Basic Concepts of Logic
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Dialetheists' Lies About the Liar
PRINCIPIA 22(1): 59–85 (2018) doi: 10.5007/1808-1711.2018v22n1p59 Published by NEL — Epistemology and Logic Research Group, Federal University of Santa Catarina (UFSC), Brazil. DIALETHEISTS’LIES ABOUT THE LIAR JONAS R. BECKER ARENHART Departamento de Filosofia, Universidade Federal de Santa Catarina, BRAZIL [email protected] EDERSON SAFRA MELO Departamento de Filosofia, Universidade Federal do Maranhão, BRAZIL [email protected] Abstract. Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, orelse because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the Logic of Para- dox (LP), which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, in- ternal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the re- sources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false. -
Logic in Action: Wittgenstein's Logical Pragmatism and the Impotence of Scepticism
This is the final, pre-publication draft. Please cite only from published paper in Philosophical Investigations 26:2 (April 2003), 125-48. LOGIC IN ACTION: WITTGENSTEIN'S LOGICAL PRAGMATISM AND THE IMPOTENCE OF SCEPTICISM DANIÈLE MOYAL-SHARROCK UNIVERSITY OF GENEVA 1. The Many Faces of Certainty: Wittgenstein's Logical Pragmatism So I am trying to say something that sounds like pragmatism. (OC 422) In his struggle to uncover the nature of our basic beliefs, Wittgenstein depicts them variously in On Certainty: he thinks of them in propositional terms, in pictorial terms and in terms of acting. As propositions, they would be of a peculiar sort – a hybrid between a logical and an empirical proposition (OC 136, 309). These are the so-called 'hinge propositions' of On Certainty (OC 341). Wittgenstein also thinks of these beliefs as forming a picture, a World-picture – or Weltbild (OC 167). This is a step in the right (nonpropositional) direction, but not the ultimate step. Wittgenstein's ultimate and crucial depiction of our basic beliefs is in terms of a know-how, an attitude, a way of acting (OC 204). Here, he treads on pragmatist ground. But can Wittgenstein be labelled a pragmatist, having himself rejected the affiliation because of its utility implication? But you aren't a pragmatist? No. For I am not saying that a proposition is true if it is useful. (RPP I, 266) Wittgenstein resists affiliation with pragmatism because he does not want his use of use to be confused with the utility use of use. For him, it is not that a proposition is true if it is useful, but that use gives the proposition its sense. -
Logic Model Workbook
Logic Model Workbook INNOVATION NETWORK, INC. www.innonet.org • [email protected] Logic Model Workbook Table of Contents Page Introduction - How to Use this Workbook .....................................................................2 Before You Begin .................................................................................................................3 Developing a Logic Model .................................................................................................4 Purposes of a Logic Model ............................................................................................... 5 The Logic Model’s Role in Evaluation ............................................................................ 6 Logic Model Components – Step by Step ....................................................................... 6 Problem Statement: What problem does your program address? ......................... 6 Goal: What is the overall purpose of your program? .............................................. 7 Rationale and Assumptions: What are some implicit underlying dynamics? ....8 Resources: What do you have to work with? ......................................................... 9 Activities: What will you do with your resources? ................................................ 11 Outputs: What are the tangible products of your activities? ................................. 13 Outcomes: What changes do you expect to occur as a result of your work?.......... 14 Outcomes Chain ...................................................................................... -
There Is No Pure Empirical Reasoning
