![Canad. Math. Bull. Vol. 24 (2), 1981 INDEPENDENT SETS OF](http://s1.studyres.com/store/data/000688774_1-b3a072b10548976a78e10ce223c659c8-300x300.png)
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 ...
On the existence of a connected component
... It is easier to build a connected component when the graph has only finitely many components. Theorem RCA0 proves that if a countable graph G has a finite set of vertices V0 such that every vertex in G is path connected to some vertex in V0 , then G can be decomposed into components. In particular, ...
... It is easier to build a connected component when the graph has only finitely many components. Theorem RCA0 proves that if a countable graph G has a finite set of vertices V0 such that every vertex in G is path connected to some vertex in V0 , then G can be decomposed into components. In particular, ...
characterization of classes of frames in modal language
... If a logic consists of K, φ → φ, φ → φ, grz, then it is characterized by the class of reflexive, transitive and antisymmetric Kripke frames which do not contain any infinite ascending chains of distinct points. S4 is valid in frames defined by grz. S4 laws in K ∪ grz were proved around 1979 by W. J ...
... If a logic consists of K, φ → φ, φ → φ, grz, then it is characterized by the class of reflexive, transitive and antisymmetric Kripke frames which do not contain any infinite ascending chains of distinct points. S4 is valid in frames defined by grz. S4 laws in K ∪ grz were proved around 1979 by W. J ...
Formal Theories of Truth INTRODUCTION
... But the situation changes fundamentally as soon as we pass to the generalization of this sentential function, i.e. to the general principle of contradiction. From the intuitive standpoint the truth of all those theorems is itself already a proof of the general principle; this principle represents, s ...
... But the situation changes fundamentally as soon as we pass to the generalization of this sentential function, i.e. to the general principle of contradiction. From the intuitive standpoint the truth of all those theorems is itself already a proof of the general principle; this principle represents, s ...
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 ...
Predicate logic - Teaching-WIKI
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
Predicate logic
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
Notes on `the contemporary conception of logic`
... formulae (or sequents) that are valid, and (ii) that they can prove all such formulae (or sequents). (pp. 8–10) Bostock here evidently takes very much the same line as Goldfarb, except that he avoids the unhappy outright identification of logical forms with schemata. And he goes on to say that not o ...
... formulae (or sequents) that are valid, and (ii) that they can prove all such formulae (or sequents). (pp. 8–10) Bostock here evidently takes very much the same line as Goldfarb, except that he avoids the unhappy outright identification of logical forms with schemata. And he goes on to say that not o ...
Basic Set Theory
... element. x. Find, with proof, identities for the operations set union and set intersection. The well ordering principle is an axiom that agrees with the common sense of most people familProblem 2.14 Prove part (ii) of Proposition 2.2. iar with the natural numbers. An empty set does not contain a sma ...
... element. x. Find, with proof, identities for the operations set union and set intersection. The well ordering principle is an axiom that agrees with the common sense of most people familProblem 2.14 Prove part (ii) of Proposition 2.2. iar with the natural numbers. An empty set does not contain a sma ...
When Bi-Interpretability Implies Synonymy
... b. U ` a,b∈S ∀xa , y b c∈S ∃z c d∈S ∀ud (ud ∈dc z c ↔ (ud ∈da xa ∨ ud =da y b )). Here ‘=da ’ is not really in the language if d 6= a. In this case we read ud =da y b simply as ⊥. It’s a nice exercise to show that e.g. ACA0 and GB are sequential. Closely related to AS is adjunctive class theory ac. ...
... b. U ` a,b∈S ∀xa , y b c∈S ∃z c d∈S ∀ud (ud ∈dc z c ↔ (ud ∈da xa ∨ ud =da y b )). Here ‘=da ’ is not really in the language if d 6= a. In this case we read ud =da y b simply as ⊥. It’s a nice exercise to show that e.g. ACA0 and GB are sequential. Closely related to AS is adjunctive class theory ac. ...
Basic Logic - Progetto e
... a conjunction is expressed as “P and Q” but can be also expressed by “P, but Q”, and “P, however Q”, “P, although Q”, “P, while Q” given that the compound meaning is just given b ...
... a conjunction is expressed as “P and Q” but can be also expressed by “P, but Q”, and “P, however Q”, “P, although Q”, “P, while Q” given that the compound meaning is just given b ...
WhichQuantifiersLogical
... reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a way that may be explained precisely in basic mathematical terms. What is taken from the inferentialists (or Zucker) is not their thesis as to meanin ...
... reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a way that may be explained precisely in basic mathematical terms. What is taken from the inferentialists (or Zucker) is not their thesis as to meanin ...
Propositional Logic: Why? soning Starts with George Boole around 1850
... The connections between the elements of the argument is lost in propositional logic Here we are talking about general properties (also called predicates) and individuals of a domain of discourse who may or may not have those properties Instead of introducing names for complete propositions -like in ...
