Second-Order Logic and Fagin`s Theorem
... Proof This is analogous to Lemma 5.31. We modify the proof of Fagin’s theorem so that instead of guessing the entire tape at every step only a bounded number of bits per step is guessed. The following relations need to be guessed. 1. Qi (t̄) meaning that the state at move t̄ is qi , 2. Si (t̄) meani ...
... Proof This is analogous to Lemma 5.31. We modify the proof of Fagin’s theorem so that instead of guessing the entire tape at every step only a bounded number of bits per step is guessed. The following relations need to be guessed. 1. Qi (t̄) meaning that the state at move t̄ is qi , 2. Si (t̄) meani ...
A SHORT PROOF FOR THE COMPLETENESS OF
... Definition 2.6. Let Σ be a set of first order clauses. Then Σ∗∗ is defined to be the following set of first order clauses Σ∗∗ = Σ∗ ∪{ f (x1 , ..., xn ) = f (x1 , ..., xn ) : f is an n-ary function symbol }. Here x1 , ..., xn are individuum variables; n may be equal to 0, in this case the equality of ...
... Definition 2.6. Let Σ be a set of first order clauses. Then Σ∗∗ is defined to be the following set of first order clauses Σ∗∗ = Σ∗ ∪{ f (x1 , ..., xn ) = f (x1 , ..., xn ) : f is an n-ary function symbol }. Here x1 , ..., xn are individuum variables; n may be equal to 0, in this case the equality of ...
comments on the logic of constructible falsity (strong negation)
... logic I describe by adding the rule D). D, however, is just as unacceptable from Nelson’s point of view as it is from that of the intuitionists. Indeed, given the constructive derivability of excluded middle for atomic (and other decidable) formulas of arithmetic, the addition of D to either intuiti ...
... logic I describe by adding the rule D). D, however, is just as unacceptable from Nelson’s point of view as it is from that of the intuitionists. Indeed, given the constructive derivability of excluded middle for atomic (and other decidable) formulas of arithmetic, the addition of D to either intuiti ...
Lecturecise 19 Proofs and Resolution Compactness for
... First-order logic allows arbitrary relations and functions (they are defined only through their axioms) Useful for modeling all of math (e.g. through set theory axioms), and thus in principle applies to all program verification problems as well. To prove whether a property holds: I describe the prop ...
... First-order logic allows arbitrary relations and functions (they are defined only through their axioms) Useful for modeling all of math (e.g. through set theory axioms), and thus in principle applies to all program verification problems as well. To prove whether a property holds: I describe the prop ...
Subintuitionistic Logics with Kripke Semantics
... Definition 6. The Canonical Model MF = hWF , ∆, R, i of F is defined by: 1. ∆ is the empty theory, 2. WF is the set of all prime theories, 3. The canonical valuation is defined by Γ p iff p ∈ Γ. In the canonical model MF = hWF , ∆, R, i, ∆ is omniscient. Because let Γ ∈ WF . If A → B ∈ ∆, then ` ...
... Definition 6. The Canonical Model MF = hWF , ∆, R, i of F is defined by: 1. ∆ is the empty theory, 2. WF is the set of all prime theories, 3. The canonical valuation is defined by Γ p iff p ∈ Γ. In the canonical model MF = hWF , ∆, R, i, ∆ is omniscient. Because let Γ ∈ WF . If A → B ∈ ∆, then ` ...
Restricted notions of provability by induction
... Proof. See Corollary 8.7 in Wilkie–Paris [25] or Theorem V.5.26 in Hájek– Pudlák [13]. As mentioned before, most results of this section are not at all specific to the induction scheme, and so are more general than stated. Here we abstract one part that may be of independent interest. This stems f ...
... Proof. See Corollary 8.7 in Wilkie–Paris [25] or Theorem V.5.26 in Hájek– Pudlák [13]. As mentioned before, most results of this section are not at all specific to the induction scheme, and so are more general than stated. Here we abstract one part that may be of independent interest. This stems f ...
Midterm Exam 2 Solutions, Comments, and Feedback
... (by strong induction hypothesis with n = k and n = k − 1) ...
... (by strong induction hypothesis with n = k and n = k − 1) ...
