Sense and denotation as algorithm and value
... The circular nature of these instructions corresponds to the self reference of the sentences, but there is nothing unusual about circular clauses like these in programs. The algorithms defined by them are recursive algorithms and (in this case, as one might expect), they do not compute any value at ...
... The circular nature of these instructions corresponds to the self reference of the sentences, but there is nothing unusual about circular clauses like these in programs. The algorithms defined by them are recursive algorithms and (in this case, as one might expect), they do not compute any value at ...
Proofs as Efficient Programs - Dipartimento di Informatica
... space [19]), etc. Moreover, we now have also systems where, contrary to lal, the soundness for polynomial time holds for lambda-calculus reduction, like dlal [6] and other similar systems. As a result, the general framework of light logics is now full of different systems, and of variants of those s ...
... space [19]), etc. Moreover, we now have also systems where, contrary to lal, the soundness for polynomial time holds for lambda-calculus reduction, like dlal [6] and other similar systems. As a result, the general framework of light logics is now full of different systems, and of variants of those s ...
REVERSE MATHEMATICS AND RECURSIVE GRAPH THEORY
... will abuse notation by denoting an edge by (x, y) rather than {x, y}. For k ∈ N, we say that χ : V → k is a k-coloring of G if χ always assigns different colors to neighboring vertices. That is, χ is a k-coloring if χ : V → k and (x, y) ∈ E implies χ(x) 6= χ(y). If G has a k-coloring, we say that G ...
... will abuse notation by denoting an edge by (x, y) rather than {x, y}. For k ∈ N, we say that χ : V → k is a k-coloring of G if χ always assigns different colors to neighboring vertices. That is, χ is a k-coloring if χ : V → k and (x, y) ∈ E implies χ(x) 6= χ(y). If G has a k-coloring, we say that G ...
4 The semantics of full first
... C∗ having properties (2), (3), and (4) described in the statement of Lemma 4.8. Then Γ∗ is satisfiable. Proof. We need to make two changes in the proof of Lemma 4.4. First we need to provide the second part of the definition of χ. We do that by setting. χ((Fin , [c1 ]R , . . . , [cn ]R )) = [c]R iff ...
... C∗ having properties (2), (3), and (4) described in the statement of Lemma 4.8. Then Γ∗ is satisfiable. Proof. We need to make two changes in the proof of Lemma 4.4. First we need to provide the second part of the definition of χ. We do that by setting. χ((Fin , [c1 ]R , . . . , [cn ]R )) = [c]R iff ...
Truth and proof
... Tennant’s answer: Sound proof suffices for truth • Any proof in PA* of a sentence in the language of L, is a ground for asserting , even though might have no proof in the weaker system . All is needed, for the assertion of , is some proof of …In particular if it turns out that there is a pr ...
... Tennant’s answer: Sound proof suffices for truth • Any proof in PA* of a sentence in the language of L, is a ground for asserting , even though might have no proof in the weaker system . All is needed, for the assertion of , is some proof of …In particular if it turns out that there is a pr ...
W. Dean. Algorithms and the mathematical foundations of computer
... a mechanical computing agent. Such characteristics inform the conception of an effectively computable function—i.e. one computable by an effective procedure—which was analyzed in distinct but extensionally equivalent ways by Church, Turing, Post, Kleene, and Gödel. Reflection on these analyses led v ...
... a mechanical computing agent. Such characteristics inform the conception of an effectively computable function—i.e. one computable by an effective procedure—which was analyzed in distinct but extensionally equivalent ways by Church, Turing, Post, Kleene, and Gödel. Reflection on these analyses led v ...
Lecture 8: Back-and-forth - to go back my main page.
... (b) Th(M ) = Th(N ) and SSy(M ) = SSy(N ). Proof. The implication (a) ⇒ (b) is trivial. For the converse, suppose (b) holds. We carry out a back-and-forth argument to find an isomorphism M → N . By recursion, we will define (rm )m∈N in M and (sm )m∈N in N such that f : rm 7→ sm is an isomorphism M → ...
... (b) Th(M ) = Th(N ) and SSy(M ) = SSy(N ). Proof. The implication (a) ⇒ (b) is trivial. For the converse, suppose (b) holds. We carry out a back-and-forth argument to find an isomorphism M → N . By recursion, we will define (rm )m∈N in M and (sm )m∈N in N such that f : rm 7→ sm is an isomorphism M → ...
