Abstract
This paper provides a programmatic overview of a conception of iconic logic from a Wittgensteinian point of view (WIL for short). The crucial differences between WIL and a standard version of symbolic logic (SSL) are identified and discussed. WIL differs from other versions of logic in that in WIL, logical forms are identified by means of so-called ideal diagrams. A logical proof consists of an equivalence transformation of formulas into ideal diagrams, from which logical forms can be read off directly. Logical forms specify properties that identify sets of models (conditions of truth) and sets of counter-models (conditions of falsehood). In this way, WIL allows the sets of models and counter-models to be described by finite means. Against this background, the question of the decidability of first-order-logic (FOL) is revisited. In the last section, WIL is contrasted with Peirce’s iconic logic (PIL).
T. Lampert—I am grateful to Wulf Rehder for many helpful comments on an earlier draft of this paper.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Algorithms that realize some of Wittgenstein’s ideas concerning logical proofs are available at the following link: http://www2.cms.hu-berlin.de/newlogic/webMathematica/Logic/home.jsp.
- 2.
- 3.
I abstain here from cumbersome references to instances of atomic formulas. Thus, I refer to P instead of “an admissible instance of P”, etc. I also abstain from specifying the trivial algorithm for paraphrasing ideal diagrams of propositional logic.
- 4.
This procedure as well as others are implemented at and can be applied via the link given in footnote 1.
- 5.
FOLDNFs are far less complex than Hintikka’s distribute normal forms of FOL; cf. [Lampert (2017a)] for details.
- 6.
- 7.
According to [Quine (1960)], Sect. 30, a predicate such as “x seeks y” does not refer to a set of pairs and, thus, does not satisfy the principle of extensionality. However, the question is how one can know this without referring to some failure of logical formalization. For our purposes, it is sufficient to note that mere instantiation of logical formulas does not guarantee that those instances behave in accordance with the laws of logic. Therefore, one must distinguish between admissible and inadmissible instances. According to WIL, instances are inadmissible if they are not judged to be true despite instantiating provable formulas.
- 8.
- 9.
Cf., in particular, [Shin (2002)] and [Dau (2006)] for detailed elaborations of PIL. [Pietarinen (2006)] provides a game-theoretic interpretation of PIL and relates this interpretation to the later work of Wittgenstein. By contrast, I refer to the early work of Wittgenstein and his conception of a logical proof as a mechanical transformation into ideal diagrams.
References
Copi, I.: Symbolic Logic. Macmillan, New York (1979)
Dau, F.: Mathematical Logic with Diagrams - Based on the Existential Graphs of Peirce (2006). http://www.dr-dau.net/eg_readings.shtml
Etechemendy, J.: The Concept of Logical Conequence. CSLI Publications, Standford (1999)
Lampert, T.: Minimizing disjunctive normal forms of first-order logic. Log. J. IGPL 25(3), 325–347 (2017)
Lampert, T.: Wittgenstein’s \(ab\)-notation: an iconic proof procedure. Hist. Philos. Log. 38(3), 239–262 (2017)
Lampert, T.: A decision procedure for Herbrand formulae without skolemization, pp. 1–30 (2017). https://arxiv.org/abs/1709.00191
Lampert, T.: Wittgenstein and Gödel: an attempt to make ‘Wittgenstein’s objection’ reasonable. Philosophia Mathematica 25(3), 1–22 (2017). https://doi.org/10.1093/philmat/nkx017
Landini, G.: Wittgenstein’s Apprenticeship with Russell. Cambridge University Press, Cambridge (2007)
Lemmon, E.J.: Beginning Logic. Hackett, Indianapolis (1998)
Montague, R.: The proper treatment of quantification in ordinary English. In: Thomason, R.H. (ed.) Formal Philosophy. Selected Papers, pp. 247–270. Yale University Press, New Haven (1966)
Peirce, C.S.: Collected Papers of Charles Sanders Peirce. In: Hartshorne, C., Weiss, P. (eds) Harvard University Press, Cambridge (1931–1958)
Peregrin, J., Svoboda, V.: Reflective Equilibrium and the Principles of Logical Analysis. Routledge, New York (2017)
Pietarinen, A.-V.: Signs of Logic. Springer, Dordrecht (2006)
Potter, M.: Wittgenstein’s Notes on Logic. Oxford University Press, Oxford (2009)
Quine, W.V.O.: Word and Object. MIT Press, Cambridge (1960)
Russell, B.: The Principles of Mathematics, 2nd edn from 1937. Routledge, London (1992)
Shin, S.J.: The Iconic Logic of Peirce’s Graphs. The MIT Press, Cambridge (2002)
Turing, A.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 2(42), 230–265 (1936)
Wengert, R.G.: Schematizing de Morgan’s argument. Notre Dame J. Form. Log. 1, 165–166 (1974)
Wittgenstein, L.: Cambridge Letters. Blackwell, Oxford (1997)
Wittgenstein, L.: Notebooks 1914-1916. Blackwell, Oxford (1979)
Wittgenstein, L.: Tractatus Logico-Philosophicus. Routledge, London (1994)
Wittgenstein, L.: Wittgenstein and the Vienna Circle. Blackwell, Basil (1979)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Lampert, T. (2018). Iconic Logic and Ideal Diagrams: The Wittgensteinian Approach. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_56
Download citation
DOI: https://doi.org/10.1007/978-3-319-91376-6_56
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91375-9
Online ISBN: 978-3-319-91376-6
eBook Packages: Computer ScienceComputer Science (R0)