How is Ontology in Aristotle related to Logic?

Philadelphia, PA

Dear Krantz,

Aristotle's "categories" are basically classifications of predicates. This can be taken as a generalized sort of "ontology," as you (and Brentano) say "modalities of being," but I expect that this may merely (or chiefly) open an historical question in philosophy. The predicates themselves of course enter into syllogisms. But on the face of things, one may be inclined to think it is not so much the predicates themselves which are relevant to ontology but the use made of them. The point is somewhat obscured by the "categorical" interpretation of the syllogism and of predicate expressions (existential presupposition) in Aristotle and the Aristotelian tradition.

Modern logic also involves classifications of predicate expressions, which may be relational or non-relations, single-place (F_), two-placed, (G_ _), transitive, & etc., but we do not think of this as generally of much interest for ontology --thought of as an answer to the question, e.g., "What sorts of things exist?"

"Category theory" strikes me as more akin to empirical lexicography than it is to logic. But that might be an question to consider.

H.G. Callaway

Thank you @H.G. Callaway for your answer.

I try to relate categories in the "philosophical" sense to categories in the "mathematical" sense. It makes the link for me more clear as I am used to the second sense. (See https://www.researchgate.net/project/The-significance-of-category-theory-in-philosophy).

Mathematics abstracts much the notion of concept and just retains its "logical" structure (Concept lattice, Formal Ontology, ...). A very living domain in mathematical category is elementary Topos theory. The underlying logic is Brouwers intuitionistic logic and you find also again the notion of lattice which reminds some aspects of ontology.

When reading about categories in Aristotle I was much impressed by the fact that the primary distinction in ontology is the one of substantial and accidental properties. There seems to be no evident equivalence in mathematical category theory for this classification in primary and subsequent distinctions, at least it is obscure to me for the moment.

Sincerely

T. Krantz

Jerry Decker

Ontology is related to Logic in that both attempt to create a system and draw conclusion established only in the intellect. Otherwise they are quite different, ontology being an exercise in philosophy and opinions derived from philosophy while logic tends toward mathematical precision drawn from postulates which might be true in the context, or might not be true.

Introducing a syllogism may apply a logic to a philosophy, possibly making a true conclusion, but also possibly making something not true.

In upper level research every proposition is evaluated with a statistical method in which the search for truth always has a chance of accepting something false, or rejecting the truth.

In recent times experimental research has been preferred for increasing the chance of finding truth without accepting something false. Some part of research still pursues truth in the intellect.

Philadelphia, PA

Dear Krantz & readers,

Many thanks for your reply. Your mathematical perspective is interesting.

The substantial in Aristotle has much to do with "essence" and definition. Aristotle has it that there can be no "science" of accidents. But the distinction is problematic insofar as it presupposes a completed science to be known.

I would say that the modern equivalent or paraphrase of the idea of "essence," is to be found in definition, and this is closely related to explanatory and comprehensive-cognitive salience: What characterization of the subject or concept best distinguishes it and/or best organizes the related subject-matter?

Might this paraphrase be of some help in mathematical category theory? I suppose that philosophical categories are going to be subject to revision in light of the growth of knowledge.

H.G. Callaway

---you wrote---

Dear H.G. Callaway ,

There might be a logical definition of essence: In Kurt Gödel's ontological proof there is a precise definition of an essential property in modal logics:

(See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof )

I wonder if other ontological distinctions could be similarly logically formalized.

Philadelphia, PA

Dear Krantz & readers,

I suspect that the "essence of modal logics," might take us pretty far afield. One might think of them as mathematical systems, I suppose, but might there not be some less controversial examples? It is not my aim to define "essence," though I have a long standing interest in lexicography and precision of terms.

If we are going to keep to the topic of your question, even as far as category theory in mathematics, then it might be best to look at the general use of category theory in mathematics --which you apparently have in mind.

My own view is that there is actually very little practical use of modal logic. It is simply too easy to get along without it. I don't discourage related developments, and I am aware that many scholars have put a great deal of work into the field. I think we should use formal methods when they are applicable and productive, but not be drawn into formalist attitudes which too easily ignore relevant details: the interesting "accidents," as Aristotle might put the matter.

