Abstract
We show how a semantics based on Aristotle’s texts and ecthetic proofs can be reconstructed. All truth conditions are given by means of set inclusion. Perfect syllogisms reveal to be valid arguments that deserve a validity proof. It turns out of these proofs that transitivity of set inclusion is the necessary and sufficient condition for the validity and perfection of a syllogism. The proofs of validity for imperfect syllogisms are direct proofs without conversion in a calculus of natural deduction. Transitivity of set inclusion turns out to be a necessary condition for the validity of imperfect syllogisms. As a consequence, it can be established what the main metalogical difference between a perfect and an imperfect syllogism is. The validity of the laws of conversion is also obtained by direct proofs. Finally, it is shown that and explained why some imperfect syllogisms satisfy the definition of a perfect syllogism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Instead of ‘belong to’, Aristotle sometimes uses ‘predicated of’.
Henceforth, the set of premisses will be written without ‘\(\{\)‘ and ‘\(\}\)’.
In all quotes about syllogisms, the letters Aristotle uses are replaced with ‘P’, ‘M’, ‘S’.
\(\varepsilon (A)^c_U\) is the set complement of \(\varepsilon (A)\) in U. If U is the universe of discourse, then \(\alpha ^c_U\) is the set \(U - \alpha \).
Usually, the truth condition of e-sentences is formulated in terms of the empty intersection \(\beta \cap \alpha \) [5, p. 103], [9, p. 121], [15, p. 225]. The two formulations are set-theoretically equivalent, i.e. ST1: \((\forall \alpha , \beta ) (\alpha \subseteq \beta ^c_U \leftrightarrow (\alpha \cap \beta = \emptyset )\).
Cf. [17].
ST2: \((\forall \gamma ) (\gamma \ne \emptyset \leftrightarrow (\exists \delta ) (\delta \ne \emptyset \wedge \delta \subseteq \gamma ))\).
ST3 (T\(\subseteq \)): \((\forall \alpha , \beta , \gamma ) (\alpha \subseteq \beta \wedge \beta \subseteq \gamma \rightarrow \alpha \subseteq \gamma )\).
Henceforth, Darapti Ferio, etc. are to be understood as abbreviations for ‘syllogisms of the form Darapti, Ferio’ etc.
The formal definition of interpretations of \(\mathbb {L}\) and s-validity are given in Sect. 3.3. Therefore, this and the next proof cannot yet be completely formalised.
Patzig believes that when Aristotle introduces his criterion of perfection, he transposes the general terms and the order of the premisses so as to have ‘evidence’ of perfection [10, p. 58] Ebert, furthermore, believes that Aristotle criterion of perfection of syllogisms is a part of his explanation of Barbara and Celarent [6, p. 359].
See p.5: ‘I call that a term [‘
’ (“hóros”)] ... both the predicate and that of which it is predicated [i.e. the extension of the predicate]’. Aristotle uses the very same letters to refer to a general term as well as to its extension.We shall not elaborate any further on Aristotle’s use of the terms ‘major’ and ‘minor’ in his discussion of the second and third figures since it is irrelevant for the purposes of this paper.
Assume that the entity exposed is an element \(m \in \varepsilon _{\hspace{-1.5pt}I}(M)\). From \(\varepsilon _{\hspace{-1.5pt}I}(M) \subseteq \varepsilon _{\hspace{-1.5pt}I}(P)\) and \(m \in \varepsilon _{\hspace{-1.5pt}I}(M)\), it follows that \(m \in \varepsilon _{\hspace{-1.5pt}I}(P)\). Such an \(\alpha = m\), however, does not satisfy the criterion of perfection, which requires that \(\alpha \subseteq \beta \). Thus, the criterion is satisfied only if the entity exposed is a subset of \(\varepsilon _{\hspace{-1.5pt}I}(M)\).
Contrary to this result, Patzig claims that ‘In the first figure alone ... the so-called “middle” term stands in the middle in such a way as to bind together the two premisses. (Let us note here ... that this fact robs the expression “middle term” of all its mystery. It is the term which in the first figure... stands in the middle in the manner stated; its extension or its “power of mediation” are perfectly irrelevant.) The greater evidence of the first figure syllogisms clearly depends on the position of their terms relative to one another.’ [10, p. 51].
ST4: \((\forall \alpha ,\beta )(\alpha \subseteq \beta ^c_U \leftrightarrow \beta \subseteq \alpha ^c_U)\).
ST5: \((\forall \alpha , \beta \subseteq U) (\alpha \subseteq \beta \leftrightarrow \beta ^c_U \subseteq \alpha ^c_U)\) (first law of reciprocity) [18, p. 15].
The s-validity proof for Baroco is the only one in which \(\beta = \varepsilon (M)^c_U\).
Ebert arrives at the conclusion that ‘there is not transitivity in syllogisms of the other figures [than the first].’ [6, p. 362].
