What Is Logic? - Sluckelman.webspace.durham.ac.uk

19 Intuitionistic and constructive logics [last19.1 Introduction to constructivity . . . . . . .19.2 Propositional intuitionistic logic . . . . . .19.2.1 Language and semantics . . . . . .19.2.2 Connections to other logics . . . .19.2.3 Proof theory . . . . . . . . . . . .19.3 Other constructive logics . . . . . . . . . .20 Paraconsistent logics [last20.1 A return to conditionals20.1.1 Connexive logics20.1.2 Relevance logicsmodified. . . . . . . . . . . . . . . .13. . . .modified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .01. . . . . . .Mar 21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .291291291291292294294Sep 18]295. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29621 Many-valued and fuzzy logics [last modified 4 Oct 18]29721.1 4-valued propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29722 Probabilistic logics [last modified 17 Jun 19]29923 Inductive and abductive logics [last modified 13 Sep 18]301V303Back matterA Exercises [last modified 10 JanA.1 Exercises for Chapter 1 . . .A.2 Exercises for Chapter 3 . . .A.3 Exercises for Chapter 4 . . .A.4 Exercises for Chapter 6 . . .A.5 Exercises for Chapter 7 . . .A.6 Exercises for Chapter 8 . . .A.7 Exercises for Chapter 10 . . .A.8 Exercises for Chapter 11 . . .A.9 Exercises for Chapter 12 . . .A.10 Exercises for Chapter 13 . . .22]. . . . . . . . . . . . . . . . . . . . .B Answers to Selected ExercisesC Errata inC.1 4.3 .C.2 4.4 .C.3 4.5 .C.4 4.6 .C.5 4.7 .C.6 4.8 .C.7 4.9 .C.8 4.10[Goldstern and Judah,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3053053053083083133213253263283283311998]. . . . . . . . . . . . . . . . . . . . . . . . .335335335335336336336336336D Glossary337Index [last modified 20 Jan 22]337Bibliography [last modified 11 Feb 21]339v

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Chapter 1What is logic? [last modified 11 Oct21]There are a plethora of introduction-to-logic textbooks out there: So why have I written another one? Itis for exactly the same reason that there are so many of them already out there. The teaching of logic isa very personal matter, in that every person who does so has individual views on what are the importantaspects to highlight, what every student taking a first-year logic course should come away knowing, whatevery student taking an upper-level logic course should come away knowing, what the scope of logic is,whether classical logic should be preferred to non-classical logic, whether there is One True Logic ormany, in addition to more mundane concerns such as preferences about notation, often inherited fromthe teacher’s own days as a student. A consequence of this is that every time I have taught logic, I havebeen dissatisfied with the available textbooks: None of them ever covered quite what I wanted them to,in the order and way I wanted them to, and with the notation I wanted. So rather than continuing totry to find the perfect textbook, I decided to write my own.But I decided to do more than write yet another introduction-to-logic book. Instead, I wanted a bookthat I could use for both the intro course and the upper level course(s) at the same time. I wanted myfirst-year students to be given peeks into what could be built upon the foundations they were learning. Iwanted my upper-level students to have all the basics available to them, in the same book, for referencewhen they were suddenly unsure whether they could, right at this very moment, perform universalgeneralization. Thus I do not expect to ever cover the entire book in one course, but hope that it canserve as a single book for a multi-year programme in logic.I also wanted to write a book that could serve as something of a translation guide between othertextbooks. To that end, I have included extensive discussions of alternative notations, and a Glossarywhich serves as both a partial index to this book but also to other books. I hope that this will preventme from rifling through my shelves in the middle of seminar to find a definition of Sahlqvist formulasbecause they have suddenly, tangentially, become relevant, and I do not have their form memorized offthe top of my head.The first question we must address is also the question that governs the book as a whole: What islogic? What is it that logicians do when they are “doing logic”? What is a logic? What are logics? Isthere more than one logic? Or is there only one?There are as many different answers to these questions as there are logicians (in fact, there areprobably more answers than there are logicians, for some logicians are pluralists and accept more thanone answer!), and unsurprisingly we will not provide a quick and easy answer here—after all, part of thepoint of this whole book is to address these questions.Nevertheless, we have to start somewhere. That starting point will be the words of Roger Bacon, amedieval philosopher and theologian who studied and taught at Oxford and Paris, and later became aFranciscan monk. Writing around 1250, Bacon says in his Art and Science of Logic (an introductorytextbook for undergraduates studying logic) thatlogic, as a science, is the habit of distinguishing what is true from what is false by means ofrules or maxims or dignities by which we can comprehend the truth of a locution throughour own efforts or with the help of others. And logic is so-called from ‘logos’, which means1