H.G. Callaway

Both Aristotle's categories and category theory in mathematics are both attempts to say what there is by providing a system of classification. Aristotle is trying to answer the question of what types of objects are there in the world and what are possible relationships between them. Category theory in mathematics is an attempt to classify mathematics by means of the functional relationships between objects. From the perspective of clarity of description, I think both types of categories are useful. It is less clear to me that either type of category really tells you what exists or not. The connection if there is one is Kant's view that (in modern language) the meaning of a proposition is a function from an ordered sequence of predicates and objects to an object. This view came explicitly from Aristotle's formulation of propositions as the premisses of syllogisms ("Socrates is a man" corresponding to the object "the man Socrates"). Frege famously changed the objects in the image of the function to truth values and the generalised logic itself at the same time. It is possible to argue that ontology (what there is) can be read from a sufficiently complete categorisation as a framework of our understanding. Personally I think this makes sense at the abstract level of "object" and "function", but an ontology makes little sense in terms of categorisations in different natural languages or usage of those languages.

Category theory in mathematics is slightly unsatisfactory in that it does not explain the notion of "object", relying on functions (or "morphisms") to do that. I also find category theory difficult to use as a learning aid, since the definitions are abstract and can lack obvious application.

Dear H.G. Callaway and readers,

I share this view - though I believe that a precise definition of essence could be useful to obtain indications how to bring philosophical category theory and mathematical category theory in accordance. I have a precise idea in mind: Kripke semantics can be transposed to topos theory, it is known as Kripke-Joyal semantics. Now I know the notion of "essence" has been studied by many thinkers, and a formalist view could make visible helpful aspects.

Philosophical arguments and mathematical arguments follow often - even if not always - similar principles. Comparative studies would be welcome.

But may be, as you suggest, we should center our discussion on Aristotle's metaphysics.

I read that since Aristotle's classification of Syllogisms there has been for a long time nearly no progress in logics. One can observe that his ontology and his logics are related systematically by what is called today "Venn diagrams". (See Aristotle in Stanford Encyclopaedia of Philosophy.) Essentially this is what one would do in an Elementary topos (for instance in the category of sets) as well.

I don't know if Aristotle's syllogistics forces "classical" logic, in the sense that the double negation of a statement coincides necessarily with the statement.

The gap between double negation and affirmation is at the source what in mathematical category theory is called a monad. Bridging the gap would correspond to giving an algebra for the monad. Remark that nearly all algebraic structures (vector spaces etc.) are algebras in this sense.

Has Aristotle been aware of this gap between affirmation and double negation (or double dualization), and did he analyze it?

I am sure that there are plenty of Aristotle scholars who could provide an account of what Aristotle said about negation. There is a view that negation is the act of denial. In those terms it is not the case that to deny a denial is the same as affirming. But syllogisms use negation in terms of predicates applying to objects, which brings to a logic of truth and falsehood rather than a logic of proof and disproof. Syllogisms were a brilliant invention, but they ossified logic until the 19th Century when Boole, Frege and Peano came along and provided a uniform notation for logic that dealt with richer predicate/object structures than syllogisms do. While I do believe in essences in the sense of natural definitions (a "point" is a part of space time with only the identity as an automorphism, for example), and I have made peace with set theory (thanks largely to Boolos's and Schoenfield's articles on the iterative conception of set), I have never known what to make of topos theory. I accept that there are set-like mathematical structures and notions of negation such that NOT-NOT-A IMPLIES A is not valid, but think that truth (as define inductively by Tarski) is clearly defined, and the complexities of intuitionistic logic are unnecessary.

Dear Andrew Powell ,

I have no doubt you know this topic, but I will recall some basics to make things clearer:

I make a small digression on logic of topoi, though it has been developed only in the 20th century and it not at all due to Aristotle. (It was my question though if Aristotle had had ideas on the double dualization theme, may in a linguistic form etc.).

The logic of topos theory is Brouwers intuistionistic logic. Out of a proof of a disjunction A \/ B you can always extract either a proof the propositions A or a proof of the proposition B. Similarly out of a proof of (there is x) A[x] you can always extract a proof for a particular term t of A[t]. For that reason the logic is called constructive: To prove something exists it is not enough to prove that non-existence is contradictory. You have to explicitly construct the object.

Classical logic in addition allows to prove existence by a proof that non existence is contradictory. From the negation of the negation of A you can deduce A. I think Aristotle an Plato used classical logic(?). Classical logic only admits a semantics of provability. This is supposedly due to Tarski.

