent possible world ...
... The model of is constructed in several steps. First, we de ne a tree structure that serves as the skeleton for our model. Next, we associate to every node of the tree a maximal consistent set of sentences of a suitable language. Then we associate to every node of the tree a dierent possible world ...
CS 103X: Discrete Structures Homework Assignment 2 — Solutions
... inequality by 1 + r without changing the sign. The second inequality follows since r 2 k ≥ 0. This proves the induction step and concludes the proof. Exercise 5 (10 points). Prove the strong induction principle from the principle of induction. Conclude that the two principles are equivalent. (That i ...
... inequality by 1 + r without changing the sign. The second inequality follows since r 2 k ≥ 0. This proves the induction step and concludes the proof. Exercise 5 (10 points). Prove the strong induction principle from the principle of induction. Conclude that the two principles are equivalent. (That i ...
solution - inst.eecs.berkeley.edu
... Suppose that I start with 0 written on a piece of paper. Each minute, I choose a digit written on the paper and erase it. If it was 0, I replace it with 010. If it was 1, I replace it with 1001. Prove that no matter which digits I choose and no matter how long the process continues, I never end up w ...
... Suppose that I start with 0 written on a piece of paper. Each minute, I choose a digit written on the paper and erase it. If it was 0, I replace it with 010. If it was 1, I replace it with 1001. Prove that no matter which digits I choose and no matter how long the process continues, I never end up w ...
Peano axioms
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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.