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[From The Continuum Companion to Plato, ed. by G. A. Press. New York: Bloomsbury

Publishing, 2012, 192-194.]

Plato’s Logic

Charles M. Young Department of Philosophy

Claremont Graduate University

Logic (following Quine 1986) is the systematic study of logical truth (or,

equivalently, valid argument). A logical truth is a sentence true in virtue of its logical

form. Logical form is relative to a set of grammatical categories. Logical systems may be

classified in terms of the categories they take to matter for logical form. Since Frege 1879,

the set of categories that matter begins with sentence and sentential operator, definitive of

sentential logic, and goes on to include term, predicate, and quantifying expression,

definitive of predicate or quantificational logic. From there one may go on, in various

ways, to develop temporal, modal, conditional, relevantist, and intuitionist systems. (See

Burgess 2009 for a helpful survey and sharp assessment of such developments.)

Plato has the ideas of logical form and logical truth. Republic 436b9, for example,

affirms:

Exclusion: It is not possible for the same thing to do or to suffer contraries in

the same respect, in relation to the same thing, or at the same time.

This is a kissing cousin of our:

Non-Contradiction: It is not possible for both a sentence and its negation to be

true.

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But whereas Non-Contradiction appeals to the grammatical category of sentence to tell

us, e.g., that if ‘Helen is beautiful’ is true, then ‘It’s not the case that Helen is beautiful’ is

not, Exclusion appeals to the grammatical category of contrary to tell us, e.g., that if

‘Helen is beautiful’ and ‘Helen is ugly’ are both true, then she is beautiful and ugly in

different respects (appearance vs. character, say), in relation to different things

(Quasimodo vs. Aphrodite, say), or at different times (youth vs. old age, say). The

divergence between Plato and Frege emerges sharply if we focus on (e.g.) ‘If justice is a

virtue, then injustice is a vice,’ presumptively a logical truth in a logic of contrariety, but

not in any obvious way a logical truth in sentential or predicate logic.

Plato wrote dialogues and not treatises, and he does not engage in any systematic

study of logical form and logical truth, in the manner, say, of Aristotle’s Prior Analytics

(or even the Topics), and thus cannot be said to have a logic in the sense specified above

(and hence ‘logic’ does not appear in the index of Kraut 1992). What we get instead is a

congeries of insights and speculations about what might matter for logical form and

logical truth, often expressed in difficult Greek, that may or may not be capable of

systematization. To give a sense of what we have and the problems we face, I take up three

texts that deal in abstract and various ways with predication.

Socrates says at Euthyphro 6e3-6 that if he learned what the idea of piety is, he

would be able to use that idea as a standard and say that an action or person conforms to

[it] is pious. Here conforms to [it] is a stab at translating the Greek toioutos, a

combination of the indefinite pronoun toios ( = of some sort) and the demonstrative

pronoun houtos ( = this); other tries include of that kind (Grube 2002), resembles [it]

(Cooper 1974), and agrees [with it] (Fowler 1999). Whatever the translation, the idea

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seems clear enough. Socrates thinks that if

1. Piety is (say) doing what the god wants done,

and if Socrates is conforms to [it] – if, that is,

2. Socrates is doing what the god wants done,

then it follows that

3. Socrates is pious.

If we note that ‘is’ marks identity in (1) and predication in (2) and (3), and that ‘doing’ is

a gerund in (1) and a participle in (2), then it follows that

4. If piety is doing what the god wants done, and Socrates is doing what the

god wants done, then Socrates is pious.

This evidently expresses a logical truth connecting noun phrases mentioning properties

with terms mentioning things that have those properties.

So far, so good. Consider now Charmides 169e1-5, where Critias, with Socrates’

endorsement, states that ‘[one] conforms to what he has: he who has quickness is quick, he

who has beauty is beautiful, he who has knowledge will know, and he who has knowledge

that is of itself will know himself.’ Here ‘[one] conforms to what he has’ is very close to a

generalization of the logical truth of the Euthyphro. But two points might give one pause.

First, the inference from ‘[one] conforms to what he has’ to ‘he who has quickness is

quick’ arguably presupposes ‘quickness is quick,’ not unlike this English exchange: –

‘You’ll have to be quick to do that’ – ‘Quick? I’m quickness itself!’ But quickness arguably

is not the sort of thing that can be quick. Second, we might balk, though again Socrates

does not, at the inference to ‘he who has knowledge that is of itself will know himself.’

Anyone still on the bus will surely get off faced with Republic 438a7-b2:

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‘Regarding things that are such as to be of something, those that are of certain sorts are of

something of a certain sort, and those that are just themselves are of something that is just

itself,’ a principle that appeals to the grammatical category of relative term to support the

thought that thirst as such is for drink as such, and not for good drink, even though all

desire is for the good. The translation in Shorey ‘But I need hardly remind you’ (of the

particles alla mentoi with which Socrates introduces the principle) anticipates the

incomprehension that Glaucon immediately expresses. The principle requires over forty

lines for its elucidation.

Whether Plato’s various scattered remarks on contrariety, predication, and

relation can be brought under control and organized into a coherent system of ideas

remains to be seen. Some worthwhile work has been done – e.g., Robinson 1966, Smith

1973, Johnson 1977, Lloyd 1992, and, most recently and systematically, Dancy 2007. But

there is still much to do.

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Bibliography

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Cooper, L., 1974. Plato on the Trial and Death of Socrates. Ithaca.

Dancy, R. M., 2007. Plato's Introduction of Forms 1st ed., Cambridge.

Frege, G., 1879. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des

reinen Denkens. Halle.

Johnson, T. M., 1977. Forms Reasons, and Predications in Plato’s Phaedo, unpublished

PhD dissertation, Claremont Graduate School, Claremont, CA.

Kraut, R., 1992. The Cambridge Companion to Plato. Cambridge.

Lloyd, G. E. R., 1992. Polarity and Analogy: Two Types of Argumentation in Early Greek

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Sprague, R. K. 1962. Plato’s Use of Fallacy. London.