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... induction will effectively prove the base case. Here’s an example of strong induction in action. Theorem 2.32 (Well Ordering Principle): Suppose A ⊆ N is non-empty. Then it has a smallest element. Proof. Let P(k) be the statement that “all subsets of N containing the integer k have a least element.” ...
... induction will effectively prove the base case. Here’s an example of strong induction in action. Theorem 2.32 (Well Ordering Principle): Suppose A ⊆ N is non-empty. Then it has a smallest element. Proof. Let P(k) be the statement that “all subsets of N containing the integer k have a least element.” ...
+ 1 - Stanford Mathematics
... One may prove this by the modified version of induction, where the base case is P2 instead of P1 . But the following is much easier: If n ≥ 2, then n2 ≥ 2n = (n + n) ≥ (n + 2) > (n + 1), so n2 > n + 1. b) Prove that n! > n2 for all integers n ≥ 4. The proof here is by induction. The basis for indu ...
... One may prove this by the modified version of induction, where the base case is P2 instead of P1 . But the following is much easier: If n ≥ 2, then n2 ≥ 2n = (n + n) ≥ (n + 2) > (n + 1), so n2 > n + 1. b) Prove that n! > n2 for all integers n ≥ 4. The proof here is by induction. The basis for indu ...
here
... • On the k th step of the process, for k > 0, perform one “A to B” sub-step and one “B to A” sub-step. The “A to B” sub-step proceeds as follows: Find the smallest a ∈ A \ Ak−1 (that is, the smallest unmatched element of A). Let K denote the vertices of Ak−1 adjacent to a, and let L denote the vert ...
... • On the k th step of the process, for k > 0, perform one “A to B” sub-step and one “B to A” sub-step. The “A to B” sub-step proceeds as follows: Find the smallest a ∈ A \ Ak−1 (that is, the smallest unmatched element of A). Let K denote the vertices of Ak−1 adjacent to a, and let L denote the vert ...
1.2 The Integers and Rational Numbers
... ages. It is remarkable that we can easily (if tediously) extend the operations of addition and multiplication of natural numbers to include 0 and the negative integers, maintaining all the fundamental laws of arithmetic. Negative numbers are necessary today because, in our society based upon the rig ...
... ages. It is remarkable that we can easily (if tediously) extend the operations of addition and multiplication of natural numbers to include 0 and the negative integers, maintaining all the fundamental laws of arithmetic. Negative numbers are necessary today because, in our society based upon the rig ...
Many-Valued Models
... This new way of looking to logico-philosophical scenario was not free of discussion, however. Stanisław Lesniewski argued that a third logical value never appears in scientific argumentation, and considered the third value as no sense, because “no one had been able until now to give to the symbol 2 ...
... This new way of looking to logico-philosophical scenario was not free of discussion, however. Stanisław Lesniewski argued that a third logical value never appears in scientific argumentation, and considered the third value as no sense, because “no one had been able until now to give to the symbol 2 ...
Mathematical Logic
... Complexity of deciding logical consequence in Propositional Logic The truth table method is Exponential The problem of determining if a formula A containing n primitive propositions, is a logical consequence of the empty set, i.e., the problem of determining if A is valid, (|= A), takes an n-expone ...
... Complexity of deciding logical consequence in Propositional Logic The truth table method is Exponential The problem of determining if a formula A containing n primitive propositions, is a logical consequence of the empty set, i.e., the problem of determining if A is valid, (|= A), takes an n-expone ...
PPT
... Let n be a counter-example Hence n is not prime, so n = ab If both a and b had prime factorizations, then n would too Thus a or b is a smaller counter-example ...
... Let n be a counter-example Hence n is not prime, so n = ab If both a and b had prime factorizations, then n would too Thus a or b is a smaller counter-example ...
Algebraic Laws for Nondeterminism and Concurrency
... two programs are equivalent when no observationscan distinguish them. Further, two subprograms or program phrasesare congruent if the result of placing each of them in any progralm context yields two equivalent programs. Then, considering the phrasesas modules, one can be exchangedfor the other in a ...
... two programs are equivalent when no observationscan distinguish them. Further, two subprograms or program phrasesare congruent if the result of placing each of them in any progralm context yields two equivalent programs. Then, considering the phrasesas modules, one can be exchangedfor the other in a ...
Dialetheic truth theory: inconsistency, non-triviality, soundness, incompleteness
... with the assumption, and note that, because PA* has a recursive proof relation (PA*derivations form a recursive set of sequences of strings on the alphabet of L), and because all recursive relations can be represented in PA (and therefore, by assumption, also in PA*), it is possible to formulate a p ...
