CC BY
35Логические исследования 2016. T. 22. № 2. C. 59-72 УДК 16
Logical Investigations 2016, Vol. 22, No. 2, pp. 59-72 DOI: 10.21146/2074-1472-2016-22-2-59-72
Философская логика
Philosophical Logic
E.G. Dragalina-Chernaya The Roots of Logical Hylomorphism
1
Dragalina-Chernaya Elena Grigor'evna
National Research University Higher School of Economics
21/4 Staraya Basmannaya St., Moscow, 105066, Russian Federation
e-mail: edragalina@gmail.com
The main purpose of this paper is to discuss the origin and the bounds of the schematic hylomorphism in ancient and medieval logic. The sub-purposes are fourfold. Firstly, various explications of the logical hylomorphism will be illustrated. Secondly, I propose to reevaluate certain interpretations of Aristotle's syllogistic. I attempt to answer the question why Aristotle was not the founder of logical hylomorphism. Thirdly, I aim to qualify the schematic hylomorphism of Alexander of Aphrodisias. Finally, I focus on the medieval discussions on syncategoremata and formal consequences.
Keywords: logical hylomorphism, logical form, logical matter, syllogistic, categore-matic term, syncategorematic term, material consequence, formal consequence
Introduction
The intuition of formality is a principle traditionally used to demarcate the boundaries of logic. While a variety of definitions of the formal have been suggested, this paper will use the dichotomy first mentioned by Edmund Husserl [24] who characterizes formal logic as both apophantic analytics and formal ontology. According to John Corcoran [12], logic as formal ontology investigates general aspects of reality while logic as
xThis study (research grant No. 14-01-0020) was supported by The National Research University-Higher School of Economics' Academic Fund Program in 20142015.
© Dragalina-Chernaya E.G.
formal epistemology describes the process of deduction. Catarina Du-tilh Novaes [18] distinguished the formal as pertaining to forms from the formal as pertaining to rules. However, 'the formal as pertaining to forms' seems tautological and 'the formal as pertaining to rules' appears too narrow. As a consequence, my approach is based on the distinction between substantial and dynamic models of formality (see [16], [17]). Dynamic formality pertains not only to rules, but also to purposes of actions. The dominant idea of this model stresses the dynamics of goal-directed activities. In essense, the distinction between the two models of the formal is based upon the dynamic of an action versus the static of a substance dichotomy rather than upon the form versus rules dichotomy.
The substantial hylomorphism presupposes the interpretation of the formal as an abstraction from matter. Alonzo Church writes:
Traditionally, (formal) logic is concerned with the analysis of sentences or of propositions and of proof with attention to the form in abstraction from the matter. This distinction between form and matter is not easy to make precise immediately [10, p. 1].
The variability of matter may concern terms or models (see [18]). Thus, various modifications of the substantial formality may be classified into two clusters, i.e. the formal as schematic (see [13], [30]) and the formal as model-theoretic invariance (see [6], [21], [22], [36]). In other words, the form of argument represents a scheme, in other words, a result of the substitution of all the non-logical terms with variables of the corresponding categories.
1. Form and matter in Aristotle's syllogistic
Logic is about the form, not the matter. Aristotle is the father of logic as a formal discipline. These axioms are veridical, but they are vague. It is generally accepted that the logical hylomorphism goes back to the Aristotelian form (morphe) versus matter (hyle) dichotomy. As Edmund Husserl tells us,
Aristotle substituted algebraic letters for the words (terms) indicating the material: that which is spoken about in the statements, that which de-
termines judgments as judgments relating to divers material provinces or single matters [24, p. 48].
Furthermore, according to Timothy Smiley,
Aristotle created mathematical logic by inventing its distinctive object of study, the formalized language [37, p. 1].
However, the role of Aristotle as the founder of logical hylomor-phism may be challenged. Although formal logic is traditionally traced back to Aristotle, he did not apply formality as a criterion for logicality. Moreover, Aristotelian form versus matter distinction is absent from the Organon (see [3, p. 39-40], [9, p. 8]). Aristotle applies this distinction to logic only twice: in Physics (195a18-19) and in Metaphysics (1013b19-20). The two passages are almost identical. Aristotle observes that the premises of an inference (hypotheses) are matter for the conclusion. These passages do not imply the logical hylomorphism because they say nothing about the logical form or the formal structure of the premises and the conclusion. Surprisingly, as John MacFarlane pointed out,
the father of both formal logic and hylomorphism was not the father of logical hylomorphism [30, p. 255].