‘Theophrast aber und Eudemos haben in einer einfacheren Weise bewiesen, daß die allgemeine verneinende (Prämisse) sich umkehren (läßt) ... Sie führen den Beweis so: Es komme A keinem B zu. Wenn es keinem (zukommt), muß A von B getrennt [...] und abgesondert [...] sein. Das Getrennte wird aber vom Getrennten getrennt. Also ist auch B ganz von A getrennt. Und falls es so ist, kommt es keinem (A) zu.’ [4, 115f].
This contradicts Barnes’ claim that ‘the idea of mereological “separation” was exploited by Theophrastus and Eudemus in their non-Aristotelian proof of E-conversion.’ [3, p. 17].
See [16, 214–219].
Łukasiewicz has also shown that the s-validity of i-conversion follows from commutativity of conjunction only [8, p. 62].
The syllogism \({P\hspace{-0.5pt}o\hspace{-0.5pt}M}{}, {S\hspace{-0.5pt}a\hspace{-0.5pt}M}{} \therefore {P\hspace{-0.5pt}o\hspace{-0.5pt}S}{}\) is perfect as well. It is the same syllogism, but the premisses are transposed. Here, we have \(\delta _1 \subseteq \varepsilon _{\hspace{-1.5pt}I}(M)\) and \(\delta _1 \subseteq \varepsilon _{\hspace{-1.5pt}I}(P)^c_U\).
Datisi is one of Łukasiewicz’s axioms [8, p. 46].
That is, four first-figure, four third-figure, and one fourth-figure syllogism.
’ (“hóros”)] ... both the predicate and that of which it is predicated [i.e. the extension of the predicate]’. Aristotle uses the very same letters to refer to a general term as well as to its extension.References
Aristotle: Metaphysics. In: Gwinn, R.P., Ross, W.D. (eds.) The Works of Aristotle, vol. 1, 2nd edn., pp. 499–626. Encyclopædia Britannica, Inc., Chicago (1992). (Trans. by Ross, W. D.)
Aristotle: Prior analytics. In: Gwinn, R.P., Ross, W.D. (eds.) The Works of Aristotle, vol. 1, 2nd edn., pp. 39–96. Encyclopædia Britannica Inc, Chicago (1992). (Trans. by Jenkinson, A. J.)
Barnes, J.: Boethus and finished syllogisms. In: Lee, M.-K. (ed.) Strategies of Argument: Essays in Ancient Ethics, Epistemology, and Logic. Oxford University Press (2014). Retrieved 14 Dec 2021, from https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780199890477.001.0001/acprof-9780199890477-chapter-8
Bochenski, J.M.: Formale Logik, 2nd edn. Verlag Karl Alber, Freiburg (1962)
Corcoran, J.: Aristotle’s natural deduction system. In: Corcoran, J. (ed.) Ancient Logic and Its Modern Interpretations, pp. 85–131. D. Reidel Pub. Co., Dordrecht (1974)
Ebert, T.: What is a perfect syllogism in Aristotelian syllogistic? Ancient Philosophy 35, 351–374 (2015)
Kalish, D., Montague, R., Mar, G.: Logic: Techniques of Formal Reasoning, 2nd edn. Harcourt Brace Jovanovich, New York (1980)
Lukasiewicz, J.: Aristotle’s Syllogistic from the Standpoint of the Modern Formal Logic, 2nd edn. Clarendon Press, Oxford (1957)
Martin, J.: Elements of Formal Semantics. Academic Press, San Diego (1987)
Patzig, G.: Aristotle’s Theory of the Syllogism: A Logico-Philological Study of Book A of the Prior Analytics. D. Reidel Pub. Co., Dordrecht (1968). (Trans. by Barnes, J.)
Rocha, M.: Semantische Untersuchung der Syllogistik. Ph.D. thesis, Universität Salzburg, Salzburg (1999)
Rocha, M.: Was ist ein aristotelischer Syllogismus? In: Born, R., Neumaier, O. (eds.) Akten des VI. Kongresses der Österreichischen Gesellschaft für Philosophie. öbv &hpt, Wien, pp. 172–177 (2000)
Ross, W.D.: Aristotle’s Prior and Posterior Analytics. Revised Text with Introduction and Commentary. Oxford University Press, New York (2001).
Schröder, E.: Vorlesungen über die Algebra der Logik, vol. 2, 2nd edn. Chelsea Pub. Co., New York (1996)
Smith, R.: Completeness of an ecthetic syllogistic. Notre Dame Journal of Formal Logic 24, 224–232 (1983)
Striker, G.: Perfection and reduction in Aristotle’s prior analytics. In: Frede, M., Striker, G. (eds.) Rationality in Greek Thought, pp. 203–219. Clarendon Press, Oxford (2007)
Thom, P.: Ecthesis. Logique et Analyse 19, 299–310 (1976)
van Dalen, D., Doets, H.C., de Swart, H.: Sets: Naive, Axiomatic and Applied. Pergamon Press, Oxford (1978)
Weingartner, P.: Wissenschaftstheorie II, 1. Grundprobleme der Logik und Mathematik. Frommann-Holberg Verlag, Stuttgart (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Rocha, M. A Study of the Metatheory of Assertoric Syllogistic. Log. Univers. 17, 347–371 (2023). https://doi.org/10.1007/s11787-023-00331-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11787-023-00331-1