... with the assumption, and note that, because PA* has a recursive proof relation (PA*derivations form a recursive set of sequences of strings on the alphabet of L), and because all recursive relations can be represented in PA (and therefore, by assumption, also in PA*), it is possible to formulate a p ...
PPT
... Let n be a counter-example Hence n is not prime, so n = ab If both a and b had prime factorizations, then n would too Thus a or b is a smaller counter-example ...
... Let n be a counter-example Hence n is not prime, so n = ab If both a and b had prime factorizations, then n would too Thus a or b is a smaller counter-example ...
Gödel`s Theorems
... we have made essential use of the notion of truth in a structure M, i.e., of the relation M |= A. The set of all closed fromulas A such that M |= A has been called the theory of M, denoted Th(M). Now instead of Th(M) we shall start more generally from an arbitrary theory T . We consider the question ...
... we have made essential use of the notion of truth in a structure M, i.e., of the relation M |= A. The set of all closed fromulas A such that M |= A has been called the theory of M, denoted Th(M). Now instead of Th(M) we shall start more generally from an arbitrary theory T . We consider the question ...
valid - Informatik Uni Leipzig
... Proof for T and T. Let F be a frame from class T. Let I be an interpretation based on F and let w be an arbitrary world in I . If 2ϕ is not true in a world w, then axiom T is true in w. If 2ϕ is true in w, then ϕ is true in all accessible worlds. Since the accessibility relation is reflexive, w is a ...
... Proof for T and T. Let F be a frame from class T. Let I be an interpretation based on F and let w be an arbitrary world in I . If 2ϕ is not true in a world w, then axiom T is true in w. If 2ϕ is true in w, then ϕ is true in all accessible worlds. Since the accessibility relation is reflexive, w is a ...
PPT - School of Computer Science
... Let n be a counter-example Hence n is not prime, so n = ab If both a and b had prime factorizations, then n would too Thus a or b is a smaller counter-example ...
... Let n be a counter-example Hence n is not prime, so n = ab If both a and b had prime factorizations, then n would too Thus a or b is a smaller counter-example ...
A Logic of Explicit Knowledge - Lehman College
... having a single state, Γ, accessible to itself, and with an evidence function such that E(Γ, t) is the entire set of formulas. In this model, t serves as ‘universal’ evidence. Also, use a valuation such that V(P ) = {Γ} and V(Q) = ∅. Then we have M, Γ t:P but M, Γ 6 t:(P ∧ Q) because, even though ...
... having a single state, Γ, accessible to itself, and with an evidence function such that E(Γ, t) is the entire set of formulas. In this model, t serves as ‘universal’ evidence. Also, use a valuation such that V(P ) = {Γ} and V(Q) = ∅. Then we have M, Γ t:P but M, Γ 6 t:(P ∧ Q) because, even though ...
Available on-line - Gert
... In 1967, Anderson [2] defined his system of relevant deontic logic as follows: take relevant system R, add a propositional constant V (“the violation” or “the bad thing”), and define O (“it is obligatory that”) by O A = ¬A → V , where → is relevant implication. This proposal naturally leads to the q ...
... In 1967, Anderson [2] defined his system of relevant deontic logic as follows: take relevant system R, add a propositional constant V (“the violation” or “the bad thing”), and define O (“it is obligatory that”) by O A = ¬A → V , where → is relevant implication. This proposal naturally leads to the q ...
The Natural Number System: Induction and Counting
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
Lecture 14 Notes
... some yet unknown element of the universe. Since we do not know this element, a should be a new parameter – this way we make sure that we don’t make any further assumptions about a by accidentally linking it to a parameter that was introduced earlier in the proof. If we were to decompose T (∀x)P x be ...
... some yet unknown element of the universe. Since we do not know this element, a should be a new parameter – this way we make sure that we don’t make any further assumptions about a by accidentally linking it to a parameter that was introduced earlier in the proof. If we were to decompose T (∀x)P x be ...
RR-01-02
... formulas in table 1 and axioms in table 2 for representing the specific problem domain of interest and for controlling deduction, and uses McCarthy’s 1986 [11] predicate circumscription 3 with forced separation as modelpreference criterion. The language of the calculus is defined in table 1. Let S1 ...
... formulas in table 1 and axioms in table 2 for representing the specific problem domain of interest and for controlling deduction, and uses McCarthy’s 1986 [11] predicate circumscription 3 with forced separation as modelpreference criterion. The language of the calculus is defined in table 1. Let S1 ...
