Technical Philosophy: Inference Rules.
Aristotle defined substance as that which possesses attributes (accidents) but is not itself an attribute. One medieval formulation helped start modern logic: an attribute or note of an attribute is an attribute of the thing itself (substance; 'Nota notae nota re ipsius est,' in Latin). So a big red car is both big and red, and we don't have to worry about whether its redness is big! Although Plato did insist that shape is that which always accompanies color.
More about metaphysics soon, now on to logic. Aristotle's Syllogistic is old and obsolete, say many modern mathematicians. Why spend much time learning Barbara, Celarent, Darii, Ferio and their kin? Now all we need are a few rules of inference like modus ponens, modus tollens and DeMorgan's Laws and we can solve any problem!
The gator in the pond is probability. Bertrand Russell once said famously that 'the objects of mathematics are clear, precise and non-existent!' Real objects in the contingent world have probabilistic relationships.
Conditional probability is very interesting. But to simplify this discussion, assume that each object of discourse is IID (independent and identically distributed) and has a high probability. Then if A implies B, and B implies C, the probability of C cannot exceed the product of the probabilities of A and B.
Suppose each probability of A and B is 0.98. Then by the product rule for IID objects, the probability of C is 0.98 x 0.98 = 0.96. If A and B have lower probabilities, the situation gets worse: 0.90 x 0.90 = 0.81. So I can be 90 percent sure going into an inference that A and B are true, but have much lower confidence that C is true.
Now extend this reasoning to typically long chains of legal or mathematical inference. Even with high but imperfect probabilities going into the chains, the probabilities of the conclusions can fall dramatically.
Mathematicians assume 100 percent probabilities going into their sequences of inference. Aristotle and Aquinas would say they can do this because they are abstracting from matter and considering only quantities. Quantity has for centuries been considered a major intelligible category of thought. So a lengthy, extended inference is normal in a mathematical proof. Such a proof can show a Platonic ideal, that is, what would be true if all our assumptions are completely true.
In legal and other professional fields, Aristotle and Aquinas would demure. Aquinas probably says it best: there is a distinction between essence and existence. Knowing the essence is not the same thing as knowing the individual. Because inferences concern the essence, it is best to keep one's inferences relatively short, with their conclusions highly probable.
Kant and other idealists would, in general, discard essence because they discard any concept of substance. What is left is the noumenal world of individuals presenting itself to the phenomenal world that we can sense, and upon which inferences are based. But for most idealists, intuition of the noumenal is ultimately out of the control of the acting subject. So metaphysics is often discarded in favor of phenomenology, and sometimes materialism (Marx).
Hegel and Kojeve take this kind of thinking to an extreme: 'The real is the rational and the rational is the real.' Real action is taken only by world-historical figures who have an artistic or other, often violent intuition of the noumenal world. Phenomenology unfolds at its own pace, and must not be impeded by an overly logical presentation. The arts and intuition rule, with classical or formal reasoning and logic becoming merely a game.
Negating Hegel involves returning to an aristotelian view. Professionals will know the individuals in their subjects well. This kind of knowledge is usually local, involving personal contact. And there is a limit, because each mind can hold limited personal knowledge.
In business, many small owners showing concern for and knowledge of their products and customers is preferable to large and impersonal structures which ultimately deal only in remote probabilities. Large organizations often generate big cost savings in production and logistics, but it is not clear that society is better off if personal knowledge of customers and their transactions is lost. A federated structure with competent local management can help alleviate this.
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For more on Bertrand Russell, his Foundations of Geometry, and Principia Mathematica are fundamental. More details about the reality of mathematical objects are here:
Stanford Metaphysics Project:
Aristotle's Presuppositions of Predication
This paper needs improvement in 2018, but I am hoping it will be helpful as posted.
Towards a Transcendental Analytic Thomism.
(A Post-Hegelian Phenomenology? )
All noted 15 March 2019.
Greek and Medieval Roots (James F. Ross)
Identity as Isomorphism (Frege, Geach and Anscombe)
Categories and Morphisms
A modern mathematical synthesis, with extensive bibliography:
Scholastic thought on Aristotle's categories usually posits an isomorphism between language and reality.
But the number of categories was controversial. Ockham reduced the number of categories to two, and doubted the above isomorphism. Francisco Suarez insisted that categories are concepts, not names. In this he seems to anticipate the views of Frege and Russell, in which sentences are mappings of names and verbs to truth values. The latter factor into reference mappings into the ontology (Frege, On Sense and Reference).
Sense and reference - Wikipedia
In the philosophy of language, the distinction between sense and reference was an innovation of the German philosopher and mathematician Gottlob Frege in 1892 (in his paper "On Sense and Reference"; German: "Über Sinn und Bedeutung"), reflecting the two ways he believed a singular term may have meaning.
Gottlob Frege, Über Sinn und Bedeutung - PhilPapers
Hegel: A somewhat sympathetic summary from Germany in early 2021. I believe Hegel's thinking is an almost complete disaster. It was a big factor in causing two world wars, at minimum.
Hegel: There's nothing else quite like modern automation to help reveal his errors.