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\title {Teaching Logic After G\"odel,\\ Tarski, Turing,\\ DeBruijn,
Gates, Wolfram,\\ Fredkin, Chomsky, Russell,\\Perot, Limbaugh, Koppel,\\
Dodgson, Smullyan, Conway, Zwicker,\\ and Marx}


\author{Jim Henle and Tom Tymoczko} 

\maketitle

\section{Disconnected, Technical, Difficult}

For over two thousand years, logic was a highly technical, specialized
discipline.  It had few connections to other pursuits.  It had few
genuine ideas.  It was applied only to the simplest arguments.  Beyond a
crude syntax, the subject consisted mostly of computational
techniques.   
 
Despite its simplicity, logic was considered extremely difficult.  The
study of logic was the mark of the educated person, but logic was studied
in isolation.  Mastering logic was an intellectual exercise.   

\section{Connected, Contextual, Difficult}

Today, the situation has changed.  The subject is actively employed
across a wide spectrum of fields: computer science, linguistics,
biology, law, engineering, political science, quantum physics, and of
course, philosophy and mathematics.  It is brimming with ideas, ideas
that plumb the depths of knowledge and truth, ideas that have created
and destroyed countless paradigms.  The very idea of ``logic'' has
undergone inconceivable expansion.  A comprehensive knowledge of logic
signifies 
something far greater today than even ten years ago.    

But logic is still considered difficult.  It remains a staple of the
proper liberal arts education (usually satisfying unpopular distribution
requirements).  It is still regarded as an intellectual exercise; it is still
``good for you.'' 

\section{Disconnected, Technical, Difficult}

Because of the intricacy and power of predicate logic, undergraduate courses
tend to focus on its syntax and grammar.  The curriculum is fundamentally
algebraic.  Connections to other fields are periferal.  Beyond a few
variations, only one logic is taught.  The course is
considered one of the most difficult in the curriculum even (or especially) by the very
students (philosophy majors) for whom it is required.  

\section{Connected, Contextual, Easy}

Despite the great expansion of material, despite the new depth and
significance of logic, logic does not have to be difficult.  The key is
to focus not on technical details, but on its expressive power: its
ability to describe the universe.

The same advances in logic that make it such a powerful tool and such a
profound subject paradoxically 
make it easier to comprehend. 

\subsection{G\"odel, Tarski, Turing}

The contributions of twentieth-century logicians show us the limitations
of formal methods.  At the same time that their discoveries enrich
logic, they free us from dependence on syntactic methods.  

\subsection{De Bruijn, Gates, Wolfram}

The contributions of twentieth-century developers and programmers 
also free us from technical methods.  In the same manner that the
pocket calculator has liberated students of mathematics, so too new
packages allow logic students to concentrate on the meaning and
significance of the subject.

\subsection{Fredkin, Chomsky, Russell}

The contributions of thinkers in a vast range of subjects have spread
the influence of logic throughout the intellectual landscape.  They have
made it increasingly easy to impress students with the importance of
logic and to explicate its nature.

\subsection{Reagan, Perot, Limbaugh, Koppel}

The contributions of twentieth-century political debate  highlight
the need for logically aware citizens.  Further, the prominence of the
law and legal issues make it easier to relate logic to the world.

\subsection{Dodgson, Smullyan, Conway, Zwicker and Marx}

We have become a culture that plays with logic.  A subject that is
inherently beautiful has become suffused with fun as well.  Nothing
is more useful in bringing logic to students.

\section{Logic at Smith College}

We have organized a logic program at Smith to take advantage of these
contributions.  Formally, it goes as far any first undergraduate course:
deduction in predicate logic.  At the same time, it embraces logical
ideas in linguistics, computer science, law, philosophy, politics, and
comic opera.  Significant time is devoted to natural language debating.
Significant time is devoted to questions that are demonstrably
unanswerable.  Significant time is devoted to fun.  The course is not
difficult. 

\subsection{Language}

The heart of logic is language.  While the languages are, in some sense, artificial, they were
constructed to reflect natural languages.  They have also grown like
natural languages, spawning slang, jargon, and dialects.  

We teach logic as a language.  We immerse students in it.  Rather than
doling it out in bits, we give it to them all at once, all of predicate
logic, in fact.  We ask that they absorb it and not worry about the grammar at
first.  We ask that they  read it,  write it, and speak it.  Only when students are acclimated, only when they have begun to
use the language (a week is sufficient) do we go back and offer careful
definitions.  

Ironically,  logical
languages are most difficult to learn when they are presented in logical
progression.  
Babies would never learn to talk if we insisted
that they conjugate verbs before speaking sentences.  

The theme of language appears over and over in the course.  We feel that
this approach has made logic considerably easier to learn.  At bottom,
the concepts of logic are not difficult.  They become difficult when
students don't appreciate that a familiar word has an unfamiliar
meaning.  Rather than discovering the discordance, students become
convinced they cannot think ``logically.''

Appendix A
contains examples of this approach.

\subsection{Application}

The languages of logic are hard at work today in numerous fields.  Every
time we teach the course, we try to incorporate a new application.  The
different contexts help students to understand the meaning of logic.  It is
helpful, for example, to see implication  used in computer
programming, to see it used in linguistic analysis, in the tax code, and so on.  

We teach logic as universal, ubiquitous.  We show logic in political
debate, in advertising, and in theological disputes.   We feel that this
encylopaedic approach makes logic easier to learn.  Ideas are
difficult when they are taught in isolation.  Psychologists have
discovered that humans are  more adept at memorizing words than random
letters and that they memorize sentences better than random words.  For
the same reason, students comprehend and retain logic when they 
use it meaningfully rather than simply master its multiple rules of
grammar and inference.

Appendix B contains examples of this approach.

\section{Enjoyment}

We teach logic as an intellectual treat.  Logic is beautiful.  It is
charming, beguiling, tantalizing.  We play, but play is as serious as it
is enjoyable.  One can argue that the paradox of Achilles and the
Tortoise is
philosophy at its most profound, and at the
same time, humor at its most cosmic.

It would be a mistake to present logic as a sequence of puzzles, no
matter how diverting.  It would also be a mistake to use amusements to
sugar-coat exercises.  We hope we do neither.  Our goal is to lead
students to appreciate the harmony of logic, its wit and its poetry.  We
expect students to use logic creatively themselves, and to see logic as
a triumph of human intelligence.

Appendix C contains examples of this approach.


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