discourse, and ‘lexis’, which means reason or understanding — as it were, the science eitherof reason joined to discourse or of discourse joined to reason [Bacon, 2009, ¶3].This definition highlights four important features of (the study of) logic:1. It is aimed at distinguishing truth from falsehood.2. It is rule-governed.3. It can be a joint venture.4. It involves discourse.Contemporary logicians would probably stop Bacon after the first two, putting the emphasis on thetruth-preserving, rule-governed nature of logic, and setting aside the discursive, cooperative aspects.One way in which this book differs from your standard 21st-century introduction to logic is that we willbe emphasizing the importance of the discursive nature of logic: The purpose to which arguments areput has a bearing on the choice of appropriate logic.1.1ArgumentsKeeping the preceding in mind, let us take as starting point the following definition of logic, which wewill revise and precisify as we proceed:Definition 1.1.1. Logic is the study of good arguments.As a definition, this is an exemplary one; it is clear, simple, and elegant. But as an explanation ofour subject matter, it is wanting, for we haven’t yet said (1) what an argument is or (2) what a goodargument is. It is easier to give a general, abstract account of what an argument is than it is to definewhich of these are good ones, and one of the things that we will see throughout the course of this bookis that there is a plurality of ways in which “good” can be cashed out.Let us begin, therefore, by giving a general definition of what we will count as an argument, and indoing so we will formally introduce our first symbolic notation:1Definition 1.1.2. An argument is a pair consisting of a finite (and possibly empty) set Γ of sentences,ϕ1 , ϕ2 , ϕ3 , . . . , called the premises, and a single sentence, ψ, called the conclusion.Sometimes we will say that an argument is a set Γ containing a distinguished sentence, ψ, which isthe conclusion, and the remainder of the sentences are the premises.It should be noted that this is a very liberal account of argument, because it does not require anysort of connection between the premises and the conclusion. On this definition, the vast majority ofarguments are going to be both (a) not recognizable as arguments on the ordinary usage of the term and(b) really, really bad.This gives us a clue as to what makes an argument good : There must be some connection betweenthe premises and the conclusion. In fact, all of the different ways of explicating “goodness” that we willexplore in this book are rooted in this idea of a connection between the premises and the conclusion. Theexistence of such a connection is tied to the guaranteed or likely transmission of some special propertyfrom the premises of the argument to its conclusion; what this property is varies from explication toexplication.A good argument will then be one in which there is a special relationship between the members ofthe set Γ and the conclusion ϕ. Again as a starting point, let us take the following definition as a generalaccount of good arguments:Definition 1.1.3. A good argument is one in which the conclusion is a consequence of the premises.This definition too is only a starting point because it doesn’t specify in what way the conclusion isa consequence of the premises, and it is by varying the type of consequence that we can come up withdifferent accounts of “good argument”. We will be covering a variety of types of consequence in thisbook, including:1 Ifyou are not familiar with the Greek alphabet, you can refresh yourself in §1.3.2

