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Erschienen in:
Diagrams and Nonmonotonic Logic: What Is the Cognitive Relation? Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

Iconic Logic and Ideal Diagrams: The Wittgensteinian Approach

verfasst von : Timm Lampert

Erschienen in: Diagrammatic Representation and Inference

Verlag: Springer International Publishing

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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).

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Vorheriges Kapitel Towards a Proof Theory for Heterogeneous Logic Combining Sentences and Diagrams
Nächstes Kapitel Aristotelian and Duality Relations Beyond the Square of Opposition
Fußnoten
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
Cf. his letters to Russell during this period, reproduced in [Wittgenstein (1997)], as well as Wittgenstein’s Notes on Logic and his Notes dictated to G.E. Moore, both printed in [Wittgenstein (1979)].
 
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
Standard logic textbooks, such as [Copi (1979)], p. 131f., or [Lemmon (1998)], p. 131f., formalize (9) by (11); by contrast, [Wengert (1974)] argues that only (8) should be formalized by (11), whereas (9) should be formalized by (12).
 
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
In fact, I have detailed a decision procedure for pure FOL without identity on the basis of a Wittgensteinian conception of proof (cf. the link given in footnote 1). For the details of a Wittgensteinian critique of undecidability proofs, cf. [Lampert (2017d)].
 
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.
 
Literatur
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Metadaten
Titel
Iconic Logic and Ideal Diagrams: The Wittgensteinian Approach
verfasst von
Timm Lampert
Copyright-Jahr
2018
Buch
Diagrammatic Representation and Inference

Print ISBN: 978-3-319-91375-9

Electronic ISBN: 978-3-319-91376-6

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-91376-6_56

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