The purpose of this section is to unravel the ground of this puzzle.
Aristotle's matter versus form dichotomy has a vast spectrum of the literature on the subject. However, Elizabeth Anscombe and Peter Geach write:
There is hardly a statement about form in the Metaphysics that is not (at least verbally) contradicted by some other statement [2, p. 75].
First, Aristotle was clear about the dichotomy between the matter and form of primary substances, but not of language entities. Second, he was not a mereological hylomorphist, that is, he did not take matter and form to be themselves parts of the whole they compose. For Aristotle, the form is not a part of a whole conjoined with its material parts but the essence of a being, the dynamic principle of its organization. As it was shown by Myles Burnyeat [9], in Metaphysics Aristotle distinguishes
between logical (logikos) and physical analyses. While logical analyses are abstract, the physical studies address to the concepts of matter and form as principles appropriate to the subject.
One immediate and obvious difficulty that we meet is a difficulty to explain why Aristotle uses letters of the alphabet, like 'A', 'B', 'C', instead of concrete terms if he did not distinguish between logical form and logical matter. Here is an exemplary formulation for the first syllogism in the first figure (Barbara) from the Prior Analytics:
if A belongs to every B and B belongs to every C, it is necessary for A to belong to every C (Pr. An., 25b37-9).
According to Jan Lukasiewicz, Aristotle's syllogisms are not inference schemata but conditional propositions. He understands Aristotle's 'schematic letters' as object language variables. Lukasiewicz wrote:
The introduction of variables into logic is one of the Aristotle's greatest inventions [29, p. 7].
Similarly, the editor of the Prior Analytics Gisela Striker notes that
the crucial innovation. . . that makes syllogistic a formal system is the introduction of letters as placeholders for the terms [39, p. xii].
In contrast, Arthur Prior was the first who claims that Aristotle's syllogisms are meta-theoretical statements about inferences ([33, p. 116], see also [7], [35]). From this perspective syllogisms are not conditional propositions (p&q) D r but metalanguage statements (p&q) r, where p, q, and r are categorical propositions (see [5]). According to Corcoran,
there is no need to postulate object language variables for Aristotle's system [11, p. 98].
For Corcoran, Aristotle's syllogistic is a theory concerned with the structure of inference, i.e. syllogistic proofs. He writes:
Aristotle nowhere refers to argument forms or propositional functions. All apparent exceptions are best understood as metalinguistic reference to 'concrete syllogisms' [11, p. 126].
As Corcoran tells us, Aristotelian grammar is too trivial while his semantics is complex enough to act as an analog to modern syntactic or semantic results. As he put it,
most of Aristotle's metasystematic results are proof-theoretic: they concern the relationship between the deductive system D and various subsystems of it [11, p. 113].
In fact, Aristotle is not interested in the syntactic structure or in the regimentations of the arguments. Aristotle does not employ a canonical language in his syllogistics (see [32]). The Prior Analytics contains different expressions for arguments, i.e. 'A is predicated of all B', 'A belongs to all B', 'A is in the whole of B' and 'A follows all of B'. Aristotle does not prescribe which expression to use. Any expression is allowable as long as it has the same meaning. For example, when Aristotle demonstrates that two premises 'M holds of every N' and 'M does not hold of some O' yields a conclusion 'N does not hold of some O' (Baroco), he adds:
And if M holds of every N but not of every X, then there will be a deduction that N does not hold of every X. (The demonstration is the same) (Pr. An., 27b1-3).
Since the Hellenistic period this small but often quoted fragment from the Prior Analytics drew special attention of commentators. Alexander of Aphrodisias (born at the end of the 2nd century A.D.), arguing against 'the moderns' ('the more recent thinkers', i.e. Stoics), asserts:
This is an argument of the sort which the more recent thinkers call sub-syllogistic: it takes something equivalent to the syllogistic premiss and deduces the same thing from it. ('Does not hold of some' has been transformed into 'does not hold of every', which is equivalent to it.) The more recent thinkers deny that such arguments are syllogisms, since they look to the words and the expression. Aristotle, however, looks to the meanings (when the same things are meant) rather than to the words, and says that the same syllogism is deduced when the expression of the conclusion is transformed in this way-granted that the conjunction is in general syllogistic [1, 84. 11-19].