Congruent subsets of infinite sets of natural numbers
... congruent n-element subsets . From these at most xo have a non-empty intersection with { 1, . . . , x0}, for their smallest element is in { 1, . . . , x o }, and two of them having the same smallest element must be identical . Hence k of the k + xo congruent subsets must be subsets of A . Remark . T ...
... congruent n-element subsets . From these at most xo have a non-empty intersection with { 1, . . . , x0}, for their smallest element is in { 1, . . . , x o }, and two of them having the same smallest element must be identical . Hence k of the k + xo congruent subsets must be subsets of A . Remark . T ...
compactness slides
... Φ̄ is a maximally satisfiable set. Adding only one new wff to Φ̄ would result in an unsatisfiable set (why?), hence the maximality. ...
... Φ̄ is a maximally satisfiable set. Adding only one new wff to Φ̄ would result in an unsatisfiable set (why?), hence the maximality. ...
071 Embeddings
... examples from an infinite list of sets. The model as a whole is any one of these, so we attempt to form the disjunction: 0,1 0,3 2,1 2,4 ... , but this is not possible as it stands because this disjunction gives the 0 of the lattice. We must treat each member of the list ...
... examples from an infinite list of sets. The model as a whole is any one of these, so we attempt to form the disjunction: 0,1 0,3 2,1 2,4 ... , but this is not possible as it stands because this disjunction gives the 0 of the lattice. We must treat each member of the list ...
Second-order Logic
... In first-order logic, we combine the non-logical symbols of a given language, i.e., its constant symbols, function symbols, and predicate symbols, with the logical symbols to express things about first-order structures. This is done using the notion of satisfaction, which relates !astructure M, toge ...
... In first-order logic, we combine the non-logical symbols of a given language, i.e., its constant symbols, function symbols, and predicate symbols, with the logical symbols to express things about first-order structures. This is done using the notion of satisfaction, which relates !astructure M, toge ...
Three Solutions to the Knower Paradox
... Montague [9] and basically is the epistemological counterpart of the Liar Paradox. I will start by presenting it in its original version, the one which treats knowledge as a predicate attached to names of sentences and uses the framework of first order arithmetic. ...
... Montague [9] and basically is the epistemological counterpart of the Liar Paradox. I will start by presenting it in its original version, the one which treats knowledge as a predicate attached to names of sentences and uses the framework of first order arithmetic. ...
Mathematical Logic Fall 2004 Professor R. Moosa Contents
... We must formalize mathematically the notions of statements, proofs, structure, and truth in a given structure, with respect to a given language. Once we have formalized these notions, we may prove theorems about them. Logic is really metamathematics. Often the study of the study of a field contribut ...
... We must formalize mathematically the notions of statements, proofs, structure, and truth in a given structure, with respect to a given language. Once we have formalized these notions, we may prove theorems about them. Logic is really metamathematics. Often the study of the study of a field contribut ...
Bounded Functional Interpretation
... urging a shift of attention from the obtaining of precise witnesses to the obtaining of bounds for the witnesses. One of the main advantages of working with the extraction of bounds is that the non-computable mathematical objects whose existence is claimed by various ineffective principles can somet ...
... urging a shift of attention from the obtaining of precise witnesses to the obtaining of bounds for the witnesses. One of the main advantages of working with the extraction of bounds is that the non-computable mathematical objects whose existence is claimed by various ineffective principles can somet ...
Propositional and Predicate Logic - IX
... Soundness - proof (cont.) Otherwise τn+1 is formed from τn by appending an atomic tableau to Vn for some entry P on Vn . By induction we know that An agrees with P. (i) If P is formed by a logical connective, we take An+1 = An and verify that Vn can always be extended to a branch Vn+1 agreeing with ...
... Soundness - proof (cont.) Otherwise τn+1 is formed from τn by appending an atomic tableau to Vn for some entry P on Vn . By induction we know that An agrees with P. (i) If P is formed by a logical connective, we take An+1 = An and verify that Vn can always be extended to a branch Vn+1 agreeing with ...
Temporal Here and There - Computational Cognition Lab
... without the syntactic restrictions made by previous approaches. The modelbased orientation of this semantics led to a paradigm suitable for constraintsatisfaction problems that is known nowadays as Answer Set Programming (ASP) [17,18] and that became one of the most prominent and successful approach ...
