Logic: a Brief Introduction Ronald L
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Aristotle's Assertoric Syllogistic and Modern Relevance Logic*
Aristotle’s assertoric syllogistic and modern relevance logic* Abstract: This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s Entailment. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic. Keywords: Aristotle, Syllogism, Relevance Logic, History of Logic, Logic, Relevance 1. -
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. -
The Peripatetic Program in Categorical Logic: Leibniz on Propositional Terms
THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 65 THE PERIPATETIC PROGRAM IN CATEGORICAL LOGIC: LEIBNIZ ON PROPOSITIONAL TERMS MARKO MALINK Department of Philosophy, New York University and ANUBAV VASUDEVAN Department of Philosophy, University of Chicago Abstract. Greek antiquity saw the development of two distinct systems of logic: Aristotle’s theory of the categorical syllogism and the Stoic theory of the hypothetical syllogism. Some ancient logicians argued that hypothetical syllogistic is more fundamental than categorical syllogistic on the grounds that the latter relies on modes of propositional reasoning such as reductio ad absurdum. Peripatetic logicians, by contrast, sought to establish the priority of categorical over hypothetical syllogistic by reducing various modes of propositional reasoning to categorical form. In the 17th century, this Peripatetic program of reducing hypothetical to categorical logic was championed by Gottfried Wilhelm Leibniz. In an essay titled Specimina calculi rationalis, Leibniz develops a theory of propositional terms that allows him to derive the rule of reductio ad absurdum in a purely categor- ical calculus in which every proposition is of the form AisB. We reconstruct Leibniz’s categorical calculus and show that it is strong enough to establish not only the rule of reductio ad absurdum,but all the laws of classical propositional logic. Moreover, we show that the propositional logic generated by the nonmonotonic variant of Leibniz’s categorical calculus is a natural system of relevance logic ¬ known as RMI→ . From its beginnings in antiquity up until the late 19th century, the study of formal logic was shaped by two distinct approaches to the subject. The first approach was primarily concerned with simple propositions expressing predicative relations between terms. -
Aristotle on Predication
DOI: 10.1111/ejop.12054 Aristotle on Predication Phil Corkum Abstract: A predicate logic typically has a heterogeneous semantic theory. Sub- jects and predicates have distinct semantic roles: subjects refer; predicates char- acterize. A sentence expresses a truth if the object to which the subject refers is correctly characterized by the predicate. Traditional term logic, by contrast, has a homogeneous theory: both subjects and predicates refer; and a sentence is true if the subject and predicate name one and the same thing. In this paper, I will examine evidence for ascribing to Aristotle the view that subjects and predicates refer. If this is correct, then it seems that Aristotle, like the traditional term logician, problematically conflates predication and identity claims. I will argue that we can ascribe to Aristotle the view that both subjects and predicates refer, while holding that he would deny that a sentence is true just in case the subject and predicate name one and the same thing. In particular, I will argue that Aristotle’s core semantic notion is not identity but the weaker relation of consti- tution. For example, the predication ‘All men are mortal’ expresses a true thought, in Aristotle’s view, just in case the mereological sum of humans is a part of the mereological sum of mortals. A predicate logic typically has a heterogeneous semantic theory. Subjects and predicates have distinct semantic roles: subjects refer; predicates characterize. A sentence expresses a truth if the object to which the subject refers is correctly characterized by the predicate. Traditional term logic, by contrast, has a homo- geneous theory: both subjects and predicates refer; and a sentence is true if the subject and predicate name one and the same thing. -
The Philosophy of Mathematics: a Study of Indispensability and Inconsistency
