Workshop Practice-based Philosophy of Logic and Mathematics

Workshop Practice-based Philosophy of Logic and Mathematics: Program and Abstracts


















Dutilh Novaes






Moktefi & Schang










van Benthem




Free slot (PhD defense of Sara Uckelman)


Final considerations and discussion










van Bendegem















16.30 Aberdein





17.15 Parikh


Löwe & Müller











Communities of Logical Practices

Andrew Aberdein


What comprises a shared community of logical practice? One answer would be that individuals belong to the same community of practice if they endorse the same system of logic. Systems of logic take many different forms: axiom systems, sequent calculi, natural deduction presentations, and so on. We may follow John Corcoran in sorting these presentations into three basic types: logistic systems, which are classified in terms of their logical truths; consequence systems, which are classified in terms of the arguments they validate; and deductive systems, which are classified in terms of the proofs they admit (Corcoran, 1969, 154). One thesis of this paper is that shared adherence to a logical system in any of these senses (or by extension, to a system specified in even more fine-grained terms) is neither necessary nor sufficient for community of logical practice.


The motivating thought is that two individuals with distinct proof systems who reliably employ the same (or at least closely analogous) proof methods in response to similar problems would exhibit a continuity of logical practice (and would perhaps do so even if they did not validate all the same inferences, or accept all the same logical truths). Conversely, sharing a specific proof system is consistent with divergent choices in the way the system is used, and thereby with membership in different communities of logical practice.


A second thesis is that any adequate classification of communities of logical practice must have regard to all aspects of argumentation, and not just to deductive logic. Firstly, not all logic is formal. Secondly, any study of formal logic which goes beyond a technical exercise must consider the process of application. This process cannot itself be purely formal, on pain of regress. Hence communities of logical practice may share a formal system but diverge sharply over how it is applied to preformal argumentation. As I shall show, such dissensus can arise not only in formalization of natural language argumentation but also in mathematics.


Since two communities may endorse the same inference schemes, but differ substantially

in which they prefer, attention must be paid to the comparative frequency of the schemes within argumentational practice. This approach raises its own methodological issues, and further progress on a positive definition of communities of logical practice requires careful study of a diverse range of examples. Some of those considered include:


• The proof methods in Euclid’s Elements. Euclid endorses superposition when it is unavoidable (I.4), but uses other, often longer, proofs when he can. And, as De Morgan observes, Euclid resists purely logical inferences. For example, he proves propositions I.6 and I.19 geometrically, even though they may be derived from their converses I.5 and I.18 by repeated application of Aristotle’s operation of contraposition (Heath, 2006, p. 130).

•The ‘vogue’ among seventeenth century Jesuits for consequentia mirabilis, of which Saccheri is the most conspicuous exponent (Kneale and Kneale, 1962, p. 347).

• Disputes amongst mathematicians about the appropriate standard of rigour often reveal competing communities of logical practice. For example, that which led to the rejection of Heaviside’s operator methods, before they were understood as anticipating the Laplace transform (Hunt, 1991, p. 91).

• Programmes advocating the adoption of nonclassical logics, whether in mathematics or more widely. (Aberdein and Read, 2009, surveys the usual suspects.).

• The heterodox argumentational behaviour reported by some ethnographers. I shall argue that this may be better explained by a preference for nondeductive schemes than by endorsement of a nonclassical logic (as has been suggested in, for example, da Costa et  al., 1998).

• Argumentational idiosyncrasies characteristic of online environments. Recent studies include that of Weger and Aakhus (2003)



Aberdein, A. and Read, S. (2009). The philosophy of alternative logics. In Haaparanta, L., editor, The Development of Modern Logic, pages 627–738. Oxford University Press, Oxford.

Corcoran, J. (1969). Three logical theories. Philosophy of Science, 36:153–177.

da Costa, N. C. A., Bueno, O., and French, S. (1998). Is there a Zande logic? History and Philosophy of Logic, 19:41–54.

Heath, T. L. (2006). The Thirteen Books of Euclid’s Elements. Barnes & Noble, New York, NY.

Hunt, B. J. (1991). Rigorous discipline: Oliver Heaviside versus the mathematicians. In Dear, P., editor, The Literary Structure of Scientific Argument. University of Pennsylvania Press, Philadelphia, PA.

Kneale, W. and Kneale, M. (1962). The Development of Logic. Clarendon, Oxford.

