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64ФИЛОСОФСКИЕ НАУКИ (PHILOSOPHY)
УДК 16 (09)
Garin S.V. Incomplete Predicates in Porphyry's Logic and
Frege's Functions
Garin Sergey Vyacheslavovich Kuban State University, Krasnodar
Abstract: The article deals with the problematic aspects of logical theory of judgment in the context of ancient theory of predication. The concept of incomplete predication (¿AArnqc KainYopia) is compared with Frege's doctrine of «unsaturated» (ungesättigt) functions. The work highlights a number of aspects of ancient theory of logical judgment and predication.
Keywords: history of logic, predication, Porphyry, Frege, semantics
When Frege had developed the theory of functions and objects in the context of predication, he certainly raised a number of issues addressed in logic since Aristotle. In particular, Frege pointed out that «what in the case of a function is called unsaturatedness, we may in the case of concept call its predicative nature [4, p. 129]. In Frege's theory, objects may be considered as complete entities, while functions are certainly dependent and incomplete. Frege uses the term «ungesättigt» (unsaturated) to convey the fact that the function must have an argument in order to get a value.
Notoriously, Frege's theory of incomplete predicates was a consequence of his position that the judgment is logically prior to concepts: «Frege's claim that concepts are unsaturated (ungesättigt) was first made in his letter to Marty of 29 August 1882, and was connected with his view that judgments are logically prior to concepts» [5, p. 113].
In fact, Frege has reinterpreted the classical Aristotle account on the nature of truth and logical judgment. Unsaturated predicate, by Frege, determined by its incompleteness as a function: «to form a predicate, one must at least associate the copula 'is' to the term. 'Is Venus' is a predicate; it is 'unsaturated' by virtue of its predicative nature. The concept denoted by the predicate inherits its unsaturated character» [8, p. 82].
Thus in modern logic, at any rate, in its Fregean vector, the nature of predicate functions has been seriously discussed. However, this problem has a long history in classical logic. The problem of «incompleteness» of predicates was firstly formulated by
ancient authors. Porphyry, in his commentary on the «Categories» considered the difficulties of the theory of Aristotelian predication in the context of what Frege later called «unsaturated functions». Although Frege attempts to eliminate predicates and subjects, usual for traditional logic, replacing them with the algebra of functions, nevertheless, he uses the concepts that are very similar to the Porphyrian terms. Thus, according to Porphyry, «a complete predication (i£A£i«c KainYopia) means a proposition having both a clear subject and predicate» [9, p. 74-75]. Porphyry also uses the concept of incomplete predicate (¿AArnqc KainYopia).
Describing Aristotle's categorical logic, Porphyry supposes that terms and predicates can be used in combination or separately. The question raised by a pupil: tiva ¿öii iä Kaiä au|inAoKqv; «which are said with combination?» [3, p. 87] (Here and after English translation is by S. K. Strange). Porphyry answering, puts forward the Aristotelian concept of complete predication (i£A£i«c KainYopia), which significantly affects the further development of traditional logic. Thus Porphyry clarifies, which statements can be considered as judgments: iä ¿k iivöv KainYopiöv i£A£i«v Suoiv q Kai nA£iov«v öUYK£i|£va, oiov 'ävOpwno; ip£x£i', 'av0p«no; ¿v AuK£iw n£pmai£i', «those that are composed of two or more complete predications, for example, 'a man runs' or 'a man is walking in the Lyceum'» [3, p. 87].
In this passage we see that the complete predicate function of judgment, according to Porphyry, has at least a binomial structure. Predicate functions which are less than binomial, are incomplete predicates (¿AArnqc KainYopia). It is the type of predicate, which Porphyry, following Aristotle, describes that they are going without any relation: iä ouiw; exovia oiov iä ö|6vu|ia navia, iä öuvwvu|ia, iä napwvu|ia, «those that are not of this type, for example all homonyms, synonyms, and paronyms» [3, p. 87].
In this case, Porphyry proposes a predicate function without subjects as an example of incomplete predication. In terms of Frege, we have here a functional expression without arguments, such as «2x + 3" or «6 ( ) — ( )», whose values are uncertain. Our inability to obtain the value of that function makes them, according to Frege «ungesättigt», (unsaturated), or by Porphyry, incomplete (¿AAinqc).
