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 (τελείως κατηγορια) means a proposition having both a clear subject and predicate» [9, p. 74-75]. Porphyry also uses the concept of incomplete predicate (ἐλλιπὴς κατηγορία).

Describing Aristotle’s categorical logic, Porphyry supposes that terms and predicates can be used in combination or separately. The question raised by a pupil: tίνα ἐστὶ τὰ κατὰ συμπλοκήν; «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 (τελείως κατηγορια), which significantly affects the further development of traditional logic. Thus Porphyry clarifies, which statements can be considered as judgments: τὰ ἐκ τινῶν κατηγοριῶν τελείων δυοῖν ἢ καὶ πλειόνων συγκείμενα, οἷον ‘ἄνθρωπος τρέχει’, ‘ἄνθρωπος ἐν Λυκείῳ περιπατεῖ’, «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 (ἐλλιπὴς κατηγορία). It is the type of predicate, which Porphyry, following Aristotle, describes that they are going without any relation: τὰ μὴ οὕτως ἔχοντα οἷον τὰ ὁμώνυμα πάντα, τὰ συνώνυμα, τὰ παρώνυμα, «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 (ἐλλιπὴς).

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: τὰ δὲ τοιαῦτα οἷον λιθολογεῖ, βουκολεῖ, ψευδοδοξεῖ, κατὰ συμπλοκήν ἐστιν ἢ χωρὶς συμπλοκῆς; «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: «оὐδὲν γὰρ αὐτῶν ἐκ τελείων συνέστηκε κατηγοριῶν».  Interestingly, Porphyry’s examples, λιθολογεῖ, βουκολεῖ, ψευδοδοξεῖ, «are each single finite compound verbs; each can stand as a complete sentence without a further specified grammatical subject» [9, p. 75]. For example, λιθολογεῖ here derived from λῐθολογέω, «build with unworked stones», ψευδοδοξεῖ from ψευδοδοξέω, «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. βουκολεῖ, 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 (πρότασις). However, when the pupil asked him about the reasons for this, Porphyry notes: ὅτι καθ’ ἑαυτὴν οὐδεμία κατηγορία πρότασίς ἐστιν, ἀλλὰ τῇ ποιᾷ συμπλοκῇ πρότασις γίνεται, «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 Interpretatione» about the relationship between judgment and truth conditions: πᾶσα γὰρ πρότασις ἤτοι ἀληθής ἐστιν ἢ ψευδής, ἑκάστη δὲ κατηγορία καθ’ ἑαυτὴν οὔτε ἀληθὴς οὔτε ψευδὴς ἐλλιπὴς οὖσα, «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: οὐκ ἂν εἴποις τὸ ζῶ ἢ περιπατῶ ἢ ὕει χωρὶς συμπλοκῆς ὄντα ἢ ἀληθῆ ἢ ψευδῆ; «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: ναὶ εἴποιμι· ἀλλὰ καὶ ταῦτα δυνάμει μετὰ συμπλοκῆς ἐστιν, εἰ καὶ μὴ μετὰ φωνῆς· τὸ γὰρ ζῶ ἴσον ἐστι τῷ ἐγὼ ζῶ, καὶ τὸ ὕει ἴσον ἐστὶ τῷ ὁ Ζεὺς ὕει, «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 ζῶ and περιπατῶ differ from ὕει, as well as from all previous examples used by Porphyry, only because they belong to the 1st person singular, while ὕει – belongs to the 3rd person singular. Porphyry makes explicit verbal predicate ζῶ, separating it from the pronoun ἐγὼ, forming a judgment ἐγὼ ζῶ. From a logical point of view, ζῶ is equivalent to the judgment ἐγὼ ζῶ. Similarly, with the predicate περιπατῶ, where we have virtually a complete judgment ἐγὼ περιπατῶ. 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 «ζῶ» shows a logical and grammatical phenomenon, when the function and argument are not separated from each other, although quite distinguishable. Thus, ζῶ and περιπατῶ have a full-fledged range of values {t, f}, which shows that they are real judgments. Regarding ὕει the situation is similar, because this verb as Porphyry says, potentially (δυνάμει) 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.