Lave, J. and Wenger, E. (1991). Situated Learning: Legitimate Peripheral Participation. Cambridge University Press, Cambridge.

Weger, Jr., H. and Aakhus, M. (2003). Arguing in internet chat rooms: Argumentative adaptations to chat room design and some consequences for public deliberation at a distance. Argumentation and Advocacy, 40(1):23–38.




Logic as a tool for building theories

Samson Abramsky


The landscape of logic has been transformed over the past half-century by the role it has come to play in Computer Science. A practice-based philosophy of logic should take account of this practice. We shall discuss some of the ways in which logic is used in the study of information, interaction and computation.

Just as the traditional machinery of continuous mathematics provides the basic language and tools in which physical theories can be formulated (although more abstract methods are also coming into play here too), so logic, broadly construed, provides tools for building informatic theories.




Model Theory and the Foundations of Mathematics

John T. Baldwin


We will discuss a shift in viewpoint among model theorists of the last half century. The focus of attention has become not logics but theories. Thus, rather than seeking a Foundation for all of mathematics, we seek foundations for various fields: algebraic geometry, real algebraic geometry, differential algebra etc. The tools for comparing the various foundations arise in stability theory.


The grounding of mathematics in ZFC set theory provides a clear account of the derivability of almost all contemporary mathematics. This base for mathematics has, however, several weaknesses. The coding into set theory loses contact with the methods and conceptual ideas of different branches of mathematics. The focus is on truth as opposed to meaning. We describe from a historical standpoint how model theory provides an alternative standpoint. We consider three types of model theoretic analysis:

1. Properties of logics (1930-1965)

2. Properties of complete theories (1950-present)

3. Properties of classes of theories (1970-present)

We will expound the differences in these viewpoints and sketch the role of stability theory in the third. We will analyze the distinctions among categoricity, categoricity in power and completeness and how they led to the modern model theoretic viewpoint. The stability classification provides a criterion for whether the models of a complete theory can be classified. When such a classification exists, there is a structure theory for the models. In particular each model of arbitrary cardinality can be decomposed into a tree of countable structures. Thus the ‘main gap’ separates theories into those which are controlled by their countable models and creative theories where new phenomena occur at arbitrary large cardinalities. A key success of this theory is its role in providing a method of analysis (a foundation) for various branches of mathematics that is sensitive to particular needs of that subject. This analysis provides support for the theses proclaimed below.


In elaborating the properties of theories and classes of theories we will mention several examples of differing foundations for different parts of mathematics.


1. algebraically closed fields (w-stable) and the notion of ‘generic’ in geometry

2. 0-minimality and real exponentiation

3. stability and definable chain conditions

4. quasiminimality and complex exponentiation

5. non-standard analysis


The survey of model theoretic work leads to the following two theses.


Thesis I: Studying the models of a (complete first order) theory provides a framework for the understanding of the methodology of specific areas of mathematics. Here are some examples:


f.o. theory



algebraic geometry


real algebraic geometry


differential algebra


(Acf,rcf, dcf are respectively algebraically, real, and differentially closed fields.)


Thesis II: Studying classes of theories provides an even more informative framework for the understanding of the methodology of specific areas of mathematics. Here are some examples:



classes of theories


stable fields

Diophantine geometry


exponential fields

excellent classes

complex exponentiation

homogeneous classes

Banach spaces


We will give an overview of the program outlined above without excessive technicality.


A Day in the Life of a Working Mathematician

Jean Paul van Bendegem


It is perhaps too early to claim that a new development is taking place in the philosophy of mathematics. But there are signs that show that at least some philosophers of mathematics are no longer interested solely in foundational studies and conceptual analysis (see, e.g., Mancosu [2008]). The basic assumption is that to understand fully what it is, if anything, mathematics is about, it is necessary to develop a full picture of the mathematician’s activity (including as a part the foundational work itself, so there need not be an opposition there). Without denying that proof is still the core concept of mathematics, we want to show that a host of elements surround this core concept. Without this more complete picture, some aspects of what mathematicians do, are hard to understand, such as why certain problems are more interesting than others, the crucial role aesthetic considerations play in the evaluation of the quality of a proof, the explanatory value of a proof, and so on. The main purpose of the presentation will be to sketch the richness of the full picture by moving through several levels: the macro-level where possibly revolutions takes place (or not, as the discussion is still going on), the meso-level where research programs are established that guide future research, and the micro-level, where the mathematician in her daily work is searching for proofs, using a whole range of techniques to help this search, including computers and other instruments.



