Keynote Speakers

Michael Beaney (York): Logical Form and Philosophical Analysis. John Cook Wilson and Gottlob Frege

On the standard account, the logical form of a proposition is the structure it has that is responsible for its inferential relations. In traditional logic, logical form was generally seen as mirroring grammatical form, but from the work of Frege onwards, logicians began stressing the distinction between logical form and grammatical form, the latter being regarded as misleading in various ways. This is where logical analysis – or philosophical analysis, as it was also called – came in: its role was to uncover the underlying logical forms of propositions.

The classic version of this account can be found in the writings of Bertrand Russell (1872-1970). Russell went through an early stage of thinking that grammar was a reliable guide to logical form, but once he had developed the theory of descriptions, he emphasized the radical divergence that typically opens up between grammatical form and logical form. This recognition of the significance of the theory of descriptions was encapsulated in Wittgenstein’s remark in the Tractatus that “Russell’s merit is to have shown that the apparent logical form of a proposition need not be its real one” (4.0031). On Wittgenstein’s early view, too, analysis was required to reveal the logical forms of propositions.

In this paper I want to raise some questions about this standard – Russellian – account by considering the work of two of his immediate predecessors: Gottlob Frege (1848-1925) and John Cook Wilson (1849-1915). Frege and Cook Wilson were almost exact contemporaries, and there are both major differences but also surprising similarities in their views and concerns. As the founder of modern logic, one would expect Frege to have endorsed the Russellian account. Certainly, Frege rejected traditional subject–predicate analysis in favour of his new function–argument analysis, but he does not talk of ‘logical form’, and unlike Russell, he stresses that propositions can have different analyses. Cook Wilson, on the other hand, has often been regarded as the archetypical logical reactionary. As Ayer said of him, he “was a fervent Aristotelian and had sat like Canute rebuking the advancing tide of mathematical logic” (Part of my Life, p. 77). Yet despite his Aristotelianism – or perhaps because of it – Cook Wilson does distinguish between grammatical and logical analysis, and his views are far more interesting and sophisticated than Ayer’s caricature would suggest. In particular, like Frege, he stresses that propositions can be analyzed in different ways.

Danielle Macbeth (Haverford): Proof and Understanding in Mathematical Practice

The mathematical practice of proving theorems seems clearly to result in improved mathematical understanding; the aim and point of proving, and reproving, theorems in mathematics is better mathematical understanding. And yet, it has become increasingly clear that proof as it is usually understood (a proof is a formal deduction of a theorem on the basis of axioms) is irrelevant to understanding. There are only two options: either mathematical understanding resides somewhere else than in proof, or a mathematical proof is something different from what the standard conception says it is. Both options have been pursued: Manders, for instance, arguing that mathematical understanding resides not in proofs per se but in the conceptual settings that mathematicians develop and deploy in their proofs, and Rav, among others, arguing that the mathematician’s proofs are essentially different from proofs as they are understood in formal logic. I will develop an alternative to both that is based on the notion of logical form that we find in Frege, a notion of logical form that does not contrast with content but instead embodies it.


Invited speakers:

Catarina Dutilh Novaes (Amsterdam): The early history of the form vs. matter distinction in logic

Thinking about logic in terms of the form vs. matter distinction is now so widespread that this particular conceptualization of logic is typically viewed as a quasi-truism to be uncritically accepted. It is, however, the product of long and winding historical paths. It is perhaps especially surprising that Aristotle, the first to formulate a full-fledged logical system (syllogistic) and the first to make systematic use of the form vs. matter dichotomy, did not apply the concepts of form or matter anywhere in his logical writings. This talk will survey some of the applications of the form vs. matter distinction to logic, covering Aristotle, the Ancient Commentators (in particular Alexander of Aphrodisias), Boethius, the early Latin Middle Ages, the later Latin Middle Ages, and the modern period up to Kant. The focus will be less on points of detail and more on the broad lines of these developments and how they are related to one another. In particular, I shall pay some attention to the neglected topic of the metaphysical status of the ‘forms’ of logic in these different applications; after all, the form vs. matter distinction is originally a metaphysical distinction, and losing sight of this aspect hinders a proper understanding of these developments.

Anssi Korhonen (Helsinki): Formality and Generality in Russell’s Logic

Like any other science, logic strives for generality. On a traditional view, generality is secured by formality, that is, by logic’s being concerned with the forms of judgment, which are ubiquitous (an example would be Kant’s ‘pure general logic’). On the face of it, Russell understood the subject-matter of logic quite differently; on his view, the generality of logic was quantificational — or ‘substantive’ — rather than ‘schematic’, as on the former view. According to his favourite explanation, logic abstracts from the particular content of propositions and issues completely general truths, truths in which the only constants belong to the class of logical constants.
Such completely general propositions are needed to guarantee the validity of valid inference. This view is a stable element in Russell’s thinking about logic. As early as The Principles of Mathematics, however, Russell comes to see that it leads to difficulties in explaining exactly how these completely general propositions are supposed to figure in inference. Reflecting on Lewis Carroll’s famous puzzle about inference (“What the Tortoise said to Achilles”, Mind 1895), he concludes that a distinction must be drawn between logical laws, which are propositions, and rules of inference (which apparently are not).
In the present paper I consider Russell’s views on valid inference in light of Lewis Carroll’s puzzle, drawing some tentative conclusions about the general shape of Russell’s conception of logic.