Notes on `the contemporary conception of logic`
... of which implies the second. (pp. 64–65) Note an initial oddity here (taking up a theme that Timothy Smiley has remarked on in another context). It is said that a ‘logical form’ just is a schema. What is it then for a sentence to have a logical form? Presumably it is for the sentence to be an instan ...
... of which implies the second. (pp. 64–65) Note an initial oddity here (taking up a theme that Timothy Smiley has remarked on in another context). It is said that a ‘logical form’ just is a schema. What is it then for a sentence to have a logical form? Presumably it is for the sentence to be an instan ...
Propositional Dynamic Logic of Regular Programs*+
... modal logic introduced by Kripke [14]. Informally, each program a defines a relation p(u) between program states: (s, t) E p(a) if and only if a executed in state s can terminate in state t. The truth of an assertion is determined relative to a program state, so we say “p is true in state s.” The fo ...
... modal logic introduced by Kripke [14]. Informally, each program a defines a relation p(u) between program states: (s, t) E p(a) if and only if a executed in state s can terminate in state t. The truth of an assertion is determined relative to a program state, so we say “p is true in state s.” The fo ...
pdf
... variables. It is clear that the truth value of b is not affected by the action p, so it would be the same before as after. But once this is established, we no longer need to know what p and b are, but only that pb = bp. It follows by purely equational reasoning in KAT that p1 b = bp1 → · · · → pn b ...
... variables. It is clear that the truth value of b is not affected by the action p, so it would be the same before as after. But once this is established, we no longer need to know what p and b are, but only that pb = bp. It follows by purely equational reasoning in KAT that p1 b = bp1 → · · · → pn b ...
Supplemental Reading (Kunen)
... set theory they use. It is generally understood which principles are correct beyond any doubt, and which are subject to question. For example, it is generally agreed that the Continuum Hypothesis (CH) is not a basic principle, but rather an open conjecture, and we are all able, without the benefit o ...
... set theory they use. It is generally understood which principles are correct beyond any doubt, and which are subject to question. For example, it is generally agreed that the Continuum Hypothesis (CH) is not a basic principle, but rather an open conjecture, and we are all able, without the benefit o ...
axioms
... Axiomatic Method • A procedure by which we demonstrate as fact (prove) results (theorems) discovered by experimentation, observation, trial and error or “intuitive insight.” • Definition: A proof is a sequence of statements, each of which follows logically from the ones (statements) before and leads ...
... Axiomatic Method • A procedure by which we demonstrate as fact (prove) results (theorems) discovered by experimentation, observation, trial and error or “intuitive insight.” • Definition: A proof is a sequence of statements, each of which follows logically from the ones (statements) before and leads ...
CHAPTER 8 One-to-One Functions and One-to
... much more important than brevity. On the other hand (OTOH), to much repetition can lead to confusion and hence decrease clarity. This, like an English composition, calls for judgement on the part of the writer. Explain clearly and completely and, to the extent possible, in your own words. As a rule ...
... much more important than brevity. On the other hand (OTOH), to much repetition can lead to confusion and hence decrease clarity. This, like an English composition, calls for judgement on the part of the writer. Explain clearly and completely and, to the extent possible, in your own words. As a rule ...
(pdf)
... 1. First Order Logic. Before making an introductory exposition of the ideas behind non-standard analysis and the related construction, it is important that the reader familiarizes himself with the theory of first order logic. This section presents an introduction to these ideas, with a few examples. ...
... 1. First Order Logic. Before making an introductory exposition of the ideas behind non-standard analysis and the related construction, it is important that the reader familiarizes himself with the theory of first order logic. This section presents an introduction to these ideas, with a few examples. ...
Program Equilibrium in the Prisoner`s Dilemma via Löb`s Theorem
... will provably halt on all inputs.) This formalism has the benefit of concreteness: we could actually program such agents, although the ones we shall deal with are often very far from efficient in their requirements. On the other hand, deducing what happens when algorithms call upon each others’ code ...
... will provably halt on all inputs.) This formalism has the benefit of concreteness: we could actually program such agents, although the ones we shall deal with are often very far from efficient in their requirements. On the other hand, deducing what happens when algorithms call upon each others’ code ...
INTRODUCTION TO THE THEORY OF PROOFS 3A. The Gentzen
... for all cases of smaller grade, and for all cases of the same grade but smaller rank. The cases where one of the ranks is > 1 are treated first, and are messy but fairly easy. The main part of the proof is in the consideration of the four cases (one for each propositional connective) where the rank ...