Jean Paul Van Bendegem: “The Creative Growth of Mathematics”. In: Dov Gabbay, Shahid Rahman, John Symons & Jean Paul Van Bendegem (eds.), Logic, Epistemology and the Unity of Science (LEUS), Volume 1, Dordrecht: Kluwer Academic, 2004, pp. 229-255.

Jean Paul Van Bendegem (co-author: Bart Van Kerkhove): “The Unreasonable Richness of Mathematics”. Journal of Cognition and Culture, vol. 4, no. 3-4, 2004, pp. 525-549.

Jean Paul Van Bendegem (editor, co-editor: Bart Van Kerkhove): Perspectives on Mathematical Practices. Bringing together Philosophy of Mathematics, Sociology of Mathematics, and Mathematics Education. Dordrecht: Springer, 2007.

Jean Paul Van Bendegem (co-author: Bart Van Kerkhove): “Pi on Earth, or Mathematics in the Real World”. Erkenntnis, vol. 68, nr. 3, 2008, pp. 421-435.

Paolo Mancosu (ed.): The Philosophy of Mathematical Practice. Oxford: OUP, 2008.



‘Writing history’: history versus practice of logic

Johan van Benthem


One of the things that strikes me in the philosophy of logic is its great distance from actual research practice. To give one example, after almost a century of model theory and recursion theory, it is still routinely claimed that logic is essentially about consequence and proof, rather than (also) about truth, meaning, and computation. And the gap gets even wider with modern logics of agency and interaction.


Another critical observation is the unthinking repetition of old agenda items, like the elusive 'boundary' between logic and other fields. Note that one of the most striking strengths of the last half century of logic as practiced has been its disregard for interdisciplinary boundaries.


And finally, by doing all this, the philosophy of logic 'deep-freezes' an old, traditional image of the field, making the (real?) logician a theorem-proving applied mathematician,  - and nostalgia for the grand old age of foundational research in the 1930s the yardstick for the 21st century. By contrast, it is striking how little philosophy of logic has been influenced by 'philosophical logic', let alone the spectacular new interfaces of logic with computer science, linguistics, and nowadays also, psychology and economics - where most research takes place.


I intend to spend half of my talk on critical observations like these, offering a taxonomy of problematic tendencies, plus some explanation why things have gone this way.


The second half is more positive, and will be devoted to issues in current research that would call for a much livelier philosophy of logic. I intend to discuss: (a) the hold of 'logical systems', their uses and non-uses, and principles for designing them,

(b) the rise of nonomonotonic reasoning and the alternative: classical dynamic logics of informational events; and triggered by this, (c) a broader move from proof and computation to information-driven rational agency as a paradigm for logic.




J. van Benthem:


1999, 'Wider Still and Wider: resetting the bounds of logic', in A. Varzi, ed., "The European Review of Philosophy", CSLI Publications, Stanford, 21-44.

2006, 'Logic in Philosophy', in Dale Jacquette, ed., "Handbook of the Philosophy of Logic", Elsevier, Amsterdam.

2007, 'Interview', inV. Hendricks & H. Leitgeb, eds., "Philosophy of Mathematics, Five Questions", Automatic Press, Copenhagen.

2009, 'Logic, Rational Agency, and Intelligent Interaction', in C. Glymour, W. Wei & D. Westerståhl, eds., "Logic, Methodology and Philosophy of Science", Proceedings of the XIIIth International Congress Beijing 2007, College Publications, London.



What is practice-based philosophy of logic? A case-study: uses of formal languages in logic

Catarina Dutilh Novaes


A practice-based philosophy of a given scientific discipline could be described as an approach that seeks to take into account the most recent developments in the discipline as well as the actual practices of those working within it. In my talk, I shall outline the main traits of a practice-based approach to logic as I see it, also stressing the methodological challenges it must face in order to become a truly fruitful approach. I argue that there are two distinct but intertwined levels that should be taken into account when doing practice-based philosophy of logic: the social level of logic as a social enterprise; and the individual, cognitive level of how the logician conducts his/her investigations. Both levels would require a suitable methodology to be properly investigated. For the latter, I believe that the analysis may be informed by findings in cognitive science, but as for the former, it is not entirely clear yet how to proceed in a methodologically robust manner – mere anecdotic evidence is not sufficient.