As Corcoran writes,
it is doubtful that Aristotle ever conceived of a language apart from its intended interpretation. In other words, it seems that Aristotle did not separate logical syntax from semantics [11, p. 104].
The freedom of paraphrase which Aristotle allows himself in representing and interchanging syntactically different arguments with the same meaning implies Lukasiewicz's verdict:
Aristotelian logic is formal without being formalistic [29, p. 15].
But logic cannot be schematically formal without being formalistic. Thus, the Aristotelian schematic hylomorphism is a mirage. Aristotle's letters are not schematic, that is to say, they are not object language variables waiting to be filled by concrete terms but 'dummy letters' which might be given a meaning (see [11], [23], [26]). As Katerina Ierodiakonou pointed out,
the only difference between examples with letters and examples with terms such as 'animal', 'man', 'horse' lies in the fact that, obviously, only in the case of letters is it irrelevant what they actually stand for. That is to say, although propositions with letters are either true or false, propositions with terms such as 'animal', 'man', 'horse' have an identifiable truth-value [25, p. 137].
Although 'dummy letters' have meanings their use indicates that the truth-values of propositions do not affect the validity of syllogistic inference rules. The Aristotelian syllogistic is concerned with the formal relations between perfect and imperfect rules of inference rather than with the canonical structures of categorical statements. According to Aristotle,
all the imperfect syllogisms are made perfect by means of the first figure (Pr. An., 29a30).
His aim is not to create a formalized language as a canon for syllogistic reasoning but rather to provide a formal criterion for determining when no assumption of syllogisms is missing. All the perfect syllogisms of the first figure "are completed through themselves" (Pr. An., 29b6-8) while all the imperfect syllogisms of the second and the third figures "are
completed by taking in addition certain things" (Pr. An., 28a5-6, see also Pr. An., 29a15-16). To sum up, the Aristotelian reductive approach to patterns of inference (i.e. to syllogistic moods in the three figures) shifts focus from the schematic towards the dynamic model of formality.
2. Alexander of Aphrodisias on logical form and logical matter
One can find schematic interpretation of formality in the insightful comments to Prior Analytics by Alexander of Aphrodisias, known as the Exegetist. Alexander's commentaries show that by his time the logical matter versus logical form dichotomy was already thoroughly familiar (see [3]). Starting from the Aristotle's mould analogy, he writes:
The figures of the syllogism are like a sort of common matrix. You may fit matter into them and mould the same form for different matters. Just as, in the case of matrixes, the matters fitted into them differ not in respect of form or figure but in respect of matter, so too is it with the syllogistic figures [1, p. 48].
According to Alexander, Aristotelian schematic letters stand for the matter of the argument:
He uses letters in his exposition in order to indicate to us that the conclusions do not depend on the matter but on the figure, on the conjunction of the premises and on the moods. For so-and-so is deduced syllogistically not because the matter is of such-and-such a kind but because the combination is so-and-so. The letters, then, show that the conclusion will be such-and-such universally, always, and for every assumption [1, p. 116].
The Exegetist attributes to Aristotle the logically significant distinction between the variable matter and invariable form:
Combinations are called syllogistic and reliable if they do not alter together with differences in the matter, i.e. if they do not deduce and prove different things at different times, but always and in every material instance preserve one and the same form in the conclusion. Combinations which change and alter configuration together with the matter and acquire different and conflicting conclusions at different times, are non-syllogistic and unreliable [1, p. 114].
Thus, it was Alexander of Aphrodisias who first offered the invariance criterion for the logical form. However, much uncertainty still exists about the Alexander's theory of logical matter. He wanted to definitively connect logical form and logical matter with metaphysical form and matter. But since form is inseparable from matter logical form is also inseparable from logical matter. As it was shown by Kevin Flannery [23], Alexander treated the logical matter as occupying an intermediate position between things and form. For Alexander, the logical matter of a syllogism is determined by the scientific or dialectical discourse in which this syllogistical inference scheme is embedded.
