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Validity as Choiceless Unification

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Diagrammatic Representation and Inference (Diagrams 2021)

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Abstract

We propose a variant of Euler’s Diagrammatic Method for Categorical Syllogistic. According to this variant, a categorical syllogism is valid without existential import if, and only if, there is one, and only one, way to unify the representations of the premises, and the representation of the conclusion follows from this unification.

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Notes

  1. 1.

    We are here exploring an analogy with Grice’s maxim of quantity: ‘Make your contribution as informative as is required (for the current purposes of the exchange) [and] do not make your contribution more informative than is required’ [3, p. 45].

  2. 2.

    Peirce’s novelty was recently examined by Bhattacharjee and Moktefi [1], Moktefi and Pietarinen [5], and Pietarinen [9].

  3. 3.

    Peirce [4, p. 1410] distinguishes two purposes of a system of logical symbols: if a system is devised for the investigation of logic, it requires economy of kinds of sign [7, p. 222-223]; and, if a system is devised as a calculus, no such a restriction is required. Thus, Peirce’s novelty is not aimed at foundational studies.

  4. 4.

    His treatment is akin to Euler’s treatment.

  5. 5.

    We deal exclusively with terms without existential import. The validity dependent on existential import can also be treated by our approach, but some cases, such as DARAPTI, FELAPTON and FESAPO, require special rules.

  6. 6.

    Peirce refers to them as ‘assertions of non-existence’ [6, p. 282].

  7. 7.

    Thus, the introduction of negative terms expands the expressive power of Categorical Syllogistic.

  8. 8.

    Peirce [6] does not mention this last assertion of non-existence.

  9. 9.

    It is not difficult to prove that these eight types are the only valid types of universal syllogisms.

  10. 10.

    As the rules in Fig. 4 express relations of logical equivalence, no priority is claimed to the first four syllogisms, at least from a logical point of view.

  11. 11.

    This procedure will be better explained and exemplified in the case of particular syllogisms.

  12. 12.

    Each row in an item represents a premise.

  13. 13.

    In a sense, universal syllogisms, represented as trios of total exclusion, also admit a type of unification, in which the middle term needs to be represented dissimilarly, in a geometric sense, in the premises, that is, it needs to be represented with a concave enclosure in one premise and with a convex enclosure in the other.

  14. 14.

    Euler is right in using indexes, a second type of representation for particular propositions, but he fails in the implementation.

  15. 15.

    In accordance with the representation of universal propositions by total inclusion, we adopt Fig. 8(c) and Fig. 8(e) as primary and, only in very special situations in the unification process, we need to resort to Fig. 8(d) and Fig. 8(f).

References

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Author information

Authors and Affiliations

  1. Universidade Federal de Santa Maria, Santa Maria, Brazil

    Frank Thomas Sautter

  2. Universidade Federal da Fronteira Sul, Erechim, Brazil

    Bruno Ramos Mendonça

Authors
  1. Frank Thomas Sautter
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  2. Bruno Ramos Mendonça
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Corresponding authors

Correspondence to Frank Thomas Sautter or Bruno Ramos Mendonça .

Editor information

Editors and Affiliations

  1. Jadavpur University, Kolkata, India

    Amrita Basu

  2. University of Cambridge, Cambridge, UK

    Gem Stapleton

  3. Lancaster University in Leipzig, Leipzig, Germany

    Sven Linker

  4. Deakin University, Burwood, VIC, Australia

    Catherine Legg

  5. Kyoto University, Kyoto, Japan

    Emmanuel Manalo

  6. Universidade Federal Fluminense, Niterói, Brazil

    Petrucio Viana

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Sautter, F.T., Mendonça, B.R. (2021). Validity as Choiceless Unification. In: Basu, A., Stapleton, G., Linker, S., Legg, C., Manalo, E., Viana, P. (eds) Diagrammatic Representation and Inference. Diagrams 2021. Lecture Notes in Computer Science(), vol 12909. Springer, Cham. https://doi.org/10.1007/978-3-030-86062-2_18

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  • DOI: https://doi.org/10.1007/978-3-030-86062-2_18

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-86061-5

  • Online ISBN: 978-3-030-86062-2

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