LOGIC: FROM TRUTH TO PROOFS AND GAMES
PROFESSOR MATHIEU MARION
Université du Québec à Montréal
marion.mathieu@uqam.ca
What is logic? The usual answer would assume that logic is of the form of an axiomatic
system, classical first-order predicate logic (with identity), to which one adds a semantics in
terms of truth conditions. This is how logic is usually taught in undergraduate courses, along
with techniques for testing validity, such as truth-table, tableaux and natural deduction, but
also the picture underlying much work in contemporary analytic philosophy. However, major
developments in logic since the 1960s have undermined this underlying picture,
developments that have been partly ignored by philosophers under the pretext that they are
only of ‘technical’, but not of ‘philosophical’ interest. The few philosopher that have sought
to revise this underlying picture of logic in light of these developments, perhaps most
famously Michael Dummett, have found themselves largely misunderstood. In this course, we
will examine some of these developments and the ways in which they force us to revise our
conception of logic.
After revising in a critical spirit the classical view of truth, logical consequence and logical
constants put forth mainly by Tarski and Quine, we will ask two related questions,
surrounding the thesis of logical ‘monism’, i.e., the idea that there is only one logic, i.e.,
classical first-order predicate logic: can logic be revised? Is there only one logic? With one
exception, however, we will not examine the arguments put forth in particular attempts at
devising alternative logics, e.g., intuitionistic or relevant logics, although these also implicate
different conceptions of the nature of logic. Doing this would require teaching an altogether
different course, but pointers will be given at the appropriate moment to the relevant issues.
Next, we will examine two new ‘use-based’ approaches to logic. As opposed to the ‘modeltheoretic’ or ‘denotational’ approach examined so far, these do not rely on axiomatic systems
or on truth-conditional semantics. There are also attempts at fleshing out a view of ‘meaning
as use’. The first of these two approaches, ‘proof-theoretical semantics’, comes from
Gentzen’s natural deduction systems (and related sequent calculi), and forms the basis of
Dummett’s and Dag Prawitz’s philosophy of logic, as well as their attempt to argue for the
superiority of intuitionistic logic over classical logic. We will thus examine its implications,
with the move from axioms to rules of inference, from truth to assertabiliy conditions, from
logical consequence to logical inference, with the resulting criticisms of Quine’s conception
of the place of logic in the web of belief, and the argument for intuitionistic logic. At this
stage, we will re-examine afresh, with new concepts and arguments taken from the point of
view of ‘substructural logics’, opened up by work on Gentzen’s sequent calculi, the question
of ‘monism’, looking at recent arguments in favour of ‘logical pluralism’.
Finally, we will examine a second ‘use-based’ approach: ‘game semantics’. It is often
portrayed as having two sources, Hintikka’s ‘game-theoretical semantics’ and Lorenzen’s
‘dialogical logic’. We will first focus on the ‘game semantics’ that evolved from Lorenzen,
explaining the basics ideas, and examining the possibility of integrating it to a pluralist
platform derived from the above ‘substructural’ standpoint. We will then present Hintikka’s
model-theoretic approach and examine some of the conceptual difficulties it faces. Finally,
we will have a look at the Curry-Howard isomorphism and the ‘propositions as types’
paradigm that it generated, that has initiated majors changes in the way logicians themselves
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see logic, changes that are, alas, largely ignored by philosophers. Of this sizeable literature,
we will look merely at philosophical papers by the logician Ruy de Queiroz, who discusses
the relation proof-theoretic semantics and game semantics from that point of view.
The reading selected are for the most part ‘classics’ that ought to be read, by anyone wishing
to speak knowingly about logic, but there will be also some papers which reflect recent work
in the field, so that students will be introduced to cutting-edge work in the field. Some of the
papers are a bit technical and admittedly difficult to read. Basic knowledge of formal logic is
presupposed, but the aim of the course is also to provide students with a pedagogical
introduction to the concepts and results they involve.
COURSE PLAN
The following provides a list of the main papers to be discussed in each class, therefore it
would be better if students read them beforehand.
PDF versions will be made available on a CD-ROM.
PART I : DENOTATIONAL APPROACH TO SEMANTICS
1. TRUTH AND LOGICAL CONSEQUENCE
Alfred Tarski, ‘The Semantic Conception of Truth and the Foundations of Semantics’, Philosophy
and Phenomenological Research, vol. 4, 1944, 341-376.
Alfred Tarski, ‘On the concept of Logical Consequence’, in Logic, Semantics, Metamathematics,
sec. ed., Indianapolis IN, Hackett, 1983, 409-420.
John Etchemendy, ‘Tarski on Truth and Logical Consequence’, Journal of Symbolic Logic, vol. 53,
1988, 51-79.