I also argue that a practice-based philosophy of logic is not to be conflated with a sociology of logic. I take the difference between sociology and philosophy of logic to pertain essentially to the descriptive vs. prescriptive divide. A sociologist of logic would essentially describe how logicians work, the kind of networks they form, mechanisms for the establishment of paradigms etc., but without questioning the legitimacy of the practices being described. A (practice-based) philosopher of logic, in contrast, ought to offer critical analysis of the conceptual foundations of actual work being done in logic, in particular attempting to clarify underlying assumptions. So a practice-based philosophy of logic should not be taken to imply that actual practices are always ‘right’; the philosopher may well identify conceptual problems underlying the practices and suggest directions for improvement.


To illustrate possible directions for a practice-based philosophy of logic, I will report on some of my preliminary findings on uses of formal languages in logic. This is an interesting case study in that it involves the two levels just mentioned, namely the social level and the individual level: formal languages are part and parcel of logic as a social enterprise, but they also seem to play a fundamental cognitive role in how logicians conduct their investigations and arrive at new insights. Formal languages as they are used in logic fulfill a double role as representational as well as operational languages, in Sybille Krämer’s fitting terms (Krämer 2003).


I present formal languages as a special kind of technology that developed from a much older form of technology, namely written languages, initially meant for specific applications. But as is often the case with technologies, formal languages turned out to offer possibilities that had not been originally foreseen in the early stages of their development, and this explains much of the phenomenal changes that logical practices underwent since it became customary to do logic with formal languages. As essentially written languages, formal languages activate cognitive mechanisms primarily related to vision, and thus have an important diagrammatic component. One suggestion for experimental work that emerges from these considerations is thus to investigate the operational (‘paper-and-pencil’) nature of uses of formal languages in logical practices, as well as the extent to which these mechanisms do indeed rely primarily on cognitive mechanisms related to vision.


To conclude, I will argue that formal languages play three main roles in logical practices, roles that are complementary but also slightly in tension with one another: i- formal languages as expressive devices; ii- formal languages as pictorial devices; iii-formal languages as calculative, operational devices.




Wittgenstein, Gödel and Turing

Juliet Floyd


In 1946, recalling his discussions with Turing in Cambridge before the war, Wittgenstein stressed that Turing’s ‘machines’ are really “humans who calculate” (RPP I 1096).  Was this intended to embrace or to reject Turing’s model of human calculative activity?  What form of anthropomorphism was (and is) at stake in regarding humans as machines, and in playing imitation games?  The question becomes even more intriguing when we reflect that while Gödel held that it was Turing’s “precise and unquestionably adequate” definition of the notion of a formal system that allowed his own incompleteness theorems to be proved rigorously for the first time, Gödel also held that Turing made a “philosophical error” in holding that human mental procedures cannot go beyond mechanical procedures.  We shall contrast the viewpoints of Wittgenstein, Gödel and Turing, emphasizing the evolution of the logical systems of notation that each one of them provided, and discussing how each viewed the philosophical significance of logic and mathematics.



Should we fear Benacerraf’s multiple reducibility challenge? Definitions and logical forms in philosophy of mathematics

Sébastien Gandon


In 1973, Benacerraf set an agenda that still plays a central role in philosophy of mathematics today. According to Benacerraf, two quite distinct kinds of concerns motivated accounts of the nature of mathematical truth : “(1) the concern for having a homogeneous semantical theory in which semantics for the propositions of mathematics parallel the semantics for the rest of the language, and (2) the concern that the account of mathematical truth mesh with a reasonable epistemology.” Benacerraf argued that one of these masters can only be served at the expense of the other.

Shapiro (see his 1997) has called Faithfulness Constraint (hereafter FC) the demand expressed in (1); FC requires that philosophers of mathematics take the surface form of the mathematical sentences as face values. Thus, to resume Benacerraf’s own example, FC demands that:

“There are at least three perfect numbers greater than seven”

should be construed as having exactly the same form as

“There are at least three large cities older than New York”.

FC raises immediately an epistemological difficulty. If we can account for the knowledge of (ii) (we have a perceptual interaction with New York and the cities older than it), it is difficult to account for the knowledge of (i): how does our mind interact with the reference of the name “seven” and with the objects in the domain over which range the existential quantifier in (i)? As Benacerraf noted, the satisfaction of FC seems to be got down only at the expense of (2). Why not dismissing FC, then?