3. Syncategoremata and the formal consequence in medieval logic
Logic was treated by the medievals as a scientia sermocinalis whose function was to describe the formal structure of language. The explicit schematic hylomorphism in medieval logic rests on two famous dichotomies: (1) categorematic terms (categoremata) versus syncate-gorematic terms (syncategoremata), and (2) material versus formal consequences. The medieval distinction between categoremata and syncategoremata goes back to Priscian (fl. 500 AD) who attributes the dichotomy to Peripatetic. Norman Kretzmann suggests that the "career of the syncategoremata" within the logica moderna falls into three stages: (1) their emergence (in the twelfth century, especially the latter half); (2) their identification as a separate treatises (from the last quarter of the twelfth century to the last quarter of the thirteenth); (3a) their assimilation into general treatises on logic; and (3b) their absorption into the sophisma-literature (from the first quarter of the fourteenth century to the disintegration of scholastic logic) (see [28, p. 215]). According to Ernest Moody, in the 14th century it became customary to call the categorematic terms the matter and the syncategorematic signs the form of propositions (see [31, p. 16-17]. Dutilh Novaes [ suggests, in turn, that
there were sporadic applications of the form vs. matter distinction to arguments in the medieval Latin tradition already in the twelfth century;
but later, the thirteenth century witnessed something of an explosion of uses of hylomorphism in logic [19, p. 345].
In any case, for the fourteenth-century logician John Buridan (d. c. 1358), categorematic terms refer to the matter of a proposition or consequence whereas all the rest, i.e. syncategorematic terms, refers to form (see [8, I.7.2]). He writes:
A formal consequence is one that holds for all terms retaining the same form, or if you wish to speak carefully. . . for which any equiform proposition which might be formed would be an acceptable consequence. For example, 'That which is A is B, so that which is B is A'. . . A material consequence is where not every proposition of the same form is valid,.. e.g., 'A man runs, so an animal runs', because it is not valid with these terms: 'A horse walks, so wood walks'. . . No material consequence is evident except by reduction to a formal consequence by the addition of some necessary proposition [8, I.4].
By consequence Buridan meant not the relations between propositions but implied proposition, i.e. implication:
Now a consequence is a molecular proposition, for it is composed from several propositions conjoined by the expression 'if' or by the expression 'therefore' or something similar [8, I.3].
On the contrary, Robert Fland , writing in Oxford around 1350, considered all the analytic consequences as formal ones (see also [34]):
General rules are given in order to appreciate when an inference is formally valid. The first is this: where the conclusion is formally understood in the premises. For example, this inference is formally valid: 'There is a man, so there is an animal' because the conclusion 'animal' is formally understood in the premise, namely, 'man' [38, p. 57].
Presumably, the 'formal understanding' of the conclusion in the premises implies the transcendental relation between the premises and the conclusion. According to scholasticism, the transcendental relation is 'anchored' in the essences of relata. For example, it is impossible for God
to create a man without creating an animal. For Rene Descartes, conversely, eternal truths do not limit God's perfection, but only our ability to understand God's perfection. In his second letter to Mersenne Descartes writes:
The eternal truths... are not known as true by God in any way which would imply that they are true independently of him [14, 3:24].
From this point of view, to discuss what is possible or impossible for God is a wrong way of doing logic (see [15]). The Paris school did not use the vague concept of 'formal understanding' but it is based on the obscure distinction between meaningful categorematic and meaningless syncategorematic terms. More than fifty different words were considered as syncategorematic terms by Latin medieval authors (see [28, p. 212]). From the beginning of the centuries-old debate, philosophers who dealt with syncategoremata explicitly offered syntactic account of them. According to syntactic criterion, syncategorematic terms cannot be subjects or predicates of propositions. Thus, the medieval syntactic approach is limited by non-universal syllogistical assumption that every proposition has one subject and one predicate. Moreover, a syncategore-matic term may be a subject when it is used as an autonymous symbol (e.g., 'No is an adverb' or 'And is a copulative conjunction'). The me-dievals try to get over the difficulty by focusing on the significative (i.e. non autonymous) use of the subject and predicate as the matter of the mental proposition. Albert of Saxony who taught at Paris from 1351 to 1362 writes in his Perutilis logica ('Very Useful Logic'):
A categorematic term is said to be one which, taken significatively, can be a subject or a predicate, or a part of the subject or a part of the distributed predicate, of a categorical proposition. For example, these terms 'man', 'animal', 'stone', are called categorematic terms because they have a definite and determinate signification. A syncategorematic term, however, is said to be one which, taken significatively, cannot be the subject or the predicate, nor a part of the subject nor a part of the distributed predicate, of a categorical proposition [31, p. 16].