1.4 The set of Real Numbers: Quick Definition and
... many of these properties for granted. Yet, without them, many of the things we do with real numbers would not be possible. First, we establish uniqueness of the identity element as well as uniqueness of the inverse under both operations. This fact is a direct consequence of the properties of each op ...
... many of these properties for granted. Yet, without them, many of the things we do with real numbers would not be possible. First, we establish uniqueness of the identity element as well as uniqueness of the inverse under both operations. This fact is a direct consequence of the properties of each op ...
thc cox theorem, unknowns and plausible value
... sufficient density of the domain to claim that continuity gives associativity which together with strict monotonicity (a requirement from ”agreement with common sense”) suffices to show that the associative multiplication is just ordinary multiplication. However, it has been well known for many many ...
... sufficient density of the domain to claim that continuity gives associativity which together with strict monotonicity (a requirement from ”agreement with common sense”) suffices to show that the associative multiplication is just ordinary multiplication. However, it has been well known for many many ...
TRUTH DEFINITIONS AND CONSISTENCY PROOFS
... criterion of soundness (or validity) for 5 according to which all the theorems of 5 are sound. In this way we obtain in S' a consistency proof for 5. The consistency proof so obtained, which in no case with fairly strong systems could by any stretch of imagination be called constructive, is not of m ...
... criterion of soundness (or validity) for 5 according to which all the theorems of 5 are sound. In this way we obtain in S' a consistency proof for 5. The consistency proof so obtained, which in no case with fairly strong systems could by any stretch of imagination be called constructive, is not of m ...
Mathematische Logik - WS14/15 Iosif Petrakis, Felix Quirin Weitk¨ amper November 13, 2014
... (B) m = 0. The corresponding (∗)-condition is that φ ∈ S(L) i.e., φ is a sentence. The definitional clauses are: (G1) A |= t1 = t2 iff t1 A = t2 A . (G2) A |= Rt1 . . . tn iff RA (t1 A . . . tn A ). (G3) A |= ¬φ iff not A |= φ. (G4) A |= (φ ∨ ψ) iff A |= φ or A |= ψ. (G5s) A |= (∃x ψ) iff there exis ...
... (B) m = 0. The corresponding (∗)-condition is that φ ∈ S(L) i.e., φ is a sentence. The definitional clauses are: (G1) A |= t1 = t2 iff t1 A = t2 A . (G2) A |= Rt1 . . . tn iff RA (t1 A . . . tn A ). (G3) A |= ¬φ iff not A |= φ. (G4) A |= (φ ∨ ψ) iff A |= φ or A |= ψ. (G5s) A |= (∃x ψ) iff there exis ...
Comparing Constructive Arithmetical Theories Based - Math
... (term) a, and also consider the formula ∀z 6 a(x + z = |a| → ∀y 6 t¬A(z, y)) as B(x). To prove P V + ¬¬N P − LIN D `i P V + coN P − LIN D, make similar changes. (iii) This is an immediate consequence of Proposition 2.2 and part (ii). Recall that the theory CP V is the classical closure of IP V an ...
... (term) a, and also consider the formula ∀z 6 a(x + z = |a| → ∀y 6 t¬A(z, y)) as B(x). To prove P V + ¬¬N P − LIN D `i P V + coN P − LIN D, make similar changes. (iii) This is an immediate consequence of Proposition 2.2 and part (ii). Recall that the theory CP V is the classical closure of IP V an ...
Decidable fragments of first-order logic Decidable fragments of first
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
... Fix any subset {b1 , b2 } of Bn of size r ∈ {1, 2} and recall that some r -table T{b1 ,b2 } from Tr is assiged to this subset. For any subset {b3 , . . . , bl+2 } of pairwise distinct elements of Bn that differ from b1 and b2 , consider the event that the table induced by b1 , . . . , bl+2 is equal ...
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... Clearly, S ⊂ S ∗ and for every A, either A ∈ S ∗ or ¬A ∈ S ∗ . To finish the proof that S ∗ is MCF we have to show that it is finitely consistent. S First, let observe that if all sets Sn are finitely consistent, so is S ∗ = n∈N Sn . Namely, let SF = {B1 , ..., Bk } be a finite subset of S ∗ . This ...
... Clearly, S ⊂ S ∗ and for every A, either A ∈ S ∗ or ¬A ∈ S ∗ . To finish the proof that S ∗ is MCF we have to show that it is finitely consistent. S First, let observe that if all sets Sn are finitely consistent, so is S ∗ = n∈N Sn . Namely, let SF = {B1 , ..., Bk } be a finite subset of S ∗ . This ...
Peano axioms
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.