... for all cases of smaller grade, and for all cases of the same grade but smaller rank. The cases where one of the ranks is > 1 are treated first, and are messy but fairly easy. The main part of the proof is in the consideration of the four cases (one for each propositional connective) where the rank ...
Second-Order Logic and Fagin`s Theorem
... CHAPTER 7. SECOND-ORDER LOGIC AND FAGIN’S THEOREM The converse of Lynch’s Theorem is an open question: ...
... CHAPTER 7. SECOND-ORDER LOGIC AND FAGIN’S THEOREM The converse of Lynch’s Theorem is an open question: ...
Automata, Languages, and Programming
... variables. It is clear that the truth value of b is not affected by the action p, so it would be the same before as after. But once this is established, we no longer need to know what p and b are, but only that pb = bp. It follows by purely equational reasoning in KAT that p1 b = bp1 ∈ · · · ∈ pn b = ...
... variables. It is clear that the truth value of b is not affected by the action p, so it would be the same before as after. But once this is established, we no longer need to know what p and b are, but only that pb = bp. It follows by purely equational reasoning in KAT that p1 b = bp1 ∈ · · · ∈ pn b = ...
The Future of Post-Human Mathematical Logic
... which few thinkers would dare to question—the foundations of mathematics and logic. He examines the reasoning of forebears, points out specific shortcomings, and offers another perspective to fulfill those shortcomings. The breadth of issues chosen by Dr. Baofu for analysis is truly astounding. In e ...
... which few thinkers would dare to question—the foundations of mathematics and logic. He examines the reasoning of forebears, points out specific shortcomings, and offers another perspective to fulfill those shortcomings. The breadth of issues chosen by Dr. Baofu for analysis is truly astounding. In e ...
The First Incompleteness Theorem
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important and seminal work on se ...
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important and seminal work on se ...
Probabilistic Theorem Proving - The University of Texas at Dallas
... a general-purpose inference engine for deterministic programming languages such as Prolog, we envision that PTP/LWMC will serve as a general-purpose inference engine for probabilistic languages. Second, our LWMC algorithms take advantage of context-specific independence (CSI) [2] and determinism [3] ...
... a general-purpose inference engine for deterministic programming languages such as Prolog, we envision that PTP/LWMC will serve as a general-purpose inference engine for probabilistic languages. Second, our LWMC algorithms take advantage of context-specific independence (CSI) [2] and determinism [3] ...
HOARE`S LOGIC AND PEANO`S ARITHMETIC
... formulae one must have, if one is to get anything proved about them in PA. (The well-structured and mechanical appearance of formal proofs in PA should always be considered a criterion for the success of a logical analysis which PA is asked to support.) We shall divide the work of this section betwe ...
... formulae one must have, if one is to get anything proved about them in PA. (The well-structured and mechanical appearance of formal proofs in PA should always be considered a criterion for the success of a logical analysis which PA is asked to support.) We shall divide the work of this section betwe ...
A Question About Increasing Functions
... Now u < v, so u − v < 0. The quantity in the square brackets on the right side of (3) is positive because it is the sum of two squares which can’t both be zero. Thus, the product on the right side of (3) is negative, and from this it follows that u3 < v 3 . Our appeal to the definition shows that th ...
... Now u < v, so u − v < 0. The quantity in the square brackets on the right side of (3) is positive because it is the sum of two squares which can’t both be zero. Thus, the product on the right side of (3) is negative, and from this it follows that u3 < v 3 . Our appeal to the definition shows that th ...
Mathematical Logic Fall 2004 Professor R. Moosa Contents
... Mathematical Logic is the study of the type of reasoning done by mathematicians. (i.e. proofs, as opposed to observation) Axioms are the first unprovable laws. They are statements about certain “basic concepts” (undefined first concepts). There is usually some sort of “soft” justification for believ ...
... Mathematical Logic is the study of the type of reasoning done by mathematicians. (i.e. proofs, as opposed to observation) Axioms are the first unprovable laws. They are statements about certain “basic concepts” (undefined first concepts). There is usually some sort of “soft” justification for believ ...
Review - Gerry O nolan
... Chapter IX, entitled 'The Myth of Formal Logic,' can and should be read separately from the rest of the book. In this chapter Stove abandons the thesis that either deductive or inductive logic is purely formal. In the latter case, this denial is used as the basis of a solution to Goodman's so-calle ...
... Chapter IX, entitled 'The Myth of Formal Logic,' can and should be read separately from the rest of the book. In this chapter Stove abandons the thesis that either deductive or inductive logic is purely formal. In the latter case, this denial is used as the basis of a solution to Goodman's so-calle ...