Linked to the constructivity of intuitionistic logic is the fact that it does not only admit a semantics of provability, but also a semantics of proofs, say a denotational semantics i.e. a semantics which distinguishes proofs. For instance the formula A=> ((A=>A) => A) has as many proofs as there are integers: A good denotational semantics will distinguish the number of times the hypothesis (A=>A) has been used in a (reduced) proof.

Set theory gives a model of such a denotational semantics:

Interprete the formula A by a set (A). Interprete the entailment relation A=>A by a mapping f from the set (A) to the set (A). Proofs of the formula A=> ((A=>A) => A) will be interpreted by the mappings f^n.

I stop the explanations here, but I don't share the idea that these are unnecessary complexities. In the complexity is hidden all the possible interpretations one can make of a proof by contra-position. In my eyes it is linked to Aristotle's distinction of essential and accidental properties...

Philadelphia, PA

Dear Krantz & readers,

I imagine a "proof" based on physical theory and reasonable empirical evidence, that there is some early galaxy, more distant from the Earth than any other. The requirement that it should be possible to name some independently identified galaxy, if such a proof is possible, suggest that some prior identification of the most distant galaxy would be necessary in order for a proof that there is such a galaxy to stand.

This seems to put the cart before the horse. But surely, any viable logic must have such empirical applications.

Again, I can imagine a reasonable "proof" based on astronomical data on the frequency of meteorite hits on Mars, that there must be a most recent hit. But the requirement that it must be possible to name some interdependently identified meteorite hit, if such a proof is possible, would seem to put the naming before the proof of the hit. All this seems quite counter-intuitive.

Right?

H.G. Callaway

---you wrote---

Dear @Thomas Krantz

I agree with everything you have said as an expression of the logic of proof and disproof. The only thing I would dispute is that Tarski's semantics for first-order formal languages is a semantics of provability. A. Tarski explicitly gave a definition of truth of a sentence by induction. The natural semantics for classical logic are interpretations which make sentences (and formulas) true. This is the realm of model theory, but clearly many models do not correspond to the natural truth semantics of a formal theory but only establish consistency. Models of course preserve provability. My personal preference for formal systems is for higher-order logic systems, such as finite type theory. Constructive logics are an interesting field of study, and are important because they generate computable witness functions, but the conclusions are weaker than formal systems using classical logic (simply because NOT-NOT-A IMPLIES A is not valid).

Dear H.G. Callaway ,

I am not very familiarized (yet) with Brouwer's texts on intuitionist philosophy. In my eyes it is very skeptical. A discussion on it is very interesting to me in particular linked to the ontological question, Aristotle's realism (is that the right word?), Plato's idealism, and the relation between both philosophies.

(I have to think about your example with the meteorites.)

K. Poppers falsification theory would be useful to examine to this respect.

Logic and Aristotle represented on the cathedral in Chartres

The dragon symbolizes fallacious arguments of Sophists using exclusively Syllogisms "Dialectica mordet".

https://www.cathedrale-chartres.fr/portails/portail_royal/baie_droite/vouss12_11.php

I studied Brouwer years ago, but fundamental to his thought is the notion of a free, creative subject. I may have misunderstood but I think Brouwer believed that a mathematician constructs mathematical objects in concrete intuition, and mathematics should be very suspicious of anything that could not be so constructed. Examples would be infinite sets (understood to exist independently of the subject) and propositions that can be proven by means of constructions. Brouwer allow the subject to use rules or to choose an object at any stage of the construction.

Philadelphia, PA

Dear Krantz & readers,

Aristotle is usually described in terms of "moderate realism." In effect, one might say that our hypotheses always go beyond the evidence available to support them. But this is pretty standard in contemporary philosophy of science. If a hypothesis were not at least subject to contrary evidence it likely wouldn't long count as empirical or scientific. This tolerance can be over-done, but few expect exhaustive proof. There is going to be "better and worse" to hypotheses, classification and definitions based in (interacting) empirical-theoretical developments.

Scientific hypotheses don't have to be judged unanimously correct, right off the bat. This fact sits rather uneasy in very hierarchical and top-down social environments.

My sense of intuitionist constructivism is that it has been an attempt to control theory by limiting its methods. Conclusions are to be guaranteed, in some sense by their sources. Something demonstrated by indirect proof (assume the negation and then show that a contradiction follows) is regarded as introducing "uncontrolled" elements or imperfectly (incompletely, inadequately) described conclusions --not fully controlled in relation to better understood sources. There is a kinship to positivism and to the traditional "Cartesian" aim of total certainty. All of this is more plausible in contexts favorable to rationalism as contrasted with empiricism. Intuitionism is more a pass time of some mathematical specialists--trying to figure out what might be done with it.

