Peirce’s Diagrammatic Solutions to ‘Peirce’s Puzzle’ Ahti-Veikko Pietarinen 1,2(B ) 1 Tallinn University of Technology, Tallinn, Estonia ahti.pietarinen@taltech.ee 2 Research University Higher School of Economics, Moscow, Russia Abstract. We present Peirce’s own solution to what is known as ‘Peirce’s Puzzle’ in formal semantics and pragmatics. In his mostly unpublished writings, Peirce analyses some sentences in the modal exten- sion of his Beta Existential Graphs (that is, in a diagrammatic system of quantified first-order logic with tinctures) and in algebraic logic. These diagrams represent the pragmatic idea of information states that sup- port or fail to support sentences with (non-existential) indefinites and modalities. The interpretation of such sentences presupposes a graphic- pragmatic criterion of cross-identification. Keywords: Existential graphs · Peirce’s puzzle · Modality · Indefinites · Tinctures · Line of identity 1 Introduction Peirce’s Puzzle states that in first-order predicate logic (and in its origins in Peirce’s general algebra of logic), the sentence A is semantically equivalent to B[1, 3, 4, 6]: A: There is a married pair and if the husband fails the wife suicides. B: If every married man fails some married woman suicides. However, the common understanding of these sentences rather is that by uttering A we mean something stronger and more specific than what we mean by uttering B, namely that “. . . if her husband fails then she suicides” (A). Peirce re-examined the puzzle, which he first presented in 1906 [3], in the Logic Notebook (LN) notes on Sept 6–7, 1908 [319r-320r], and in letters to P.Carus in 1908 and to J.H.Woods in 1913. He uses quantification over “states of information”. He takes the meaning of A to be that there is some married couple of which, under all “conceivable states of the universe,” 1 the wife will suicide or else her husband would not have failed [4]. 1 Epistemic phrases such as “states of information”, “circumstances”, “states of knowl- edge”, “states of affairs”, “certain recognized states” were all used in Peirce’s late writings, in the order of prevalence. Supported by the Basic Research Program of the HSE University and the TalTech grant SSGF21021. c  Springer Nature Switzerland AG 2021 A. Basu et al. (Eds.): Diagrams 2021, LNAI 12909, pp. 1–5, 2021. https://doi.org/10.1007/978-3-030-86062-2_23 Author Proof