Three Arguments for the Ideas:
1. In the Phaedo, Socrates suggests that true knowledge is only possible of supersensible beings uncontaminated by “all contact and association with the body.” (66a-67b) In support of this possibility, Plato presents the One-Over-Many Argument (74a) that knowledge of any term, such as ‘equality’, requires some recognition of the perfect exemplar, or paradigm, of that term. Socrates comments that there is one single perfect paradigm for the "various multiplicities to which we give the same name." (Reb. 596a) The many sensible instances of equality, such as between “stick to stick and stone to stone”, are not perfectly equivalent, but at best only adequate imitations that imperfectly approximate this paradigm. (74e) Sensible instances of equality are imperfect because of the infinite divisibility of sensible magnitudes: since, any magnitude of either extension in space or the duration in time may be infinitesimally subdivided, and afterwards inconspicuously increased or diminished, the possibility that sensible objects are less than perfectly equal can never be completely guaranteed. Since, then, all sensible objects are possibly unequal, each may be imperfect and none can be certainly known to be perfectly equal. The perfect paradigm of equality is, however, distinct from all sensible instances of equality simply because it is, not a sensible object, but a totally supersensible Idea. (74c)
2. To explain the possibility of knowledge of perfect paradigms beyond all sensibility, Plato presents the Transcendental Argument. Transcendental arguments claim that some antecedent condition is necessary for the possibility of a consequence, and since the consequence is assumed to be true, the antecedent necessary condition must also be true. Socrates conjectures: “we must somewhere have acquired the knowledge that there is such a thing as absolute equality. Otherwise we could never have realized, by using it as a standard for comparison, that all equal objects of sense are desirous of being like it, but are only imperfect copies.” (75b) Since we assume it is possible to use the concept of equality as a standard of comparison between equivalent things, but this would be altogether impossible without knowledge of the perfect paradigm of equality, we must necessarily assume knowledge of the Ideas. Yet since knowledge of the Ideas of “beauty, goodness, and virtue” cannot be acquired from sensible objects in space and time, it must be somehow acquired from an innate and immediate mental intuition. (75d) Socrates thus describes, in the Theaetetus (187a), that “we have progressed so far, at least, as not to seek for knowledge in perception at all, but in some function of the soul.”
3. In the Republic, Socrates further reifies the Ideas by presenting an Argument from Science. Scientific knowledge is defined, in terms reminiscent of Parmenides, as "naturally related to that which is, to know that and how that which is is" (477b) and "the condition of that which is." (478a). Opinion is, to the contrary, distinguished as that which "partakes of both, of what is, and of what is not." (478e) Where scientific knowledge describes the existence and conditions of real beings, opinion merely describes the inconsistent mixture of being and non-being. Since the objects of scientific knowledge are pure beings, with no admixture of non-being, and - following Parmenides - being is prior to non-being, the objects of scientific knowledge must be logically prior and onto-logically superior to the shadow-play of opinions. Socrates thus calls the men who "have opinions about all things but know nothing of the things they opine" the lovers of opinion, or doxaphilists, but the men who "contemplate the very things themselves in each case, ever remaining the same and unchanged" the lovers of wisdom, or philosophers. (479e-480a)
Aristotle's lost work On Ideas is reported by Alexander of Aphrodisias to have summarized these three arguments for Plato's theory of Ideas: (a) the One-Over-Many Argument (80.8-81.22) states that if many subjects truly share the same predicate, then this shared predicate must be one perfect paradigm that is logically prior to the various instances in which it is predicated of many other subjects; (b) the Transcendental Argument (also called the Argument from Thought) (81.25-82.7) states that the possibility of thinking any term of thought necessarily requires this object to invariantly endure as one everlasting Idea; and (c) the Argument from Science (79.3-80.6) states that, since truth is a correspondence between propositions and reality, and every scientific deduction begins with one axiomatic proposition, the truth of every axiom, and the very possibility of scientific knowledge, necessarily requires real Ideas. (cf. Fine 1993) These three arguments are cumulative and produce a single theory of Ideas: (a) the One-Over-Many Argument argues from many semantic predicate instances to the epistemic concept of one possible universal paradigm; (b) the Transcendental Argument argues from the possibility of this universal paradigm to the necessity of an everlasting Idea; and (c) the Argument from Science argues from the logical necessity of the everlasting Idea for any scientific knowledge to the ontic reality of the Ideas. The arguments for Ideas thus lead the “soul’s ascension” (517b) from (a) possible semantics, to (b) necessary epistemic concepts, to (c) the reality of the Ideas themselves.
See my lecture Plato, Logic, and Ontology for more details:
Six Criticisms against the Ideas:
1. Extent of the Ideas Argument (130a-e): Can there be an Idea of ignoble things like mud? Yes, there are Ideas of all predicate-properties, including muddiness.
2. Part-Whole Dilemma Argument (130e-131e): Are Ideas whole or parts? No, Ideas are neither parts nor wholes because partness and wholeness are spatial, topological, or mereological concepts that do not properly apply to non-spatial simple Ideas.
3. Third Man Argument(132a-b): If there is one universal Idea for every plurality of particular predicate instances, and this universal Idea is self-predicated, then must there not be an infinite regress of universal Ideas over all self-predicated universal Ideas? No, the participation-relation of universal Ideas to particular instances is an intrinsic 'tree-type' or pros heauto predication in which the universal Idea is self-predicated without any real distinction requiring additional universal Ideas. (See part III of my essay Plato's Contest)
4. Conceptualism Argument (132b-c): If all universal Ideas are concepts, then are not the particular instances of universal Ideas also concepts? Yes, the particular instances are analogous to concepts of universal Ideas in Mind, or Nous.
5. Resemblance Regress Argument (132c-133a): Can the Third Man Argument be avoided by reformulating universal Ideas as resemblances of common meanings? No, the same regress results regardless of whether one common meaning is substituted for one universal Idea, but the Third Man Argument can be easily answered by distinguishing 'tree-type' pros heauto from 'garden variety' pro ta alla predication.
6. Greatest Difficulty Argument (133a-134a): If universal Ideas and particular instances are two distinct sets, and these sets are not related, then how can we have knowledge of gods and Ideas, or the gods have knowledge of humans and appearances? Platonic participation implies that the set of particular instances are participants of the being of the universal Ideas. Yet the distinct sets of universal Ideas and particular instances results in a paradoxical set that both includes and doesn't include itself. This paradox can possibly be answered by the construction of a set theoretic hierarchy from the dialectical hypotheses of the second part of the Parmenides. (See part IV of my essay Plato's Contest)