Claremont Colleges Scholarship @ Claremont Scripps Senior Theses Scripps Student Scholarship 2016 The hiP losophy of Mathematics: A Study of Indispensability and Inconsistency Hannah C. Thornhill Scripps College Recommended Citation Thornhill, Hannah C., "The hiP losophy of Mathematics: A Study of Indispensability and Inconsistency" (2016). Scripps Senior Theses. Paper 894. http://scholarship.claremont.edu/scripps_theses/894 This Open Access Senior Thesis is brought to you for free and open access by the Scripps Student Scholarship at Scholarship @ Claremont. It has been accepted for inclusion in Scripps Senior Theses by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. The Philosophy of Mathematics: A Study of Indispensability and Inconsistency Hannah C.Thornhill March 10, 2016 Submitted to Scripps College in Partial Fulfillment of the Degree of Bachelor of Arts in Mathematics and Philosophy Professor Avnur Professor Karaali Abstract This thesis examines possible philosophies to account for the prac- tice of mathematics, exploring the metaphysical, ontological, and epis- temological outcomes of each possible theory. Through a study of the two most probable ideas, mathematical platonism and fictionalism, I focus on the compelling argument for platonism given by an ap- peal to the sciences. The Indispensability Argument establishes the power of explanation seen in the relationship between mathematics and empirical science. Cases of this explanatory power illustrate how we might have reason to believe in the existence of mathematical en- tities present within our best scientific theories. The second half of this discussion surveys Newtonian Cosmology and other inconsistent theories as they pose issues that have received insignificant attention within the philosophy of mathematics. -
The History of Logic
c Peter King & Stewart Shapiro, The Oxford Companion to Philosophy (OUP 1995), 496–500. THE HISTORY OF LOGIC Aristotle was the first thinker to devise a logical system. He drew upon the emphasis on universal definition found in Socrates, the use of reductio ad absurdum in Zeno of Elea, claims about propositional structure and nega- tion in Parmenides and Plato, and the body of argumentative techniques found in legal reasoning and geometrical proof. Yet the theory presented in Aristotle’s five treatises known as the Organon—the Categories, the De interpretatione, the Prior Analytics, the Posterior Analytics, and the Sophistical Refutations—goes far beyond any of these. Aristotle holds that a proposition is a complex involving two terms, a subject and a predicate, each of which is represented grammatically with a noun. The logical form of a proposition is determined by its quantity (uni- versal or particular) and by its quality (affirmative or negative). Aristotle investigates the relation between two propositions containing the same terms in his theories of opposition and conversion. The former describes relations of contradictoriness and contrariety, the latter equipollences and entailments. The analysis of logical form, opposition, and conversion are combined in syllogistic, Aristotle’s greatest invention in logic. A syllogism consists of three propositions. The first two, the premisses, share exactly one term, and they logically entail the third proposition, the conclusion, which contains the two non-shared terms of the premisses. The term common to the two premisses may occur as subject in one and predicate in the other (called the ‘first figure’), predicate in both (‘second figure’), or subject in both (‘third figure’). -
Validity and Soundness
1.4 Validity and Soundness A deductive argument proves its conclusion ONLY if it is both valid and sound. Validity: An argument is valid when, IF all of it’s premises were true, then the conclusion would also HAVE to be true. In other words, a “valid” argument is one where the conclusion necessarily follows from the premises. It is IMPOSSIBLE for the conclusion to be false if the premises are true. Here’s an example of a valid argument: 1. All philosophy courses are courses that are super exciting. 2. All logic courses are philosophy courses. 3. Therefore, all logic courses are courses that are super exciting. Note #1: IF (1) and (2) WERE true, then (3) would also HAVE to be true. Note #2: Validity says nothing about whether or not any of the premises ARE true. It only says that IF they are true, then the conclusion must follow. So, validity is more about the FORM of an argument, rather than the TRUTH of an argument. So, an argument is valid if it has the proper form. An argument can have the right form, but be totally false, however. For example: 1. Daffy Duck is a duck. 2. All ducks are mammals. 3. Therefore, Daffy Duck is a mammal. The argument just given is valid. But, premise 2 as well as the conclusion are both false. Notice however that, IF the premises WERE true, then the conclusion would also have to be true. This is all that is required for validity. A valid argument need not have true premises or a true conclusion. -