The reason why FC should be adhered to is forcefully illustrated in Benacerraf 1965. In ZF theory, 2 is defined as {{Æ}}, while, in von Neumann’s approach, 2 is defined as {Æ, {Æ}}. Which is the genuine 2? This is not a question mathematicians ask, and, according to Benacerraf, a good philosophy of mathematics should exclude this question from its agenda.  Now, the only way to set aside this kind of question is to stick with FC. Indeed, as soon as you dismiss the surface form of the mathematical sentences, you are confronted with the problem of choosing the real structure of the mathematical propositions, without having any mathematical criterion to make your choice. For Benacerraf, Shapiro and many others as well, FC is a then good principle: it helps to distinguish philosophy of mathematics from general metaphysics. The multiple reduction challenge is the impasse reached by every approach which reject it.


My talk aims at contesting this appraisal. The idea is not to say that we should renounce FC because of the epistemological difficulties it provokes. My claim is that we should renounce FC because it makes us loose Benacerraf’s multiple reductions challenge. That is, I would like to show that the main argument usually given in favour of FC provides us with the main reason to dismiss it. The multiple reduction challenge is not a spectre that philosophers of mathematics should avoid at any price – on the contrary, multiple reductions are an essential ingredient of the mathematical practices, and hence, they are something that philosophers of mathematics should account for.


I will argue for that in two steps. First, I will claim that the issue raised by Benacerraf is related to the nature of mathematical definitions and that any good philosophy of mathematics should deal with this question. Mathematicians never stop to rewrite their mathematical propositions. To take their statements as face value is therefore to shut one’s eyes to this very important process. In other words, to be faithful to what mathematicians say is certainly not the best way to be faithful to what mathematicians do.

Benacerraf’s alternative definitions of 2 is not, it is true, an issue for the mathematicians. But I claim that this is not because the question raised is an essentially metaphysical issue, but because Benacerraf chooses a bad (because too trivial) example of a phenomenon which is very frequent and very important in mathematics.  In order to counter the effect of this unfortunate choice, I will present another, mathematically richer, illustration: Pieri’s construction of the separation relation between two couples of points on a projective line in terms of incidence relations (see Pieri 1898). This example has been very important for Russell (see Russell 1903), who is the first to have defended the idea that doing philosophy is uncovering the real logical form of the propositions (and then, the first to have dismissed FC).



Necessity in mathematics II

Wilfrid Hodges


The paper is a second installment of a trawl I am making through mathematical textbooks, finding expressions that appear in theorems, definitions and exercises but can't be translated literally into set theory.  So what are these expressions doing there?  The title reflects the fact that many of these expressions are modal.  But when the talk gets written, I think I may concentrate on expressions that are used for talking about functions.



Formalization is idealization: Uncovering the subtle nuances of logical and mathematical practices

Benedikt Löwe and Thomas Müller

In the formal sciences, we use formalization as a method to give representations of human activities. For instance, logic can be seen as a formalization of human reasoning, and deductive mathematics (as developed in the late 19th and early 20th century) can be seen as a formalization of mathematical activity, which of course existed long before the definition of a formal derivation.

Formalization is idealization: we lose information and possibly distort the subject of investigation by formalization since we identify different instances of the practice under consideration with a unique – often symbolic -- "normal form".

Based on the mathematization of logic, the goal of logic in the first half of the 20th century was to find the one true formalization of human reasoning. Anti-psychologist sentiments supported a strongly normative view of logic. The past decades have seen a liberalization of logic, now considered as a toolkit of many different formal tools, to be applied adequately in varying contexts. This opened the opportunity to look at
actual human reasoning (formerly considered a taboo by the anti-psychologists) with empirical methods, leading to the thriving and interesting research area of "Psychology of Reasoning".

A similar phenomenon can be observed in philosophy of mathematics: The deductive model of mathematics had gained almost universal acceptance as _the_ adequate model of mathematical reasoning.  Despite the fact that logicism failed as a philosophy of mathematics, its ideology stuck.) In analogy to a normativist view of logic, different mathematical activities were identified by appeal to the deductive formal form, hiding a number of philosophically important phenomena. A practice-based philosophy of
mathematics has to uncover these subtle differences in mathematical practice. In our talk, we shall explore this parallel, focusing on examples from our own work in philosophy of mathematics.



