# 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.