Rediscovering Plato and the Mystical Science of Dialectic
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This multitude of contradictions threatens to trivialize the dialectical exercises and render them totally incapable of consistently of answering Parmenides' challenge. Contradictions can, however, be resolved in either a hard or a soft way: the hard way is to reject one of the contradictory propositions e. The soft solution is to show how, because the contradictions are merely an apparent degree of contrary opposition, the more contrary hypotheses may be subsumed by the less contrary hypotheses.
Only the soft solution of subsuming the less into the more consistent hypotheses allows for the construction of a satisfactory hierarchical model of sets. He argues, for example, that H3 is semantically equivalent to H2 because the many of H3 partake of the one of H2. Through this method of subsuming the subset hypotheses into superset hypotheses, and pairs of hypotheses into tetrads of hypotheses, we can construct a hierarchical set theoretic model that formally represents a semantic interpretation of the dialectical exercises. This method of subsuming subset hypotheses into superset hypotheses further implies that the subordinate hypotheses may be predicated of the superordinate hypotheses simply because any subset is analytically contained in its superset and whatever is analytically contained may be predicated of the containing subject.
Dialectic - New World Encyclopedia
Consequently, each of the individual hypotheses, i. Hn, may be predicated of their containing individual, pair, or tetrad of hypotheses e. Hnx… Hnb Hna. Since Platonic participation is both ontic and epistemic, the construction of such a hierarchical set theoretic model of the sequence of pro heauto participative predication also signifies a grand tree-branch ontology in which the being of each of the subordinate hypothesis participates in the being of the superordinate hypotheses e. Lynch viii Answering the challenge of the Parmenides minimally requires a demonstration of the possibility of a single non-contradictory interpretation.
Plato could have responded to this challenge by constructing a hierarchy of sets, in which each superset hypothesis restricts the predicative scope of its subset hypotheses and the inconsistent set is universally replicated at every level as a basic constituent of the set theoretic model. Logic Aristotle is traditionally credited with the invention of formal logic.
Yet Platonists have traditionally asserted the primacy of Plato: Plethon even accused Aristotle of sophistry for not acknowledging his debt to Plato. He also planted the seeds for a viable alternative to classical logic.
In the Prior Analytics, Aristotle defined a syllogism as "an argument in which, certain things having been assumed, something other than these follows of necessity from their truth. This priority of deduction further presupposes the existence of some formal laws of deductive reasoning, or the laws of logic. The classical laws of logic are the Law of Identity that states that any subject term is equal to itself i.
Any skeptic who questions these laws of logic confronts a fundamental dilemma: either the laws of logic are absolutely prior to every use of logic, and so must be dogmatically but non-logically assumed with no prior logical deduction, or they are posterior to and deduced through the use of logic. Yet since any use of logic must assume the laws of logic, it seems that any logical deduction that purports to generate these laws must fallaciously assume the very laws that they purport to generate petitio principii.
The traditional answer to this skeptical challenge has instead been, first, to claim that any demonstration must assume certain axioms, of which the most indubitable are the laws of logic; and, second, to claim that every judgment of reason must use the laws of logic, so that there is no possibility of even raising this skeptical dilemma without having first assumed the very laws which it purports to question: for example, one must suppose the Law of Excluded Middle to distinguish between the two horns of the dilemma, in which the laws of logic exist either prior and posterior to the use of logic.
Yet this claim, that every judgment of reason must use the laws of logic, cannot alone answer the skeptical dilemma because it also assumes petitio principii the existence and universality of the very laws of logic that are called into question. This skeptical dilemma might be resolved by affirming the first horn of the dilemma, hypothesizing a substantial identity between the laws of logic and the Ideas of Plato, and grounding the non-logical assumptions of the laws of logic in the Ideas of Platonic ontology. Yet there is an immediate difficulty with this straight-forward identification of the laws of logic with universal Ideas: if the laws of logic are hypothesized to exist prior to the use of logic, but the Ideas are meant to be logically constructed, then these Ideas of the laws of logic must also somehow exist prior to both logic and Ideas.
This would require the laws of logic to be hypothesized as Ideas beyond Ideas, or 'Meta-Ideas' at the limits of intelligibility b. In the Sophist d Plato enumerates three opposed couplets of six Meta-Ideas: being and relative non-being; rest and motion; identity and difference. Since the One-Over-Many Argument requires every multiplicity to be united in one universal Idea, even these Meta-Ideas must be further united in one supreme principle of all Ideas.
The principles of the Limit and the Unlimited seem to have been derived from an earlier Pythagorean tradition. In the Republic, Plato formulated an epistemic version of the Law of Non-Contradiction, which extended Parmenides' prohibition on the coincidence of being and non-being to the coincidence of contrary opposites in one and the same thought.