Ampliative Deductive Proof: A Case Study

Danielle Macbeth


It can seem obviously contradictory to suppose that one might extend one’s knowledge by reason alone, that is, by deductive proof. And yet, Frege claims, such proofs can extend our knowledge. Indeed, Frege took himself to have demonstrated in his Begriffsschrift proof of theorem 133, in the 1879 logic, that a strictly deductive proof can nonetheless be ampliative: “from this proof [of theorem 133] it can be seen that propositions that extend our knowledge can have analytic judgments for their content” (Grundlagen §91). As Frege further indicates (in section 88 of Grundlagen), definitions, at least those that he describes as fruitful, play an essential role in such deductive yet ampliative proof. The problem, then, is to explain how a strictly deductive proof from definitions can constitute a real extension of our knowledge.


If we think of deductive reasoning the usual way, that is, as we think of it in mathematical logic (e.g., proof or model theoretically), the problem admits of no solution. I pursue a different strategy. Following Kant and Peirce, I take mathematical reasoning, as contrasted with reasoning in natural language, to be a paper-and-pencil activity that is essentially constructive and (in a broad sense) diagrammatic; and I read Frege’s proof as an instance of this sort of paper-and-pencil construction. That is, I read the proof of theorem 133 as continuing the tradition of mathematical reasoning begun with Euclid’s diagrammatic reasoning and carried forward by Descartes’ constructive, algebraic problem solving. The task is to see theorem 133 “grow” from Frege’s definitions as a plant grows from a seed.



Wittgenstein and Social Constructivism

Mathieu Marion


In the early 1970s, both Peter Winch’s critique in [12] of Evans-Pritchard on the Azande [6] and David Bloor’s Strong Sociology of Knowledge (SSK) programme [1], linked Wittgenstein with different forms of relativism. (See David Bloor’s [3], [4] for a full interpretation.) Traditionally, sociological analysis was deemed inapplicable to mathematics and logic, but Bloor (and others) moved into this territory with SSK. The resulting social constructivism about mathematics and logic has been recently subjected to much critical scrutiny [5], [9], [10], [11], in light of which I want to re-examine in this paper the claim that Wittgenstein’s remarks in [13], [14] and [15], ranging from his critique of Frazer’s Golden Bough to his rule-following argument, provide support for cultural relativism or social constructivism. My conclusions will be that they can only provide such support at the price of distortion. Wittgenstein’s input is better construed as undermining both ‘relativist’ and ‘universalist’ claims (see, e.g., [7], [8] for a similar conclusion). Of course, this has nothing directly to do with the current debate about logical monism, so the very limited conclusion of this paper will simply be that those who would like to take the easy road to pluralism through cultural relativism will not find support from arguments in Wittgenstein, which is not to say that monism has been vindicated.


[1] B. Barnes & D. Bloor, ‘Relativism, Rationalism and the Sociology of Knowledge’, in M. Hollis & S. Lukes (eds.), Rationality and Relativism, Cambridge MA, MIT Press, 1982, 21-47.

[2] D. Bloor, ‘Wittgenstein and Mannheim on the Sociology of Mathematics’, Studies in the History and Philosophy of Science, vol. 4, 1973, 173-191.

[3] D. Bloor, Wittgenstein: A Social Theory of Knowledge, New York, Columbia University Press, 1983.

[4] D. Bloor, Wittgenstein, Rules and Institutions, London, Routledge, 1997.

[5] P. Boghossian, Fear of Knowledge. Against Relativism and Constructivism, Oxford, Clarendon Press, 2006.

[6] E. E. Evans-Pritchard, Witchcraft Oracles among the Azande, Oxford, Clarendon Press, 1937.

[7] C. Greiffenhagen & W. Sharrock, ‘Mathematical Relativism: Logic, Grammar, and Arithmetic in Cultural Comparison’, Journal for the Theory of Social Behaviour, vol. 396, 2006, 97-117.

[8] C. Greiffenhagen & W. Sharrock, ‘Logical Relativism: Logic, Grammar, and Arithmetic in Cultural Comparison’, Configurations, vol. 14, 2006, 275-301.

[9] I. Hacking, The Social Construction of What?, Cambridge MA, Harvard University Press, 1999.

[10] A. Kukla, Social Constructivism and the Philosophy of Science, London, Routledge, 2000.

[11] S. Sismondo, ‘Some Social Constructions’, Social Studies of Science, vol. 23, 1993, 515-553.

[12] P. Winch, ‘Understanding a Primitive Society’, in B. R. Wilson (ed.), Rationality, Oxford, Blackwell, 1970, 78-111.

[13] L. Wittgenstein, Philosophical Investigations, Oxford, Blackwell, 1953.

[14] L. Wittgenstein, Remarks on the Foundations of Mathematics, sec. ed., Oxford, Blackwell, 1978.