1 Critical Thinking: the Very Basics
1 CRITICAL THINKING: THE VERY BASICS - HANDBOOK Dona Warren, Philosophy Department, The University of Wisconsin – Stevens Point I. RECOGNIZING ARGUMENTS An argument is a unit of reasoning that attempts to prove that a certain idea is true by citing other ideas as evidence. II. ANALYZING ARGUMENTS 1. Identify the ultimate conclusion (the main idea that the argument is trying to prove). Sometimes, this is unstated. 2. Determine which other ideas are important. An idea is important if it helps the argument to establish the truth of the ultimate conclusion. 3. Figure out how these other ideas work together to support the ultimate conclusion. Ideas work together according to four basic patterns of cooperation. Basic Patterns: i. Premise / Ultimate Conclusion Idea Premise - an idea that the argument assumes to be true % without support Inference - the connection that holds between the idea(s) at the top of the arrow and the idea at the bottom of the arrow when the truth of the idea(s) at the top is supposed % to establish the truth of the idea at the bottom - often indicated by conclusion indicator expressions like “therefore” and reason indicator expressions “because.” Idea Ultimate Conclusion – what the argument is ultimately % trying to prove. 2 ii. Subconclusions Idea Idea Subconclusion – an intermediate idea on the way from the % premises to the ultimate conclusion Idea iii. Dependent Reasons Idea + Idea Dependent Reasons – neither idea can support the % conclusion alone but together they can support the conclusion Idea iv. Independent Reasons Independent Reasons – each idea can support the Idea Idea conclusion on its own % Idea This gives us independent lines of reasoning. -
Review of Philosophical Logic: an Introduction to Advanced Topics*
Teaching Philosophy 36 (4):420-423 (2013). Review of Philosophical Logic: An Introduction to Advanced Topics* CHAD CARMICHAEL Indiana University–Purdue University Indianapolis This book serves as a concise introduction to some main topics in modern formal logic for undergraduates who already have some familiarity with formal languages. There are chapters on sentential and quantificational logic, modal logic, elementary set theory, a brief introduction to the incompleteness theorem, and a modern development of traditional Aristotelian Logic: the “term logic” of Sommers (1982) and Englebretsen (1996). Most of the book provides compact introductions to the syntax and semantics of various familiar formal systems. Here and there, the authors briefly indicate how intuitionist logic diverges from the classical treatments that are more fully explored. The book is appropriate for an undergraduate-level second course in logic that will approach the topic from a philosophical (rather than primarily mathematical) perspective. Philosophical topics (sometimes very briefly) touched upon in the book include: intuitionist logic, substitutional quantification, the nature of logical consequence, deontic logic, the incompleteness theorem, and the interaction of quantification and modality. The book provides an effective jumping-off point for several of these topics. In particular, the presentation of the intuitive idea of the * George Englebretsen and Charles Sayward, Philosophical Logic: An Introduction to Advanced Topics (New York: Continuum International Publishing Group, 2013), 208 pages. 1 incompleteness theorem (chapter 7) is the right level of rigor for an undergraduate philosophy student, as it provides the basic idea of the proof without getting bogged down in technical details that would not have much philosophical interest. -
What Is a Logical Argument? What Is Deductive Reasoning?