Therefore, the medieval logicians have to distinguish semantically between different uses of terms, i.e. between different kinds of suppositions
(see [27]). In addition, according to medieval semantic criterion, syn-categoremata have no meaning. They signify nothing outside the mind but merely co-signify, that is to say, they modify the semantic functions of categorematic terms. For Buridan, a mental proposition is generated by adding a form, i.e. a complexive concept (conceptus complexiuus) signified by a copula, to a pair of simple concepts, that is, to a subject and a predicate. As is the case with physical things, the form of a mental proposition cannot exist without matter (see [40]). However, in modern model-theoretic semantics, 'syncategorematic' terms receive independent semantic values (e.g., generalized quantifiers are interpreted as sets of subsets of the domains).
4. Conclusions
Logical hylomorphism is based upon the distinction between logical form and logical matter. Curiously enough, the status of logical form was not exactly determined in ancient and medieval logic. On the one hand, the logical matter versus logical form dichotomy cannot be considered as an extrapolation of metaphysical form versus matter dichotomy. On the other hand, ancient and medieval authors have failed to propose a coherent syntactic or semantic criterion for the demarcation of logical terms and formal consequence. Shifting focus from the static of a substance towards the dynamic of an action offers some important insights into the demarcation of the bounds of logic as a formal art.
References
[1] Alexander of Aphrodisias. "On Aristotle's Prior Analytics 1.1-7", trans. by J. Barnes, S. Bobzien, K. Flannery, and K. Ierodiakonou; London: Duckworth, 1991. 252 pp.
[2] Anscombe, G.E.M. and P.T. Geach. "Three Philosophers: Aristotle, Aquinas, Frege". Oxford: Blackwell, 1961. 162 pp.
[3] Barnes, J. "Logical Form and Logical Matter", Logica, Mente e Persona, ed. by A. Alberti. Florence: Leo S. Olschki, Editore, 1990, pp. 1-119.
[4] Beziau, J.-Y. "What is 'Formal Logic'?", Proceedings of the XXII World Congress of Philosophy, Vol. 13, Logic and Philosophy of Logic, 2008, pp. 9-22.
[5] Bocharov, V.A. and V.I. Markin. "Sillogicheskie teorii" [Syllogistic theories]. Moscow: Progress-traditsiya Publ., 2010. 336 pp. (In Russian)
[6] Bonnay, D. "Logicality and Invariance", Bulletin of Symbolic Logic, 2008, Vol. 14, pp. 29-68.
[7] Boger, G. "Aristotle's Underlying Logic", Handbook of the History of Logic, eds. by D.M. Gabbay, J. Woods, and A. Kanamori. Elsevier, 2004, pp. 1-101.
[8] Buridan, J. "Tractatus de Consequentiis", ed. by Hubert Hubien. Lou-vain: Publications Universitaires, 1976. 138 pp.
[9] Burnyeat, M. "A Map of Metaphysics Zeta". Pittsburgh: Mathesis Publications, 2001. 176 pp.
10] Church, A. "Introduction to Mathematical Logic". Princeton: Princeton University Press., 1996. 392 pp.
11] Corcoran, J. Aristotle's Natural Deduction System, Ancient logic and its modern interpretations, ed. by J. Corcoran. Boston, 1974, pp. 85-132.
12] Corcoran, J. "The Founding of Logic", Ancient Philosophy, 1994, Vol. 14, pp. 9-24.
13] Corcoran, J. "Schemata: The Concept of Schema in the History of Logic", The Bulletin of Symbolic Logic, 2006, Vol. 12, No. 2, pp. 219-240.
14] Descartes, R. "The Philosophical Writings of Descartes", eds. by J. Got-tingham, R. Stoothoff, and D. Murdoch. Cambridge: Cambridge University Press, 1991, Vol. 3. 432 pp.