... The connections between the elements of the argument is lost in propositional logic Here we are talking about general properties (also called predicates) and individuals of a domain of discourse who may or may not have those properties Instead of introducing names for complete propositions -like in ...
The disjunction introduction rule: Syntactic and semantics
... example, Braine and O´Brien (1998b) assign it a limited role in the basic model linked to primary skills of the mental logic theory, and Braine and O´Brien (1998c) acknowledge that they thought of the possibility of introducing some restrictions to this rule. Nonetheless, this new problem can be eas ...
... example, Braine and O´Brien (1998b) assign it a limited role in the basic model linked to primary skills of the mental logic theory, and Braine and O´Brien (1998c) acknowledge that they thought of the possibility of introducing some restrictions to this rule. Nonetheless, this new problem can be eas ...
DOC - John Woods
... Metatheory of CPL A big question is, “Why do we bother with proof theory?” After all, its principal concepts – axiom, theorem, deduction, proof – have no intuitive meaning there. What’s the point? Suppose we could show that for each of these uninterpreted properties of CPL’s proof theory theory is a ...
... Metatheory of CPL A big question is, “Why do we bother with proof theory?” After all, its principal concepts – axiom, theorem, deduction, proof – have no intuitive meaning there. What’s the point? Suppose we could show that for each of these uninterpreted properties of CPL’s proof theory theory is a ...
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY? 1. Introduction
... statements. Even this answer is not quite satisfactory: in theories of pairing we also have partial satisfaction predicates. See e.g. [Vau67] and [Vis10]. However, these satisfaction predicates are, in a sense, more finitistic than the ones provided by sequentiality. So, we could add to our specific ...
... statements. Even this answer is not quite satisfactory: in theories of pairing we also have partial satisfaction predicates. See e.g. [Vau67] and [Vis10]. However, these satisfaction predicates are, in a sense, more finitistic than the ones provided by sequentiality. So, we could add to our specific ...
Negative translation - Homepages of UvA/FNWI staff
... Excluded Middle ϕ ∨ ¬ϕ). However, the opposite point of view makes sense as well: one could also think of intuitionistic logic as an extension of classical logic. The reason for this is that there is a faithful copy of classical logic inside intuitionistic logic: such a copy is called a negative tra ...
... Excluded Middle ϕ ∨ ¬ϕ). However, the opposite point of view makes sense as well: one could also think of intuitionistic logic as an extension of classical logic. The reason for this is that there is a faithful copy of classical logic inside intuitionistic logic: such a copy is called a negative tra ...
Curry`s Paradox. An Argument for Trivialism
... paradoxical sentences obtained from self-reference are dialetheiae. Priest’s dialetheism has been extensively criticized in the literature (for an overview of criticism see Berto 2007, part IV). In this paper we will not discuss the crucial problem concerning the acceptance of a dialetheia. Rather, ...
... paradoxical sentences obtained from self-reference are dialetheiae. Priest’s dialetheism has been extensively criticized in the literature (for an overview of criticism see Berto 2007, part IV). In this paper we will not discuss the crucial problem concerning the acceptance of a dialetheia. Rather, ...
A MODAL EXTENSION OF FIRST ORDER CLASSICAL LOGIC–Part
... disjoint from the original. For example, A1 ...
... disjoint from the original. For example, A1 ...
Lesson 2
... The truth-value valuation of propositional symbols is a mapping v that to each propositional symbol p assigns a truth value, i.e., a value from the set {1,0}, which codes the set {True, False}: {pi} {1,0} The truth-value function of a PL formula is a function w, which for each valuation v of propo ...
... The truth-value valuation of propositional symbols is a mapping v that to each propositional symbol p assigns a truth value, i.e., a value from the set {1,0}, which codes the set {True, False}: {pi} {1,0} The truth-value function of a PL formula is a function w, which for each valuation v of propo ...
Exam 1 Solutions for Spring 2014
... 4. (10 points) A number n is a multiple of 3 if n = 3k for some integer k. Prove that if n2 is a multiple of 3, then n is a multiple of 3. Graded by Stacy Note: This question should have specified that n is an integer. To help compensate for this omission, the lowest score you can receive on this qu ...
... 4. (10 points) A number n is a multiple of 3 if n = 3k for some integer k. Prove that if n2 is a multiple of 3, then n is a multiple of 3. Graded by Stacy Note: This question should have specified that n is an integer. To help compensate for this omission, the lowest score you can receive on this qu ...
Logic for Gottlob Frege and Bertrand Russell:
... Propositions are the semantic content of thought, and propositions stand essentially in inferential relations to each other. “The” logico-philosophical question: how are inferential relations essential to thought? I. Frege: formal logic can answer this question by developing a logical notation (Begr ...
... Propositions are the semantic content of thought, and propositions stand essentially in inferential relations to each other. “The” logico-philosophical question: how are inferential relations essential to thought? I. Frege: formal logic can answer this question by developing a logical notation (Begr ...