But I suppose that if some suspicious "uncontrolled" mathematical system got through the professional journals, still lack of useful applications would likely doom it. If a mathematical system finds no useful application (this may take some time of course, as with non-Euclidean geometries), then it is just going to sit there one the library shelves doing little damage to science.

H.G. Callaway

---you wrote---

Juan Weisz

A syllogism is just the observation that what holds for a larger set also holds for a subset, or particular member.

This logic is too limited for science, where induction becomes more important.

Similarly intuitionism in mathematics seems too limited, though ironclad; only sticking to constructible elements.

My impresion that those in analytic philosophy are too limited in scope, competing with science and math.;

one needs to be creative in returning to metaphysics, in some form. Philosophy in its proper setting.

Philadelphia, PA

Dear Weisz & readers,

I think that if you broach "metaphysics" in the philosophy of science and in discussions with scientists you will chiefly raise skeptical eyebrows and doubts. As a sub-discipline of philosophy, metaphysics is regularly linked to epistemology or the theory of knowledge--"How do we know?"; and otherwise is often chiefly historical --in good part a compendium of historical approaches (and mistakes) concerning the nature of reality. The origin of the word "metaphysics" is not without significance. Originally it was the title given to a work of Aristotle's regularly organized "after" his book on Physics, that is the original meaning --"after the Physics."

Analytical philosophy cannot "compete" with the sciences or with mathematics. It specializes in clarification of related concepts and their relationship to the common sense of the educated public. It seeks to understand the sciences--not second guess them. (there are rare exceptions.) This practice is certainly open to creative expression and developments. If you doubt of this, I can recommend some readings.

Deductive logic belongs to scientific theory and inquiry, though it does not exhaust them. More generally it belongs to the topic of scientific methods.

H.G. Callaway

Dear H.G. Callaway ,

Thank you for this nice explanation. In my mind "meta" had the only meaning of "beyond", and metaphysics would precede in some sense physics.

Is the following a valuable etymological source?

https://www.etymonline.com/word/meta-

Philadelphia, PA

Dear Krantz & readers,

Your link looks like a good source on the etymology of "meta-" and its modern compounds.

Clearly, by now, "meta-" also means "beyond" or "above," even "superintending," in various compounds, say, "metacriticism" --addressing the grounds of criticism?

H.G. Callaway

Philadelphia, PA

Dear Krantz & readers,

Webster's says:

Definition of metanoia

: a transformative change of heart especially : a spiritual conversion.

---End quotation.

Sounds very "Reformation," Protestant, Calvinist.

H.G. Callaway

Dear H.G. Callaway ,

you looked it up in an english dictionnary!

I saw the french version of the corresponding entry in wikipedia.

Anyhow, already in ancient Greece it meant 'giving oneself a (supposedly) better norm of behaviour' or 'changing one's mind (or heart(!)) towards a higher ideal'.

Afterwards in a Christian context 'repentance' or 'conversion' for all confessions.

So in the very profane context of our metaphysical discussion it could have different meanings:

1) I learned today that ontology is for Aristotle really primary and accessible to mind. His physics stumbles already on definition difficulties of phusis ('nature').

2) I hardly know anything about Aristotle's foundation of theology in metaphysics so I will not try to say something about it.

Finally it could also mean more basically not only trying to find truth in syntactic rules or categorical norms. Aristotles criticism of the Sophists praxis already showed limitations of syntax. In a theological debate phusis plays again a role.

Here Plato and Aristotle see things differently I think.

Now seeking truth would then mean not only adhering informally to an abstract ideal but also to convert one's heart.

Juan Weisz

Ask a scientist where he got his original idea from.

Some will say a dream, others struck when walking across a park,

none knows.

So thing just seem to descend from metaphysics into the concrete. As if knowledge were there, but it is just beyond reach...something like the irrational mind, 10 times the rational.

Again, deductive logic only plays a minor role for us. You dont find something new through deduction, or only very rarely. Analogy and induction mostly apply.

Firstly you draw out something very sketchy, then you start completing it.

When you are finally done it sounds more like you deduced it, but not so.

The scientist today is like the modern alchemist, blind as a bat.

H.G.

Sounds like philosophers converted into science writers for the general public,

your description. It may be true.