[15] L. Wittgenstein, The Big Typescript TS 213, Oxford, Blackwell, 2005.



On the role of practices in logical disagreements

Amirouche Moktefi

Fabien Schang


The practical turn in philosophy of science (1970s-1980s) established itself as a fruitful approach to the understanding of scientific knowledge. Such a viewpoint however is much easier to apply in the case of the natural sciences than in the formal ones, among which one includes logic. There seems to be several reasons for that. Obviously, the logic’s concern with the laws of thought rather than the laws of nature makes it difficult to study from the viewpoint of practices, as no “physical” experiments are expected to be observed in the field of logic. Another reason that makes the practical approach less used in the philosophy of logic might be the ambiguous position of logic itself, between mathematics and philosophy, which makes its aim and methods difficult to grasp, as it borrows from both scientific and philosophical (different) practices. One might add the dominance of the natural sciences in the field of the sociology of science in the 1960s-1970s, didn’t encourage the development of a practices-oriented philosophy of logic. One has to wait for the 1900s-2000s for a strong “practical” trend in the philosophy of mathematics, and it is hoped that this conference and forthcoming events will lead to a similar trend in the philosophy of logic.

It is obvious that logical investigation, as well all scientific and intellectual research, is a social activity as logicians are human beings working within (and thus influenced by) a social context. These social factors influence the way they conduct their logical investigations, how they publish their results, as well as the exchanges they would have with their colleagues. It is less obvious however to understand how logicians can disagree if we assume logical laws to be universal. It has been noted that there is little variety in mathematical practices (J. Azzouni, “How and why mathematics is unique as a social practice”, in B. Van Kerkhove et J. P. van Bendegem (eds.), Perspectives on Mathematical Practices, Dordrecht: Springer, 2007, pp. 3-23). There seems to be however much more disagreement between logicians. In this paper, we will discuss two examples (one fictional and one real) of logical disagreements on (a priori) basic logical principles.

The first study concerns Lewis Carroll’s well-known Achilles and the Tortoise infinite regress, published in the journal Mind in 1895. The text is a dialogue between the two characters on the justification of a basic logical inference:

(A) Things that are equal to the same are equal to each other.

(B) The two sides of this Triangle are things that are equal to the same.

(Z) The two sides of this Triangle are equal to each other.

How can we force “logically” someone, who accepts the premises A and B, to accept the conclusion Z, if he does note accept the hypothetical: (C) If A and B are true, Z must be true. Carroll shows that if we insert C in the premises, we will be dragged into an infinite regress. The problem has been widely discussed by twentieth century philosophers and logicians. The generally admitted moral of the story is that an inference should not include its own Hypothetical proposition in its set of premises. This view has been defended by Gilbert Ryle: “The principle of an inference cannot be one of its premises or part of its premises. Conclusions are drawn from premises in accordance with principles, not from premises that embody those principles. The rules of evidence do not have to be testified to by the witnesses.” (“‘If,’ ‘So,’ and ‘Because’”, in M. Black (ed.), Philosophical analysis, Englewood Cliffs, N. J.: Prentice-Hall, 1963, pp. 306-307). In our presentation, we will interpret the problem with a special attention to the source of disagreement and the role that the characters play in the controversy.  The aim of this first study is to identify what can prevent the establishment of a consensus in a logical discussion. 

The second case study concerns a real historical disagreement between two logicians: Lewis Carroll and John Venn about one particular question: should universal affirmative propositions assert the existence of their subject? The interest of this study is to explain how different practices may prevent the establishment of an agreement, and what it tells us about logical principles. This is particularly interesting in our example, as the authors are working within the same social context. Both championed symbolic logic at the end of the nineteenth century, inventing a new logical symbolism and a new diagrammatic scheme. We will discuss, using both printed and manuscript material, how the two logicians developed each his own theory of existential statement, and how they treated the issue within their diagrammatic methods. We will then show how they reached contradictory conclusions: Carroll considered that propositions in A do assert the existence of their subject, while Venn stated that they do not.

Thanks to these two case studies, we will be able to question how social consensus and disagreement arise in the domain of logic, and what that teaches us about the nature of logical principles.



Belief, Behavior and Bisimulation

Rohit Parikh


A traditional account of belief takes our beliefs to be stored in our brains, rather like aphorisms on a scroll.   How do I know what I believe?  I look inside the brain and read the scroll (which only I can read at present).   The scroll tells me what I believe.  This account is still popular with some philosophers, for it is appealing, although quite unrealistic. 