2. WHAT ARE LOGICAL CONSTANTS?
Alfred Tarski, ‘What are Logical Notions?’, History and Philosophy of Logic, vol. 7, 1986, 143–154.
W. v. Quine, ‘Grammar, Truth and Logic’, in S. Kanger & S. Öhman (eds.), Philosophy and
Grammar, Dordrecht, D. Reidel, 1980, 17-28.
John Etchemendy, ‘The Doctrine of Logic As Form’, Linguistic and Philosophy, vol. 6, 1983, 319334.
3. THE STATUS OF LOGIC.
W. v. Quine, ‘Two Dogmas of Empiricism’, Philosophical Review, vol. 60, 1951, 20-43.
W. v. Quine, ‘Introduction’, in Methods of Logic, London, Routledge & Kegan Paul, 1956, xi-xvii.
Lewis Carroll, ‘What the Tortoise Said to Achilles’, Mind, n.s., vol. 4, 1895, 278-280.
Stewart Shapiro, ‘The Status of Logic’, in P. Boghossian & C. Peacocke (eds.), New Essays on the A
Priori, Oxford, Clarendon Press, 2000, 333-366.
4. IS THERE ONLY ONE LOGIC ?
W. v. Quine, Philosophy of Logic, sec. ed., Cambridge MA, 1980, chapter 6.
Susan Haack, Deviant Logic, Fuzzy Logic. Beyond the Formalism, Chicago, University of Chicago
Press, 1996, chapters 1 & 2.
Michael Resnick, ‘Ought There to be but One Logic?’, in B. J. Copeland (ed.), Logic and Reality.
Essays on the Legacy of Arthur Prior, Oxford, Clarendon Press, 1996, 489-517.
PART II : USE-BASED APPROACHES TO SEMANTICS
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5. FROM ‘TONK’ TO THE INTUITIONIST CHALLENGE
Arthur N. Prior, ‘The Runabout Inference-Ticket’, Analysis, vol. 21, 1960, 38-39.
Nuel Belnap, ‘Tonk, Plonk and Plink’, Analysis, vol. 22, 1962, 130-134.
Ian Hacking, ‘What is Logic?’, Journal of Philosophy, vol. 76, 1979, 285-319.
Dag Prawitz, ‘Meaning and Proofs: On the Conflict Between Classical and Intuitionistic Logic’,
Theoria, vol. 43, 1977, 1-40.
6. SUBSTRUCTURAL LOGIC AND LOGICAL PLURALISM
Kosta Dosen, ‘Logical Constants as Punctuation Marks’, Notre Dame Journal of Formal Logic, vol.
30, 1989, 362-381.
J. C. Beall & Greg Restall, ‘Logical Pluralism’, Australasian Journal of Philosophy, vol. 78, 2000,
475-493.
Greg Restall, ‘Carnap’s Tolerance, Meaning, and Logical Pluralism’, Journal of Philosophy, vol. 99,
2002, 426-443.
7. DIALOGICAL LOGIC
Laurent Keiff, ‘Dialogical Logic’, Stanford Encyclopedia of Philosophy. Available at:
http://plato.stanford.edu/entries/logic-dialogical/
Helge Rückert, ‘Why Dialogical Logic? ‘, in H. Wansing (ed.), Essays on Non-Classical Logic,
Singapore, World Scientific, 165-185.
8. DIALOGICAL LOGIC AND LOGICAL PLURALISM
Shahid Rahman & Laurent Keiff, ‘How to be a Dialogician’, in D. Vanderveken (ed.), Logic,
Thought and Action, Dordrecht, Springer, 2005, 359-408.
Patrick Blackburn, « Modal Logic and Dialogical Logic », Synthese, vol. 127, 2001, 57-93.
9. GAME-THEORETICAL SEMANTICS
Jaakko Hintikka & Gabriel Sandu, ‘Game-Theoretical Semantics’, in J. van Benthem & A. ter
Meulen (eds.), Handbook of Logic and Language, Amsterdam, Elsevier, 1997, 361-410.
Jaakko Hintikka,, ‘Language Understanding and Strategic Meaning’, Synthese, vol. 73, 1987, 497529.
Neil Tennant, ‘Language Games and Intuitionism’, Synthese, vol. 42, 1979, 297-314.
10. THE CURRY-HOWARD ISOMORPHISM
W. A. Howard, ‘The Formulae-As-Types Notion of Construction’, in J. P. Seldin & J. R. Hindley
(eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism,
London, Academic Press, 1980, 480-490.
Ruy de Queiroz, ‘Normalisation and Language-Games’, Dialectica, vol. 48, 1994, 83-123.
Ruy de Queiroz, ‘On Reduction Rules, Meaning-as-Use and Proof-Theoretic Semantics’, Studia
Logica, vol. 90, 2008, 211-247.