Donald Davidson describes 7 how Plato experimented with successively more refined methods of positive justification, including the methods recollection, or anamnesis, in the Meno; of hypotheses of Ideas, or noesis, in the Phaedo and the Republic; and the method of collection and division, or diaresis, in the Phaedrus e the Sophist a and the Statesman d.
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Each successive method produced more sophisticated logical machinery for the construction of Plato's ontology: recollection received innate Ideas in memory; hypotheses re-produced these Ideas as mental postulates; and collection and division constructed an ideal taxonomy of different species within identical genera. These methods were meant to establish the non-contradictory ground of justified knowledge, yet the challenge of the Parmenides demonstrated how even the Ideas must possess contrary properties.
University of Texas Professor T. An indeterminate continuum Seung's logic of paradigms is first an eidetic, or formal, construction of concepts, and second an ontic construction of beings. These two modes of construction, eidetic and ontic, together constitute the theory of Platonic construction. Since the terms of particular instances necessarily instantiate the terms of universal Ideas, the instantiation of particular instances in universal Ideas may also function as a necessary deduction from universal Ideas to particular instances. Such a Platonic logic of Ideas would not be abstractly formal but concretely ontological, because the simple unity of thought and being in Platonic Ideas re-unites the form of valid reason with the content of being signified by terms.
Since any analytic deduction requires the consequence to be contained within the antecedent, Platonists could only respond by hypothetically impregnating the One with multiplicity and being.
Platonists have responded by multiplying their hypotheses and mystifying their hypostases: for example, Speucippus, the second scholararch of the Old Platonic Academy, responded by multiplying the One into as many as five different 'ones' Dillon 44 , while his successor Xenocrates responded by mythologizing the supreme Principles into many different divinities Later, Plotinus would affirm the ineffability of the One, and Proclus would distinguish ordinary negation from hyper-negation so as to deny the applicability of the laws of classical logic to the supreme principle altogether.
Dialectical logic is principally distinguished from classical logic by its use of contrariness to motivate the division and reunion of contrary opposite concepts into a higher, dynamic, and synthetic unity-of-opposites. Phaedrus e Since the laws of logic are derived from the supreme Principles, which are the supremely generic universals of all specific universal Ideas, the laws of logic neither properly apply to the Principles nor the derivative relation between universal Ideas and particular instances. Since the Law of Non-Contradiction does not apply to the paradigmatic relation between universals and particulars, Plato can potentially resolve the apparent contradiction of deriving many from the One by constructing various unified mixtures of contrary opposites in universal Ideas.
Cosmogony Harold F. The knowledge of these things is not entirely communicable as other knowledge, but after much discussion about these things, and after a communality of living together, suddenly, like a flame which is lighted from a spark which springs forth, it is born in the soul and from itself is nourished. Since the differences within this absolute identity are distinguished by their contrary opposition to the One, the One also virtually contains all contrary opposition of the Dyad within itself. And since all contrary opposition remains united within the simple self-identity of the One, every instance of contrary opposition must also ultimately resolve into a further Mixture.
The Unitary One The One is hypothetically postulated as the perfect self-exemplar and paradigmatic cause of both its own self- identical unity and, by the negation of its other, of all determinate division of itself from its other, i. Thus, not only unity, but all divisibility, and also the mixture of unity and divisibility is thought to be virtually contained in the One as its potential effects.
Traditionalism as a Theory: Sophia, Plato and the Event
Hypothetical axiom of supreme unconditioned condition b. Singular Aspect: simple self-identical unity c. Multiple Aspect: absolutely complex virtually contained universality II.
The Oppository Dyad The One first determines itself for itself in its own self-identical unity. By determining itself for itself, it simultaneously determines itself as not what is other than itself; as not its other; and in opposition to otherness. In this oppositing determination, the One distinguishes itself from the dyad that is not the One because it is both itself and other than itself.
Where the One is the perfect exemplar and paradigmatic cause of unity and self-identity, the Dyad is the opposite perfect exemplar and paradigmatic cause of divisibility and difference. Opposition of I. Oppositing of the definite One into the indefinite Dyad c. This equalization of diverse elements results in the particularization of the indefinite continuum of an infinite set into an infinite profusion of particular infinitely larger supersets and infinitesimally smaller subsets.
The infinite and infinitesimal profusion of sets continue to participate in the one, both in the unity of their self-identity and by the indefinite unity of the set of all sets. Indefinite extension of dyad II. Great and Small II. The infinitely multiplied i. Equalization of the Great and Small in the Numerical Dyad a.
Equalization of particular infinite set i. Further particularization to divide infinite set into particular subsets i. First particularization of the numerical Dyad, i. The Oppository Halving and Doubling towards all Even natural numbers a. Indefinite extension of particularization B. Particularization of multiple doubles of the numerical dyad, i. Particularization of multiple halves of the numerical dyad, i.