However, an account which borrows from behaviorism takes beliefs to be revealed by how we act.    This account is fairly explicit in the logician Frank Ramsey and the statistician Leonard Savage and is becoming more popular among philosophers as well.  One need not be a diehard Skinnerian  to find such accounts appealing.


A related matter is the issue of computational theories of the mind.  Such theories tend to take Turing machines as a representation, but more flexible models which take observable processes and bisimulation to be the important concepts are more suitable to such a representation.  Such models allow us to "understand" others without having to "look inside" their brains.


Various issues like logical omniscience, and absent that, the issue of explaining the coherence of our beliefs, can then be handled.


References (list is partial):


Arthur Collins,  Behaviorism and belief, Annals of Pure and Applied Logic, March 1999, 75-88.

Rohit Parikh, Sentences, belief and logical omniscience, or what does deduction tell us?, in The Review of Symbolic Logic, volume 1, 2008.

Frank P Ramsey, The Foundations of Mathematics and Other Logical Essays (Kegan Paul, London, 1931).

Leonard J Savage, The Foundations of Statistics (John Wiley and Sons, New York, 1954).

Eric Schwitzgebel, Belief, in the Stanford Encyclopedia of Philosophy 



Obligations and Disputations

Stephen Read


The real objective of the medieval theory of obligations is still a matter of dispute. This genre of medieval treatise has intrigued scholars for many years. In an obligational disputation, there is an Opponent and a Respondent. If the Respondent accepts a hypothesis and proposition put forward by the Opponent, he is then logically obligated to respond to a succession of propositions put to him by the Opponent according to specific rules. But what was the purpose of the disputation? It was clearly not to establish the truth of the proposition or propositions put forward by the Opponent, or of anything else. A brief comment in the Obligationes Parisiensis suggests that the aim was pedagogical, to train students in the art of reasoning logically. The Tractatus Sorbonnensis, however, declared that the purpose was to solve sophisms and logical puzzles. Since then, a raft of different suggestions has been made as to the goal behind this strange and unparalleled genre.


Further confusion is cast by the examination of a novel theory of obligations presented by Roger Swyneshed in the 1330s. His nova responsio had two striking and iconoclastic consequences: that the Respondent could deny a conjunction each of whose conjuncts he had already granted, and equally could grant a disjunction each of whose disjuncts he had denied. What sort of logic could underlie such rules? Put differently, what sort of logical training would these rules inculcate? Why did Swyneshed propose such surprising rules? How were they received and what further implications do they have?


Much work on obligations has concentrated on one particular species of obligation, positio. Walter Burley, arch exponent of the antiqua responsio, distinguished five further types of obligation, Swyneshed just two others, though for both of them, positio seems to have played a central role. Its description takes up two-thirds of Burley’s treatise and at least half of Swyneshed’s. Nonetheless, it is important to consider these other species of obligation when considering the purpose of obligational disputations. It is also important to consider the relation between Obligations treatises and other logical treatises, both treatises on Consequences and those on Insolubles. My conclusion is that obligations are indeed pedagogical exercises, albeit ones whose familiarity to medieval thinkers led to other treatises being shot through with obligational terminology which coloured and framed their method and approach to other issues. I will also argue that Swyneshed, provocateur extraordinaire though he was, was not as logically radical as he might seem.



Logic in semantics

Martin Stokhof

Logic has been applied systematically in the study of natural language meaning since the late sixties of the twentieth century. One question that has hitherto received relatively little attention is whether, and if so what kind of, ontological commitments this brings along. Taking a lead from Maddy's approach (in Naturalism in Mathematics), we will investigate how such issues should be dealt with in a practice-based approach, focusing in particular on the role that abstraction is (supposed to be) playing in defining the proper object of formal semantics.



"Im Anfang war die Tat": Inference, not consequence, is the key-notion of logic
Göran Sundholm


An inference is a deed, an act, something one can do, make, or perform. Consequence, on the other hand, is a relation that may, or may not, hold (even logically, "in all terms") between suitable relata; it is not something that one can do or perform. In my talk, I will elaborate on this difference: in particular, the use of assumptions in connection with inferences versus the role of antecedent propositions in consequences, and the explication of validity of inference versus the (logical) holding of consequence will be treated of. Writings of Bolzano, Frege, and Gentzen will be nodded at as well as those of some medievals.