What is a logical argument? What is deductive reasoning? Fundamentals of Academic Writing Logical relations • Deductive logic – Claims to provide conclusive support for the truth of a conclusion • Inductive logic – Arguments support a conclusion, but do not claim to show that it is necessarily true Deductive logic • Categorical propositions – Deductive arguments are either valid or invalid. – Premises are either true or not true. – If the argument is valid and the premises are true, then the conclusion is true. Deductive logic • Categorical propositions – All S is P. – No S is P. – Some S is P. – Some S is not P. • Quantity: Some or All • Quality: Positive or Negative (no, not) Deductive logic • Categorical propositions – All dogs are mammals. – No dogs are fish. – Some mammals are carnivores. – Some mammals are not carnivores. • The truth of a proposition is determined by its “fit” with the world. – “Some mammals are carnivores” is true if and only if there are some mammals that eat meat. Deductive logic • Categorical propositions – “Physicians licensed to practice in Japan must pass the National Medical Licensing Board Exam.” – All licensed physicians in Japan are people who passed the Licensing Board Exam. – All S is P. Exercise • Create one or more categorical statements of the following types. Compare your statements with your group members’. – All S is P. – No S is P. (= All S is not-P.) – Some S is P. – Some S is not P. Syllogisms • Syllogism: A conclusion inferred from two premises All Cretans are liars. All liars are dishonest. ∴ All Cretans are dishonest. Syllogisms All Cretans are liars. -
Introduction to Logic and Critical Thinking
Introduction to Logic and Critical Thinking Version 1.4 Matthew J. Van Cleave Lansing Community College Introduction to Logic and Critical Thinking by Matthew J. Van Cleave is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Table of contents Preface Chapter 1: Reconstructing and analyzing arguments 1.1 What is an argument? 1.2 Identifying arguments 1.3 Arguments vs. explanations 1.4 More complex argument structures 1.5 Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form 1.6 Validity 1.7 Soundness 1.8 Deductive vs. inductive arguments 1.9 Arguments with missing premises 1.10 Assuring, guarding, and discounting 1.11 Evaluative language 1.12 Evaluating a real-life argument Chapter 2: Formal methods of evaluating arguments 2.1 What is a formal method of evaluation and why do we need them? 2.2 Propositional logic and the four basic truth functional connectives 2.3 Negation and disjunction 2.4 Using parentheses to translate complex sentences 2.5 “Not both” and “neither nor” 2.6 The truth table test of validity 2.7 Conditionals 2.8 “Unless” 2.9 Material equivalence 2.10 Tautologies, contradictions, and contingent statements 2.11 Proofs and the 8 valid forms of inference 2.12 How to construct proofs 2.13 Short review of propositional logic 2.14 Categorical logic 2.15 The Venn test of validity for immediate categorical inferences 2.16 Universal statements and existential commitment 2.17 Venn validity for categorical syllogisms Chapter 3: Evaluating inductive arguments and probabilistic and statistical fallacies 3.1 Inductive arguments and statistical generalizations 3.2 Inference to the best explanation and the seven explanatory virtues 3.3 Analogical arguments 3.4 Causal arguments 3.5 Probability 3.6 The conjunction fallacy 3.7 The base rate fallacy 3.8 The small numbers fallacy 3.9 Regression to the mean fallacy 3.10 Gambler’s fallacy Chapter 4: Informal fallacies 4.1 Formal vs. -
Traditional Logic and Computational Thinking
philosophies Article Traditional Logic and Computational Thinking J.-Martín Castro-Manzano Faculty of Philosophy, UPAEP University, Puebla 72410, Mexico; [email protected] Abstract: In this contribution, we try to show that traditional Aristotelian logic can be useful (in a non-trivial way) for computational thinking. To achieve this objective, we argue in favor of two statements: (i) that traditional logic is not classical and (ii) that logic programming emanating from traditional logic is not classical logic programming. Keywords: syllogistic; aristotelian logic; logic programming 1. Introduction Computational thinking, like critical thinking [1], is a sort of general-purpose thinking that includes a set of logical skills. Hence logic, as a scientific enterprise, is an integral part of it. However, unlike critical thinking, when one checks what “logic” means in the computational context, one cannot help but be surprised to find out that it refers primarily, and sometimes even exclusively, to classical logic (cf. [2–7]). Classical logic is the logic of Boolean operators, the logic of digital circuits, the logic of Karnaugh maps, the logic of Prolog. It is the logic that results from discarding the traditional, Aristotelian logic in order to favor the Fregean–Tarskian paradigm. Classical logic is the logic that defines the standards we typically assume when we teach, research, or apply logic: it is the received view of logic. This state of affairs, of course, has an explanation: classical logic works wonders! However, this is not, by any means, enough warrant to justify the absence or the abandon of Citation: Castro-Manzano, J.-M. traditional logic within the computational thinking literature.