# Preface

These notes have been a long time in the making. From 1983 to early in 1988 I taught a one-semester introductory logic course at the University of Edinburgh, basing the core of the course on natural deduction and using as a textbook the classic Beginning Logic Lemmon by John Lemmon. At the same time, I taught a more advanced introduction to mathematical logic, which included not only traditional core metatheory but also material on non-classical paradigms. The typescript handouts for those courses grew over time into a sheaf of notes intended to supplement the textbooks. During that period, I also wrote a proof-editing program called Lemmon-Aid which was used at Edinburgh from about 1985 and was prepared for release, running on personal computers of the day, in 1989. Part of the user manual for that software was an introduction to logic itself, constructed from my old logic notes. Lemmon-Aid is long gone, but the notes live on. In a later millennium, I found myself again with the opportunity to teach the basics of logic to students at the Australian National University, and did so from 2011 to 2020, using my upgraded notes in place of a textbook. They were moved in 2013 from a print format to an online resource, which has continued to be improved and extended ever since. Support tools in the form of proof and tableau editors were added in 2021, and it now seems time to let the entire artefact go forth into the world, in the hope that people in search of a serious but beginner-oriented introduction to logic will find it and perhaps gain from it.

The Notes are in several respects different from the standard one-semester logic textbook, and a few words are in order concerning the special features of the account they provide.

Why does the first order language have no free variables?
There is no need for free variables. As in Lemmon's book, their place is taken by arbitrary names. Unlike Lemmon, I do not distinguish between arbitrary and proper names, as from the logical point of view this is a distinction without a difference. Managing without free variables somewhat complicates the definition of a formula, but has the compensating advantage that the deduction rules for quantifiers are not hedged about with requirements that "t is free for v in A" (whatever students are supposed to make of that) and the great virtue that every formula makes sense as a sentence on its own, and has a truth value on interpretation.

Why are quantifiers presented as binary operators, taking two predicates to a sentence, whereas virtually all logic textbooks follow Frege in treating them as unary?
I make no apology for that: quantifiers are binary operators, and to pretend otherwise is to opt for an account on which the only real quantifiers are numerical ones, and "Most people like ice cream" is an anomaly or a puzzle rather than a standard case on the same level as "Some people like ice cream". Be it noted firstly that the unary quantifiers are in any case definable in terms of the binary ones (though not vice versa) and secondly that quantifiers in practical use, in knowledge representation or engineering for example, are nearly always binary. Students should be taught the truth on this point.

Why are such advanced topics, notably in Chapter 7 on metalogic and Chapter 8 on non-classical logics, included?
This material would indeed normally thought to lie outside the scope of an introduction—these notes originated from more than one course, after all. As might be expected, I disagree with the thought that there is a clear boundary to what counts as "introductory", and can report that the topics to be found here have been included very successfully in first logic courses over many years. Students need to see that the orthodox theory of logic is one theory among many, and can be challenged. The non-classical systems presented briefly in Chapter 8 are tough material, unlike the more usual soft options of 3-valued logic or the like, because they represent deep and sophisticated alternative paradigms, not merely relaxations of a few dogmas.

Why introduce the basic connectives and quantifiers of standard logic via natural deduction, when semantic tableaux, for instance, are so much easier for the average student to master?
The simplicity and convenience of tableaux is superficial at best. While they are easy to learn as a nice way to reason with truth tables, they do not look so good when quantifiers and function symbols enter the scene, are virtually unusable with identity (another essential of elementary logic), cannot easily accommodate non-classical reasoning, and do not readily connect with uses of logic as the basis for axiomatic theories, for example. In any case, if students have learnt the basic concepts using natural deduction, they can add tableaux to the toolkit in a week, whereas if they have grown up on tableaux, there is no chance of adding natural deduction in a week.

In a similar vein, why is there no grounding in traditional syllogistic logic, which is after all historically important and easy to grasp?
In my view, syllogistic theory does not belong anywhere near a real introduction to logic. Not only is it now of merely historical interest, but, to echo Russell, it resembles the British royal family in that it has been allowed to persist so long in the erroneous belief that it does no harm. It is restricted to a tiny fragment of a first order language, and even in that fragment it does not address the question of which are the valid arguments, but at best of which are the valid syllogisms. This is the wrong question, independently of the suspicion that the theory may provide the wrong answer. Logic has moved beyond this.

OK, but if substructural logics are included, why not modal logic?
In designing the content, I was much more ambivalent on this point. Modal logic has always been an interest of mine, and many students find it quite approachable. Moreover, it connects with important issues in fields such as temporal reasoning and non-monotonic reasoning as well as in reasoning with propositional attitudes. However, it is essentially an extension of the basic theory rather than an alternative to it, so in the interests of drawing the boundary somewhere, I decided that it belongs squarely in a possible extension of the notes. If these pages constitute Volume 1, I reserve the right to compose Volume 2.

The online format makes possible certain features not found in conventional books. Most obvious are the interactive tools for constructing proofs and tableaux, and the multimedia items such as animations, with or without voice-over. Also important is the fact that the text does not have to be seen as linear. There are hyperlinks cross-referencing text segments, and the reader is at liberty to decide whether to follow links to more mathematical pages containing metatheorems and the like. There is a glossary containing short notes on a hundred or so technical expressions. This is highly cross-referenced, so it is possible to study by surfing the glossary, reading material from the main pages only where this seems to be useful.

It remains the case that there is no universally agreed notation for logical connectives and quantifiers. An advantage of the online format is that notation is not fixed. Is material implication written as '→', '⇒', '⊃' or something else? The user decides. It is possible to select symbols for connectives, set scope conventions and more via the preferences menu. The selected notation is applied to all pages of the site, including the interactive tools. This flexibility is my attempt to please everyone regarding such matters as notation. At any rate, it seems better than imposing my choices in the matter.