... without the syntactic restrictions made by previous approaches. The modelbased orientation of this semantics led to a paradigm suitable for constraintsatisfaction problems that is known nowadays as Answer Set Programming (ASP) [17,18] and that became one of the most prominent and successful approach ...
Document
... us from something we have to something we want. • Both require a kind of experimentation to determine not only what rule to apply but, in cases in which content is to be added, what it is useful to add. • And although the derivation of 5 does not, other derivations in Part II involve inferences that ...
... us from something we have to something we want. • Both require a kind of experimentation to determine not only what rule to apply but, in cases in which content is to be added, what it is useful to add. • And although the derivation of 5 does not, other derivations in Part II involve inferences that ...
Slide 1
... 1. Lexicographically enumerate sound proofs. 2. Check each proof as it is created. If it succeeds in proving w, halt and accept. ...
... 1. Lexicographically enumerate sound proofs. 2. Check each proof as it is created. If it succeeds in proving w, halt and accept. ...
Induction and the Well-Ordering Principle Capturing All The Whole
... that every n-good set has a least element. We are ready to define our set A to this end: A = {n | n ∈ W and every n-good set has a least element}. “We prove that A = W by Induction. First, let’s see why 0 ∈ A. Suppose B is a 0-good set. We must show B has a least element. Being 0-good means that B h ...
... that every n-good set has a least element. We are ready to define our set A to this end: A = {n | n ∈ W and every n-good set has a least element}. “We prove that A = W by Induction. First, let’s see why 0 ∈ A. Suppose B is a 0-good set. We must show B has a least element. Being 0-good means that B h ...
pdf
... the inductive clause says that any n-terms can be combined with an n-ary function symbol to make a new term. 2. The set of L formulas is defined inductively. 3. The set of valid sequents of L is defined inductively. Notice here that what is actually defined is a subset of the set of pairs F (S) × S, ...
... the inductive clause says that any n-terms can be combined with an n-ary function symbol to make a new term. 2. The set of L formulas is defined inductively. 3. The set of valid sequents of L is defined inductively. Notice here that what is actually defined is a subset of the set of pairs F (S) × S, ...
Lesson 12
... 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 ...
Section 1.2-1.3
... 3. Let P (n) be a statement where n stands for a natural number. Suppose that the following two conditions hold: (a) P (1) is true, and (b) for all n 1, if P (n) holds then P (n + 1) also holds. Use the well-ordering principle to prove that P (n) must be true for all natural numbers n 1. Exercise No ...
... 3. Let P (n) be a statement where n stands for a natural number. Suppose that the following two conditions hold: (a) P (1) is true, and (b) for all n 1, if P (n) holds then P (n + 1) also holds. Use the well-ordering principle to prove that P (n) must be true for all natural numbers n 1. Exercise No ...
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction
... Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes theorems being its final products. The starting points are called axioms of the system. We distinguish two kinds of axioms: logical LA and specif ...
... Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes theorems being its final products. The starting points are called axioms of the system. We distinguish two kinds of axioms: logical LA and specif ...
Completeness or Incompleteness of Basic Mathematical Concepts
... concept of set”; of the possibility that new axioms will be found via “more profound understanding of the concepts underlying logic and mathematics.”11 There is nothing to suggest that perception of sets could help in finding new axioms or played a role in finding the old ones. A second relevant-loo ...
... concept of set”; of the possibility that new axioms will be found via “more profound understanding of the concepts underlying logic and mathematics.”11 There is nothing to suggest that perception of sets could help in finding new axioms or played a role in finding the old ones. A second relevant-loo ...
Mathematical Induction Proof by Weak Induction
... • Notice that the base case in the last proof is when n = 1, not when n = 0. How then is an induction principle applicable? It is because we can restate the statement of the theorem as “for any natural number n and for any tree T , if n ≥ 0 and T is of order n then T has size n − 1.” Since no tree o ...
... • Notice that the base case in the last proof is when n = 1, not when n = 0. How then is an induction principle applicable? It is because we can restate the statement of the theorem as “for any natural number n and for any tree T , if n ≥ 0 and T is of order n then T has size n − 1.” Since no tree o ...
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.