There Is No Pure Empirical Reasoning 1. Empiricism and the Question of Empirical Reasons Empiricism may be defined as the view there is no a priori justification for any synthetic claim. Critics object that empiricism cannot account for all the kinds of knowledge we seem to possess, such as moral knowledge, metaphysical knowledge, mathematical knowledge, and modal knowledge.1 In some cases, empiricists try to account for these types of knowledge; in other cases, they shrug off the objections, happily concluding, for example, that there is no moral knowledge, or that there is no metaphysical knowledge.2 But empiricism cannot shrug off just any type of knowledge; to be minimally plausible, empiricism must, for example, at least be able to account for paradigm instances of empirical knowledge, including especially scientific knowledge. Empirical knowledge can be divided into three categories: (a) knowledge by direct observation; (b) knowledge that is deductively inferred from observations; and (c) knowledge that is non-deductively inferred from observations, including knowledge arrived at by induction and inference to the best explanation. Category (c) includes all scientific knowledge. This category is of particular import to empiricists, many of whom take scientific knowledge as a sort of paradigm for knowledge in general; indeed, this forms a central source of motivation for empiricism.3 Thus, if there is any kind of knowledge that empiricists need to be able to account for, it is knowledge of type (c). I use the term “empirical reasoning” to refer to the reasoning involved in acquiring this type of knowledge – that is, to any instance of reasoning in which (i) the premises are justified directly by observation, (ii) the reasoning is non- deductive, and (iii) the reasoning provides adequate justification for the conclusion. -
A Philosophical Treatise on the Connection of Scientific Reasoning
mathematics Review A Philosophical Treatise on the Connection of Scientific Reasoning with Fuzzy Logic Evangelos Athanassopoulos 1 and Michael Gr. Voskoglou 2,* 1 Independent Researcher, Giannakopoulou 39, 27300 Gastouni, Greece; [email protected] 2 Department of Applied Mathematics, Graduate Technological Educational Institute of Western Greece, 22334 Patras, Greece * Correspondence: [email protected] Received: 4 May 2020; Accepted: 19 May 2020; Published:1 June 2020 Abstract: The present article studies the connection of scientific reasoning with fuzzy logic. Induction and deduction are the two main types of human reasoning. Although deduction is the basis of the scientific method, almost all the scientific progress (with pure mathematics being probably the unique exception) has its roots to inductive reasoning. Fuzzy logic gives to the disdainful by the classical/bivalent logic induction its proper place and importance as a fundamental component of the scientific reasoning. The error of induction is transferred to deductive reasoning through its premises. Consequently, although deduction is always a valid process, it is not an infallible method. Thus, there is a need of quantifying the degree of truth not only of the inductive, but also of the deductive arguments. In the former case, probability and statistics and of course fuzzy logic in cases of imprecision are the tools available for this purpose. In the latter case, the Bayesian probabilities play a dominant role. As many specialists argue nowadays, the whole science could be viewed as a Bayesian process. A timely example, concerning the validity of the viruses’ tests, is presented, illustrating the importance of the Bayesian processes for scientific reasoning. -
In Defence of Folk Psychology
FRANK JACKSON & PHILIP PETTIT IN DEFENCE OF FOLK PSYCHOLOGY (Received 14 October, 1988) It turned out that there was no phlogiston, no caloric fluid, and no luminiferous ether. Might it turn out that there are no beliefs and desires? Patricia and Paul Churchland say yes. ~ We say no. In part one we give our positive argument for the existence of beliefs and desires, and in part two we offer a diagnosis of what has misled the Church- lands into holding that it might very well turn out that there are no beliefs and desires. 1. THE EXISTENCE OF BELIEFS AND DESIRES 1.1. Our Strategy Eliminativists do not insist that it is certain as of now that there are no beliefs and desires. They insist that it might very well turn out that there are no beliefs and desires. Thus, in order to engage with their position, we need to provide a case for beliefs and desires which, in addition to being a strong one given what we now know, is one which is peculiarly unlikely to be undermined by future progress in neuroscience. Our first step towards providing such a case is to observe that the question of the existence of beliefs and desires as conceived in folk psychology can be divided into two questions. There exist beliefs and desires if there exist creatures with states truly describable as states of believing that such-and-such or desiring that so-and-so. Our question, then, can be divided into two questions. First, what is it for a state to be truly describable as a belief or as a desire; what, that is, needs to be the case according to our folk conception of belief and desire for a state to be a belief or a desire? And, second, is what needs to be the case in fact the case? Accordingly , if we accepted a certain, simple behaviourist account of, say, our folk Philosophical Studies 59:31--54, 1990. -