15] Dragalina-Chernaya, E. "L'interprétation performative du Cogito cartésien", Cahiers de philosophie de l'Université de Caen, 2013, Vol. 50, pp. 121-139.
16] Dragalina-Chernaya, E. "Logical Hylomorphism Revisited", Philosophy, Mathematics, Linguistics: Aspects of Interaction 2014 (PhML-2014). St. Petersburg: The Euler International Mathematical Institute, 2014, pp. 6-11.
17] Dragalina-Chernaya, E. "Neformal'nye zametki o logicheskoi forme" [Informal notes on logical form]. St. Petersburg: Aleteiya, 2015. 202 pp. (In Russian)
18] Dutilh Novaes, C. "The Different Ways in which Logic is (Said to Be) Formal", History and Philosophy of Logic, 2011, Vol. 32, pp. 303-332.
19] Dutilh Novaes, C. "Reassessing logical hylomorphism and the demarcation of logical constants", Synthese, 2012, Vol. 185, No. 3, pp. 387-410.
[20] Dutilh Novaes, C. "Form and Matter in Later Latin Medieval Logic: The Cases of Suppositio and Consequentia", Journal of the History of Philosophy, 2012, Vol. 50, No. 3, pp. 339-364
[21] Dutilh Novaes, C. "The Undergeneration of Permutation Invariance as a Criterion for Logicality", Erkenntnis, 2014, Vol. 79, No. 1, pp. 81-97.
[22] Feferman, S. "Set-theoretical invariance criteria for logicality", Notre Dame Journal of Formal Logic, 2010, Vol. 51, pp. 3-20.
[23] Flannery, K. L. "Ways into the Logic of Alexander of Aphrodisias". Leiden: Brill, 1994. 170 pp.
[24] Husserl, E. "Forma,l and Transcendental Logic". Dordrecht: Springer, 1969. 340 pp.
[25] Ierodiakonou, K. "Aristotle's Use of Examples in the 'Prior Analytics"', Phronesis, 2002, Vol. 47, No. 2, pp. 127-152.
[26] Kirwan, C. "Logic and Argument". New York: University Press, 1978. 299 pp.
[27] Klima, G. "Syncategoremata", Encyclopedia of Language & Linguistics, Second Edition, ed. by K. Brown. Oxford: Elsevier, 2006, Vol. 12, pp. 353-356.
[28] Kretzmann, N. "Syncategoremata, exponibilia, sophismata", Cambridge History of Later Medieval Philosophy,eds. by N. Kretzmann, A. Kenny & J. Pinborg. Cambridge, 1982, pp. 211-245.
[29] Lukasiewicz, J. "Aristotle's Syllogistic from the Standpoint of Modem Formal Logic", 2nd ed., Oxford, 1957. 222 pp.
[30] MacFarlane, J. "What does it mean to say that logic is formal?" PhD dissertation, Pittsburgh University, 2000.
[31] Moody, E. "Truth and Consequence in Mediaeval Logic", North-Holland, 1953. 113 pp.
[32] Morison, B. "What was Aristotle's Concept of Logical Form?", Episteme, etc. Essays in Honour of Jonathan Barnes, eds. by B. Morison & K. Ierodiakonou. Oxford: Oxford University Press, 2012, pp. 172-88.
[33] Prior, A. N. "Formal Logic", 2nd ed., Oxford, 1962. 341 pp.
[34] Read, S. "The Medieval Theory of Consequence", Synthese, 2012, Vol. 187, pp. 899-912.
[35] Rose, L. E. "Aristotle' s Syllogistic". Springfield, 1968. 148 pp.
[36] Sher, G. "The Foundational Problem of Logic", The Bulletin of Symbolic Logic, 2013, Vol. 19, No. 2, pp. 145-198.
[37] Smiley, T. "The Schematic Fallacy", Proceedings of the Aristotelian Society, 1982-3, Vol. 83, pp. 1-17.
[38] Spade, P. "Robert Fland's consequentiae: An edition", Mediaeval Studies, 1976, Vol. 38, pp. 54-84.
[39] Striker, G. "Aristotle's Prior Analytics: Book 1". Oxford: Clarendon Press, 2009. 268 pp.
[40] Thom, P. "Logical Form", The Oxford Handbook of Medieval Philosophy, ed. by J. Marenbon. Oxford University Press., 2012, pp. 271-288.