logical2 probabilistic inductive dialogical epistemicBecause of the variety of ways in which ‘good’ is defined, we will give specific definitions for each contextwe investigate in this book.1.2Logic vs. a logic vs. logicsWe can now distinguish logic, as a field of study—the study of good arguments—from a logic, which isa specific system of argumentation designed to make explicit certain features of the arguments of thatsystem. One such feature that is often highlighted is the form or structure of a given argument, as there isoften a close link between the goodness of an argument and the structure or form that it has. Individuallogics will each be accompanied with mechanisms which allow us to make this form or structure explicit,and we will further use these mechanisms to prove general results about the logics (systems of argument)in question. These logics differ in the features that they highlight, and the contexts in which they areapplied, but each logic can be characterized by three things: (1) its language, (2) its proof theory, and(3) its semantics. Thus, for every logic that we consider in this book, we will begin by identifying:1. The language, that is, the set of logical and non-logical symbols and the ways in which they canbe combined into syntactically (i.e., grammatically) correct strings.2. The proof-theory, that is, a system of formal proof, or the ways in which strings of symbols can bemanipulated in that system.3. The semantics, that is, what it is that gives meaning to the logical and non-logical symbols.As soon as we have a language and a proof-theory, we are able to prove results at two levels: At theobject level, we use a system’s proof theory to prove results within the system. At the meta level, wereason in English, outside of the formal system, to prove results about the system, in particular, aboutthe relationship between the proof-theory and the semantics.An important goal of this book is to introduce even the novice logic student to the distinction betweenthe object- and meta-level and the methods of proof which are used in both. Once an English-languageargument has been translated into a particular logical language, we can characterise the goodness of theargument via either proof-theoretic or semantic means (cf. Table 1.1). In the best case scenario, we willgive both types of characterizations, and prove that these two characterizations coincide, i.e., that everyargument which is “good” according to the proof-theoretic characterization is also “good” according tothe semantic characterization (when this is the case, we say that the proof theory is “sound”), and thatevery argument which is “good” according to the semantic characterization is also “good” according tothe proof-theoretic characterization (when this is the case, we say that the proof theory is “complete”.3 )1.3NotationIn this section, we introduce and define notation that will be used throughout this book, as well asnotation used in other logical textbooks to facilitate translation between texts.2 This is a poor choice of word, because it has the implication that the other types of consequence are not logical, butthis goes against our view of logic as being the study of good arguments, and these good arguments can involve any of avariety of types of consequences. The reason for using this word here is primarily historical: If one thinks that the onlygood arguments are the ones where the strongest notion of consequence holds between the premises and the conclusion,then logic is the study of that notion of consequence alone, and hence that notion can be called ‘logical’. In this book, wetake a much more pluralistic approach to what counts as ‘good’, and hence what falls under the study of logic.3 The notions of soundness and completeness are introduced in more detail in §7.6.3

entationsyntaxrules of inferenceTable 1.1: The two faces of logic1.3.1The Greek alphabetUpper and lower case Greek letters are often used to stand for arbitrary logical sentences, or sets ofsentences. The alphabet is reproduced in Table 1.2, and the names and forms of the letters should bemastered as soon as hiChiPsiOmegaUppercaseΑΒΓ αβγδ ζηθικλµνξοπρστυϕχψωTable 1.2: The Greek alphabet1.3.2Logical notationNotation is not fixed across logical and philosophical literature. Here we give a brief summary of thedifferent types of notation and the symbols used. Operators that take one argument are called monadicor unary operators. Operators that take two arguments are called dyadic or binary operators. Thestandard logical operators and connectives are all either monadic or dyadic. Negation (§7.3), universaland existential quantification (§8.2), and necessity and possibility (§11.2) are all monadic operators.Conjunction, disjunction, implication, and equivalence (§7.3) are all dyadic operators.InfixThe most common type of notation is infix notation, wherein the binary connectives are inserted betweenthe formulas that they connect. In the list below, the symbol in red is the symbol used in this book.4

Negation: ϕ, ϕ, ϕ̄. Conjunction: ϕ ψ, ϕ&ψ, ϕ ψ, ϕ ψ. Disjunction: ϕ ψ, ϕ ψ. Implication, material: ϕ ψ, ϕ

Durham University s.l.uckelman@durham.ac.uk January 27, 2022. 2. Contents 1 What is logic? [last modi ed 11 Oct 21] 1 . 8 Predicate logic [last modi ed 02 Mar 21] 103 . 12 Modal logic: applications [last modi ed 19 Jan 21] 225