In Porphyry's question-answer commentary, a pupil raises the problem of incomplete predication using new quite offbeat examples. In particular, the question was to clarify a number of expressions: iä 5e loiaöia oiov Ai0oAoY£i, ßouKoA£i, ^£u5o5o^£i, Kaiä öu^nAoKqv ¿öiiv q x^pic öU|nAoKqc; «what about expressions like 'stonebuilder', 'cowherd', and 'false-believer'? Are they said with combination or without combination?» [3, p. 87]. Porphyry responds negatively. These concepts don't belong to the class of complete predicates, because they have no predicate relation: «ouöev Yäp auiwv ¿k i£A£iwv öuv£öinK£ KainYopiöv». Interestingly, Porphyry's examples, Ai0oAoY£i, ßouKoA£i, ^£u5o5o^£i, «are each single finite compound verbs; each can
stand as a complete sentence without a further specified grammatical subject» [9, p. 75]. For example, Ai0oAoY£i here derived from aiooaoye«, «build with unworked stones», ^£u5o5o^£i from ^£u5o5o^£«, «entertain a false opinion or notion». These expressions belong to the same grammatical category. Porphyry does not accept these expressions as a complete predicate, despite the fact that they belong to the 3rd person singular verbs. It is known that Ancient Greek verbs, due to their nature, may be considered as complete sentences because «Greek verbs have the properties of person, number, voice, mood, and tense. Related to tense are the grammatical concepts of aspect and time» [6, p. 13]. Thus, the predicate verb, e.g. poukoa£i, contains a 3rd person singular pronoun. However, according to Porphyry, these examples «nevertheless do not count as expressions 'said with combination', because they need to have their subject specified in order for the proposition they express to be fully understood» [9, p. 75].
Considering the logical boundaries of predication, Porphyry uses the Aristotelian division of categories as that from which the judgment may be formed. Thus, considering various combinations of categories, Porphyry shows that the result is always concrete - the formation of judgment (npoiamc;). However, when the pupil asked him about the reasons for this, Porphyry notes: oil Ka0' £auTqv ou5£|ia KaTnYopia npoiaai; £aiiv, aAAa ifl noia au|nAoKfl npoiaai; Yiv£Tai, «because no predicate by itself is a proposition resulting from a certain sort of combination of such predicates» [3, p. 87]. Porphyry further develops the Aristotelian doctrine from «De Interpretation» about the relationship between judgment and truth conditions: naaa Yap npoiaai; htoi aAq0nc £aiiv q ^£u5qc, £K&ain 5£ KaiqYopia Ka0' £auiqv oui£ aAq0nc oui£ ^£u5qc £AArnqc otiaa, «every proposition is either true or false, but no predicate by itself is either true or false, since it is incomplete» [3, p. 87]. The incompleteness of the predicate as it was aforementioned is related to our inability to evaluate the truth conditions of the predicate.
Developing the theme further, the pupil asks the question connected with the lexical and grammatical problems of predication: ouk av £moi; to q n£pinai« q u£i X«pi; au|inAoKqc ovTa q aAq0q q ^£u5q; «would you not say that 'alive', 'walking', and 'raining' are true or false, even though they are things said without combination?» [3, p. 87]. It is noteworthy that Porphyry, despite his negative answer to the previous example, in this case agrees with the student, interpreting verbal predication quite differently: vai £inoi|v aAAa Kai TauTa 5uva|£i |£Ta au|inAoKqc £otiv, £i Kai |iq |£Ta ^«vq; to Yap i'aov £oti t« £y« Kai to u£i iaov £aTi t« o z£u; u£i, «Yes I would, but each of these is implicitly said with combination, even if this is not expressed in words. For 'alive' is equivalent to 'I am alive', and 'raining' is equivalent to 'It is raining'» [3, p. 87].
Predicates and n£prnaT& differ from u£i, as well as from all previous examples used by Porphyry, only because they belong to the 1st person singular, while
Ü£i - belongs to the 3rd person singular. Porphyry makes explicit verbal predicate Z&, separating it from the pronoun ey«, forming a judgment ey« From a logical point of view, is equivalent to the judgment ey« Z«. Similarly, with the predicate nEpinai«, where we have virtually a complete judgment ey« nEpmara. Speaking in terms of Frege, in Porphyry's verbal predicates we have such an expression of the function, which already includes the argument. Of course, this is not the case when the function includes its own argument. As we know, Wittgenstein formulated an alternative to Russell's theory of types, based on the fact that the function cannot be its own argument. Porphyry's example of the judgment «Z«» shows a logical and grammatical phenomenon, when the function and argument are not separated from each other, although quite distinguishable. Thus, Z« and nEpmara have a full-fledged range of values {t, f}, which shows that they are real judgments. Regarding ÜEi the situation is similar, because this verb as Porphyry says, potentially (5uvá|Ei) contains all the other parts of the judgment - namely the subject (the singular 3rd person pronoun).
These aspects of Porphyry's logic, in our opinion, require a separate comprehensive study.
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