Relative Interpretations and Substitutional Definitions of Logical
Relative Interpretations and Substitutional Definitions of Logical Truth and Consequence MIRKO ENGLER Abstract: This paper proposes substitutional definitions of logical truth and consequence in terms of relative interpretations that are extensionally equivalent to the model-theoretic definitions for any relational first-order language. Our philosophical motivation to consider substitutional definitions is based on the hope to simplify the meta-theory of logical consequence. We discuss to what extent our definitions can contribute to that. Keywords: substitutional definitions of logical consequence, Quine, meta- theory of logical consequence 1 Introduction This paper investigates the applicability of relative interpretations in a substi- tutional account of logical truth and consequence. We introduce definitions of logical truth and consequence in terms of relative interpretations and demonstrate that they are extensionally equivalent to the model-theoretic definitions as developed in (Tarski & Vaught, 1956) for any first-order lan- guage (including equality). The benefit of such a definition is that it could be given in a meta-theoretic framework that only requires arithmetic as ax- iomatized by PA. Furthermore, we investigate how intensional constraints on logical truth and consequence force us to extend our framework to an arithmetical meta-theory that itself interprets set-theory. We will argue that such an arithmetical framework still might be in favor over a set-theoretical one. The basic idea behind our definition is both to generate and evaluate substitution instances of a sentence ' by relative interpretations. A relative interpretation rests on a function f that translates all formulae ' of a language L into formulae f(') of a language L0 by mapping the primitive predicates 0 P of ' to formulae P of L while preserving the logical structure of ' 1 Mirko Engler and relativizing its quantifiers by an L0-definable formula. -
CSE Yet, Please Do Well! Logical Connectives
administrivia Course web: http://www.cs.washington.edu/311 Office hours: 12 office hours each week Me/James: MW 10:30-11:30/2:30-3:30pm or by appointment TA Section: Start next week Call me: Shayan Don’t: Actually call me. Homework #1: Will be posted today, due next Friday by midnight (Oct 9th) Gradescope! (stay tuned) Extra credit: Not required to get a 4.0. Counts separately. In total, may raise grade by ~0.1 Don’t be shy (raise your hand in the back)! Do space out your participation. If you are not CSE yet, please do well! logical connectives p q p q p p T T T T F T F F F T F T F NOT F F F AND p q p q p q p q T T T T T F T F T T F T F T T F T T F F F F F F OR XOR 푝 → 푞 • “If p, then q” is a promise: p q p q F F T • Whenever p is true, then q is true F T T • Ask “has the promise been broken” T F F T T T If it’s raining, then I have my umbrella. related implications • Implication: p q • Converse: q p • Contrapositive: q p • Inverse: p q How do these relate to each other? How to see this? 푝 ↔ 푞 • p iff q • p is equivalent to q • p implies q and q implies p p q p q Let’s think about fruits A fruit is an apple only if it is either red or green and a fruit is not red and green. -
Conditional Statement/Implication Converse and Contrapositive
Conditional Statement/Implication CSCI 1900 Discrete Structures •"ifp then q" • Denoted p ⇒ q – p is called the antecedent or hypothesis Conditional Statements – q is called the consequent or conclusion Reading: Kolman, Section 2.2 • Example: – p: I am hungry q: I will eat – p: It is snowing q: 3+5 = 8 CSCI 1900 – Discrete Structures Conditional Statements – Page 1 CSCI 1900 – Discrete Structures Conditional Statements – Page 2 Conditional Statement/Implication Truth Table Representing Implication (continued) • In English, we would assume a cause- • If viewed as a logic operation, p ⇒ q can only be and-effect relationship, i.e., the fact that p evaluated as false if p is true and q is false