Contributed Talks

Johan Blok (Groningen): Logical and Mathematical Notions of Form in Bolzano’s Early Work (long abstract)

In his early work, Bolzano uses the term ‘form’ both in his reform of mathematics and of logic. In my paper I will shortly comment on the use of ‘form’ in logic and discuss the role of ‘form’ in Bolzano’s new conception of mathematics more in detail. In both cases, it will be shown how Bolzano modified existing notions of ‘form’. With regard to the logical use it will become clear that Bolzano relies on his fundamental distinction between kinds of copula’s. Accordingly, the term ‘form’ does neither refer to a syntactic structure nor to another aspect of the modern notion of a formal language. With regard to the use of ‘form’ in relation to mathematics I will argue that ‘form’ refers to a conditional structure. Having defined mathematics as the science of conditions for the possibility of things, Bolzano is able to regard a mathematical object as a proper object of study regardless whether it can be realized.

Sergio Gallegos (CUNY): Notions of Formality and Notions of Apriority (long abstract)

Anna-Sophie Heinemann (Paderborn): An Analysis of Analyses of the Laws of Thought: Notions of Form in 19th Century British Logic (long abstract / paper)

This talk makes a contribution to research on historical debates over the question whether an operative notion of logic can be reduced to aspects of ‘form.’ The inquiry takes recourse to 19th century British logicians to survey their notions of logical ‘forms.’ It is guided by their answers to the question whether and how these are to be interpreted. There are at least three senses in which 19th century logicians’ statements on the analysis of ‘forms’ is to be understood. Firstly, ‘form’ is used in the sense of ‘laws’ or ‘conditions’ (‘form’ versus ‘matter’). Secondly, ‘form’ can mean a ‘force’ or ‘principle’ of operation (‘form’ versus ‘content’). Thirdly, talk of ‘form’ can come close to what in contemporary language may be termed a ‘syntax’ as opposed to its ‘semantic’ models (‘form’ versus ‘interpretation). The underlying notions of form and their interrelation are analyzed with special regard to Sir William Hamilton and Augustus De Morgan.

Ansten Mørch Klev: Variables and Hilbert’s Grundlagen der Geometrie (long abstract)

I will look at two senses of `formal’. The one sense applies to mathematical works such as Hilbert’s Grundlagen der Geometrie. The other sense is that surfacing in `formal system’. One can see these two senses of the formal as arising from two kinds of modification of symbols. One modification is the extraction of material, or substantive, content, a modification we might term formalization. I think of this modification as resulting in a `categorial grid’, which in its turn is one way of understanding the notion of a variable or schematic letter; for grasping a variable seems to consist in grasping certain categorial restrictions on possible substituents of the variable. Another modification is objectification; this results in formal objects, the elements of formal systems.  The claim is then that if you start out, say, with the Elements and apply formalization you end up with something that is formal in the sense of the Grundlagen, whereas if you apply objectification you get a formal system.

Stefan Roski (VU Amsterdam), Antje Rumberg (Utrecht): Bernard Bolzano’s Notion of Form and his Concept of Formal Grounding (long abstract)
§ 12 of Bernard Bolzano’s Theory of Science contains an interesting discussion of the doctrine that logic is a merely formal science,which Bolzano mostly associates with logicians from the Kantian tradition. Despite of several confusions Bolzano finds in that doctrine, he also admits that it contains a kernel of truth: logic is not concerned with particular propositions, but rather with whole collections of propositions that exhibit a certain structural similarity. Formality is to a certain extend identified with a type of generality. By means of his so-called method of variation, Bolzano defines a notion of form that captures precisely this intuition. In our talk, we will present Bolzano’s definition and point out how it relates to the consequence relations that are essential to his logic: deducibility and grounding. While deducibility, which is frequently considered as an ancestor of Alfred Tarski’s concept of logical consequence, constitutes a formal — albeit not necessarily logical — consequence relation in Bolzano’s sense, the grounding relation, a ground-consequence relation among true propositions, is, first of all, a material one. Nevertheless, deducibility and grounding are compatible with each other. With the concept of formal grounding, Bolzano provides a notion that is simply a blend of deducibility and grounding: a grounding relation that rests solely on the form of the premises and conclusions involved. This relation turns out to play a crucial role for the axiomatic buildup of scientific theories, as it induces a unique order among true propositions and allows for science-specific inferences, which must not be logical. Moreover, we will briefly point to the resemblance proof systems ordered via the logical variant of the formal grounding relation bear to normal proofs in natural deduction (and their role in proof-theoretic semantics).

Simon Summers (UEA Norwich): Form, Content and Unity of Russellian Propositions

In his Principles of Mathematics (henceforth PofM) published in 1903, Bertrand Russell proposes a theory of propositions as (i) structured complexes made up of constituents (individuals, properties and relations) and as (ii) the semantic values of sentences. A problem arises for this proposal: a structured proposition can be analysed into its constituent parts, but such an analysis seems to rob the proposition of its characteristic unity as a complex whole. However, it is in virtue of its unity that a proposition fulfils its role as (i) the bearer of truth and falsity, (ii) the semantic value (the ‘meaning’ of a sentence) and (iii) the object of a propositional attitude. So, the following question arises for any structured-proposition account: what is it that binds the constituents together, such that a proposition, and the sentence expressing it, is able to fulfil these roles? To answer this question satisfactorily is to provide a solution to the unity of the proposition problem (UP).

Russell’s PofM theory is the progenitor of a range of contemporary neo-Russellian accounts of structured propositions.The present paper presents a detailed account of the content, form and unity of Russellian propositions as presented in PofM, and explains how and why this conception gives rise to UP. The paper also presents and assesses a number of contemporary neo-Russellian solutions to UP.

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