**21 October 2011
Leiden, The Netherlands**

Frege in Leiden Friday, October 21, 2011 Room C006, Kamerling Onnes Gebouw, Steenschur 25, Leiden Programme 10.00 Maria van der Schaar (Leiden) Judgement and the judgemental force 11.00 Harm Boukema (Nijmegen) Frege's missing link 14.00 Göran Sundholm (Leiden) Making sense of truth-values: Frege in mysterious circumstances 15.00 Ansten Klev (Leiden) On Frege's notion of pseudo-axiom 16.30 Kai Wehmeier (Irvine) Frege's proof of Hume's Principle (joint work with Robert May) Abstracts Maria van der Schaar (Leiden) Judgement and judgemental force Today, most writings on Frege?s notion of judgement focus on the term ?acknowledgement? (?Anerkennung?) in Frege?s assertion that judgement is the acknowledgement of the truth of a proposition. One should not forget, though, that judgement, for Frege, is the logically primitive activity (logische Urtätigkeit) that is indefinable. Judging is therefore not to be understood as a special case of the genus acknowledging, and one should not take the term ?acknowledgement? too literal. The aim of my talk is to elucidate Frege?s notion of judgement by means of examples of those judgements that are inferences. Thus hoping to make clear how, for Frege, the logical laws, the laws of truth, may constitute a norm for our practice of judgement and inference. Harm Boukema (Nijmegen) Frege's missing link In his article Funktion und Begriff, Frege illustrates his distinction between functions and objects by means of a very simple and remarkable geometrical example: the division of a segment. Even in such a case, where at first sight there seems to be involved no dissimilarity at all, its parts actually differ in form. For if the dividing point, say C, is not counted twice, it can only belong to one of the two parts. The other part will be a half open segment. If [A,B] is assumed to be the whole, it may be divided in [A,C] and (C,B] or in [A,C) and [C,B]. In other words: the two parts only fit if one of them lacks C. No link between them is needed, because the missing itself is the link. In this lecture, it will be argued that, although this piece of analysis seems to be impeccable, something of mathematical and logical importance has been overlooked, namely that, nevertheless, the point C has to be mentioned twice. Awareness of the serious nature of this failure constitutes the missing link with a form of analysis different from the one applied by Frege and prevailing in analytic philosophy. Göran Sundholm (Leiden) Making sense of truth-values: Frege in mysterious circumstances It is hard to conceive of a notion more central to today's philosophy of language and logic that that of a truth-value. Frege's explanations, when introducing the notion of a truth-value around 1890 are examined and found wanting; his formulation in terms of "circumstances" will be rejected and, in connection with it, an almost universal mistranslation corrected. It is suggested that the origin for the notion of truth-value does not lie within the "philosophy of language". On the contrary, the reason for its introduction is purely technical. It resides in Frege's (and everyone else's) use of the notion of ordered pair in attempts to prove Dedekind's theorem on definitions by recursion over the natural numbers. Ansten Klev (Leiden) On Frege's notion of pseudo-axiom In the second series of his Über die Grundlagen der Geometrie (1906) Frege interprets Hilbert's axiomatic presentation of geometry (1899) as an allgemeiner Lehrsatz, that is, as a conditional whose antecedent is the conjunction of the axioms, whose consequent is a conjunction of theorems, and in which the primitive terms are replaced by free variables. Frege calls the axioms as they occur in such a Lehrsatz `pseudo-axioms'. In the secondary literature on Frege and Hilbert it seems to me univocally assumed that these axioms are pseudo solely in virtue of their containing free variables. This assumption, however, is in conflict with how Frege viewed free variables throughout his career, namely as signs that bestow generality upon the sentences in which they occur. It is also in conflict with Frege's explanation of why he uses the pre-fix `pseudo'. An Hilbertean axiom is pseudo, he says, because the free variables that occur in it can exercise their function, namely that of bestowing generality, only within the larger context of the Lehrsatz. Hence, as separated from this whole, a pseudo-axiom is meaningless; Frege therefore also describes pseudo-axioms as dependent parts of theLehrsatz. Kai Wehmeier (Irvine) Frege's proof of Hume's Principle In §53 of Grundgesetze, Frege begins to prove what he calls the "basic laws of cardinal number". We explore the initial part of his development of arithmetic through theorems (32) and (49), which when taken together essentially constitute what we call today Hume?s Principle (HP). Frege outlined a proof of HP already in Grundlagen §73. The informal exposition there has a characteristic asymmetry that is carried over to the formal presentation of Grundgesetze: In both, pride of place is given to the proof of the "right-to-left direction", F ~ G -> #F = #G. In Grundlagen, this direction is discussed in the main body of the text, while the provability of the reverse direction is merely mentioned in a footnote. In Grundgesetze, the right-to-left direction, i.e. Theorem 32, is accorded the status of a Basic Law of Cardinal Number, a status that Theorem 49, the left-to-right direction, lacks. Why does Frege differentiate the two directions of HP in this way, and why does he never bring them together into a biconditional? The symposium is kindly supported by the Vereniging voor Logica.