is true would force q to be true. • This does not say that p causes q • If “it is snowing,” then “3+5=8” is • Truth table meaningless in this regard since p has no p q p ⇒ q effect at all on q T T T • At this point it may be easiest to view the T F F operator “⇒” as a logic operationsimilar to F T T AND or OR (conjunction or disjunction). F F T CSCI 1900 – Discrete Structures Conditional Statements – Page 3 CSCI 1900 – Discrete Structures Conditional Statements – Page 4 Examples where p ⇒ q is viewed Converse and contrapositive as a logic operation •Ifp is false, then any q supports p ⇒ q is • The converse of p ⇒ q is the implication true. that q ⇒ p – False ⇒ True = True • The contrapositive of p ⇒ q is the –False⇒ False = True implication that ~q ⇒ ~p • If “2+2=5” then “I am the king of England” is true CSCI 1900 – Discrete Structures Conditional Statements – Page 5 CSCI 1900 – Discrete Structures Conditional Statements – Page 6 1 Converse and Contrapositive Equivalence or biconditional Example Example: What is the converse and •Ifp and q are statements, the compound contrapositive of p: "it is raining" and q: I statement p if and only if q is called an get wet? equivalence or biconditional – Implication: If it is raining, then I get wet. -
Verifying the Unification Algorithm In
Verifying the Uni¯cation Algorithm in LCF Lawrence C. Paulson Computer Laboratory Corn Exchange Street Cambridge CB2 3QG England August 1984 Abstract. Manna and Waldinger's theory of substitutions and uni¯ca- tion has been veri¯ed using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an exam- ple of interaction with LCF. Translating the theory into LCF's domain- theoretic logic is largely straightforward. Well-founded induction on a complex ordering is translated into nested structural inductions. Cor- rectness of uni¯cation is expressed using predicates for such properties as idempotence and most-generality. The veri¯cation is presented as a series of lemmas. The LCF proofs are compared with the original ones, and with other approaches. It appears di±cult to ¯nd a logic that is both simple and exible, especially for proving termination. Contents 1 Introduction 2 2 Overview of uni¯cation 2 3 Overview of LCF 4 3.1 The logic PPLAMBDA :::::::::::::::::::::::::: 4 3.2 The meta-language ML :::::::::::::::::::::::::: 5 3.3 Goal-directed proof :::::::::::::::::::::::::::: 6 3.4 Recursive data structures ::::::::::::::::::::::::: 7 4 Di®erences between the formal and informal theories 7 4.1 Logical framework :::::::::::::::::::::::::::: 8 4.2 Data structure for expressions :::::::::::::::::::::: 8 4.3 Sets and substitutions :::::::::::::::::::::::::: 9 4.4 The induction principle :::::::::::::::::::::::::: 10 5 Constructing theories in LCF 10 5.1 Expressions :::::::::::::::::::::::::::::::: -
Against Logical Form
Against logical form Zolta´n Gendler Szabo´ Conceptions of logical form are stranded between extremes. On one side are those who think the logical form of a sentence has little to do with logic; on the other, those who think it has little to do with the sentence. Most of us would prefer a conception that strikes a balance: logical form that is an objective feature of a sentence and captures its logical character. I will argue that we cannot get what we want. What are these extreme conceptions? In linguistics, logical form is typically con- ceived of as a level of representation where ambiguities have been resolved. According to one highly developed view—Chomsky’s minimalism—logical form is one of the outputs of the derivation of a sentence. The derivation begins with a set of lexical items and after initial mergers it splits into two: on one branch phonological operations are applied without semantic effect; on the other are semantic operations without phono- logical realization. At the end of the first branch is phonological form, the input to the articulatory–perceptual system; and at the end of the second is logical form, the input to the conceptual–intentional system.1 Thus conceived, logical form encompasses all and only information required for interpretation. But semantic and logical information do not fully overlap. The connectives “and” and “but” are surely not synonyms, but the difference in meaning probably does not concern logic. On the other hand, it is of utmost logical importance whether “finitely many” or “equinumerous” are logical constants even though it is hard to see how this information could be essential for their interpretation. -
7.1 Rules of Implication I
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification.