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LOGIC IN ACTION

an open course in logic developed by

Johan van Benthem, Hans van Ditmarsch, Jan van Eijck, Jan Jaspars

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This is the homepage of the Logic in Action Open Course Project. This project
was developed to provide a modern introduction to the field of logic with topics
reflecting both its mathematical essentials and a broad view of its
interdisciplinary role.

The material was written for elementary and intermediate courses in logic, to
assist teachers and also intrepid readers engaging in self-study. All material
is freely available.

For an overview of the ideas behind the course as well as some experiences of
its developers and teachers, see the paper Logic in Action: An Old Discipline
With a New Twist.

Below you will find the individual chapters of our self-contained introduction
to logic.

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Academic courses that have used and tested this material were given at:

 * Amsterdam University College, NL

 * Tsinghua University in Beijing, CN

 * Stanford University, USA

 * University of Sevilla, SP

The course is also part of the Stanford Online Highschool Curriculum.

The book is used as a course text in the Discrete Structures course at the
Department of Computer Science, National University of Ireland, Maynooth, Co.
Kildare.

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A pdf version of the textbook can be downloaded from here.

Links to individual chapters of the manuscript are given below.

There are also slides for teaching available, in English and in Spanish (made by
Fernando Velazquez Quesada).

Further exercise material for the chapters (courtesy of Dora Achourioti, AUC)
can be found here.

Online exercises in modal logic (an important topic in the course) can be found
here.

The development of this site is co-funded by the Center for Creation, Content
and Technology, which is part of the Netherlands Network for Humanities, Social
Sciences and Technology, with a subsidy from Platform Beta Techniek.

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WHAT IS LOGIC?

Gottfried Wilhelm Leibniz

In science, logic means "the study of valid reasoning", and for centuries it has
been a philosophical discipline. Nowadays, in the information era, it is part of
the scientific background assumed by researchers from a wide diversity of
scientific backgrounds such as mathematics, computer science, linguistics,
cognitive science and economics. In this first chapter a variety of examples of
valid and invalid reasoning are presented and informally discussed. The material
is presented against a historical canvas of the development of logic, both in
the western and in the eastern world.

Download the Chapter.


CLASSICAL SYSTEMS


PROPOSITIONAL LOGIC



Propositional logic is a purely sentential system originating with the Stoic
philosophers. The first completely formal version of propositional logic was
presented around 1850 by George Boole (picture), and published in his famous
'The Laws of Thought'. Although the language of propositional logic is simple,
the logic itself is very expressive and powerful. This is witnessed by the fact
that all digital computation can be modeled as propositional reasoning
(although, admittedly, this is a rather low-level representation).

Download the Chapter.


SYLLOGISTIC REASONING

Aristotle

In this chapter we go back to antiquity. Syllogistics is a logic of quantified
expressions which was introduced by Aristotle (400BC). Syllogistics has
dominated the study of logic in the western world until the 19th century. Since
the second half of the 19th century syllogistics is mathematically
well-understood. We introduce and explain a system which makes use of so-called
Venn-diagrams (after the British mathematician John Venn). In addition to this,
we discuss how syllogistics can be modeled as propositional reasoning.
Syllogistics corresponds to a natural fragment of propositional logic for which
validity can be computed at low computational cost. Whether an efficient
validity testing method for full propositional logic exists is still unknown,
although it is widely believed that this is not the case.

Download the Chapter.


THE WORLD ACCORDING TO PREDICATE LOGIC

Gottlob Frege

In the system of predicate logic, quantification is combined with the sentential
operations of propositional logic. Unlike syllogistics, predicate logic allows
multiple quantification, combined with the expression of relations between
objects. The German philosopher Gottlob Frege proposed this formalism in 1878,
as a Begriffsschrift (concept language) in which, so he hoped, all mathematical
reasoning could be represented in a precise formal manner. Nowadays, predicate
logic is the most common logical system, for representing reasoning in
mathematics and in many other fields.

Download the Chapter.


KNOWLEDGE, ACTION AND INTERACTION


KNOWLEDGE AND INFORMATION FLOW

Jaakko Hintikka

This chapter introduces the logic of knowledge, otherwise called epistemic
logic. The essential difference with the classical systems of part 1 is that in
epistemic logic the reasoner(s), or agent(s), are incorporated within the
system. Epistemic logic allows one to formalize the reasoning of an agent about
his own knowledge and ignorance, as well as about the knowledge and ignorance of
other agents. Epistemic logic goes back to the work of the Finnish philosopher
Jaakko Hintikka (picture) in the early 1960s. In this chapter we also discuss
recent developments, culmi nating in systems that are able to model various acts
of communication between agents and their effects.

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LOGIC AND ACTION

Vaughan Pratt

This chapter introduces dynamic logic, a system that was introduced in computer
science for reasoning about the behaviour of computer programs, by Vaughan Pratt
and others, in the 1970s. Later on, it turned out that dynamic logic has other
applications as well. Dynamic logic can be viewed as a general logic of actions
and their effects. It allows us to not only reason about what agents know and
say, but also about what they do, and what the effects of these actions are.

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LOGIC, GAMES AND INTERACTION

Ernst Zermelo

Games have always played an important role in the history of logic. Many sorts
of interactive games, of two opposing players, have been used as tools for
demonstrating the validity or invalidity of inferences, or for determine the
truth or falsity of utterances (in so-called evaluation games). We discuss
several examples of applications of games to the classical systems of part 1, as
well as to the systems in this part. Next, we focus on game theory, as studied
in economics and the social sciences as formal models of human social behaviour,
and explain how this relates to logic games.

Download the Chapter.


METHODS


VALIDITY TESTING

Evert Willem Beth

This chapter explains the so-called tableau-system of Evert Willem Beth (1955,
picture). Tableaus were meant as a general method to decide validity of
inferences, in propositional logic and predicate logic. We show such systems
work, for propositional logic, for predicate logic, and for single agent
epistemic logic. The tableau method is a systematic search and construction
method for counter-models. If such a counter-model can be constructed, this
demonstrates the invalidity of an inference. Validity of an inference can be
demonstrated by showing that counter-models do not exist. Tableau construction
is a complete method for the logics which are discussed in this chapter: for
every valid inference in propositonal logic, predicate logic and single agent
epistemic logic, the tableau method can demonstrate that no counter-models
exist.

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PROOFS

Gerhard Gentzen

The tableau method for deciding the validity of logical arguments is a useful
analysis tool, but it is not quite faithful to the way humans reason in
practice. In this chapter we introduce a system of reasoning called natural
deduction, with rules that are meant to reflect the formal structure of
mathematical proofs as you can find them in textbooks. These rules are also
quite close to the way sound reasoning proceeds in daily life. We discuss
systems for propositional logic, predicate logic and a well known system of
arithmetic as an extension of predicate logic. The systems discussed in this
chapter go back to the work of the influential German logician and mathematician
Gerhard Gentzen (picture) in the 1930s, and were streamlined later by the
Swedish logician Dag Prawitz (1960s).

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COMPUTATION

Alain Colmerauer

In this chapter it is explained how predicate logic can be used for programming.
The chapter gives an introduction to concepts used in automating logical theorem
proving. This leads to computational tools underlying the well known logic
programming language Prolog, popular in AI an in computational linguistics. The
inferential mechanism behind Prolog is called SLD-resolution. This is the
reasoning engine for a natural fragment of predicate logic (the so-called
Horn-fragment).

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APPENDIX


SETS, RELATIONS AND FUNCTIONS

Georg Cantor

This chapter explains the basics of formal set notation, and gives an
introduction to relations and functions. The chapter ends with a short account
of the principle of proof by mathematical induction.

Download the Chapter.