Plato's Sophist and Type Theory (older post)

Suppose we had a type $A \rightarrow B$. Then application $a : A, f : A\rightarrow B \vdash f a : B$ can be 'internalized' as a type \[ A\rightarrow (A\rightarrow B) \rightarrow B \] But application of this type can likewise be internalised as\[ A \rightarrow (A \rightarrow B) \rightarrow (A\rightarrow (A\rightarrow B) \rightarrow B ) \rightarrow B \] and so forth. This is similar to the 'third man' argument of the Parmenides. Take $B$ to be the 'truth-value' type $\Omega$. The 'canopy' argument in fact seems to herald the idea that a type (i.e. a 'form' or 'unsaturated' propositional function) should be seen not only a 'set' but as a 'space' as in homotopy type theory. In a passage of the Sophist 237 there is a remarkable discussion on 'non-being'. You cannot talk about non-being because by doing so you already attribute it implicitly the mark of a something, an ought - both unity and being. Now this passage is in many ways an anticipation of the rule of contradiction (for $\bot$) in natural deduction as well as its logical deployment in axiomatic set theory, specially dealing with $\emptyset$ in formal proofs. But most interesting is the connection to Martin-Löf type theory: the zero type or empty type $\mathbb{O}$. This type is not inhabited by anything. But yet to use this type, to reason with it, you must assume that it is inhabited $ a : \mathbb{O}$.  Martin-Löf type theory allows us to flesh out Plato's intuition about the connection between falsehood, nothingness and absurdity. The empty set is a paradigm of the initial object of a category. There is a unique set theoretic function $f : \emptyset \rightarrow X$ for a given set $X$. Correspondingly the type $\mathbb{O}\rightarrow A$ is inhabited where $A$ is for example some non-empty inductive type. Thus we can speak meaningfully about nothing. Consider the very difficult passage Sophist 243-246 that clearly picks up on and rectifies the Parmenides. But for Plato what are 'Being', 'Unity' and the 'Whole' ?  We can consider the unit type $\mathbb{I}$ as corresponding to 'Unity' and think of it as either a set-theoretic singleton $\{u\}$ or as a contractible type where equality is interpreted as a homotopy path. In this sense it is a homogenous space much like the 'sphere' in Parmenides' poem. Plato does indeed distinguish between pure unity and a whole participating of unity. The singleton set is a paradigm of a terminal object in a category.
The 'Whole' is clearly a 'universe' type $U$ or $Type$.  It is a difficult problem to relate the 'Whole' to 'Unity'. What does it mean even for the 'Whole' to participate of 'Unity' ? That there is one supreme universe $Unity$ with only one inhabitant $Type : Unity$ ? But then we could not have accumulativity : $ a : Type$ implying that $ a : Unity$. Or, categorically, how do we interpret that all objects $A$ admit a unique morphism $A \rightarrow 1$ ? Category theoretically a ($\infty-\enspace$) groupoid is a candidate for a category 'participating in unity'. There is a tension between unital being $\mathbb{I}$ and the being-whole, the being spread out and shared by all beings $Type$. The logic in this passage is apparently 'type-free' and impredicative. 'Being' $: \Pi ( X: Type), \Omega$ and 'Unity': $\Pi (X : Type), \Omega$ seem to be able to be applied meaningfully to anything (indeed $Type : Type$). The passages on 'names' and 'reference' is quite striking. Specially when Plato conjures up 'names that refer to names' and a name referring only to itself (imagine a type only inhabited by itself). Part of Plato's argument is that no matter what in reality 'is' the fact is that there is a plurality of \names. In the later section on the five suprema genera we may ask: why is there not also a form for 'participation' itself ?What are some fundamental kinds of types ? The empty type $\mathbb{O}$, the unit type $\mathbb{I}$, the universe(s)  $\mathcal{U} $ (or $Type_i$), equality $ \Pi ( X\enspace Y: Type), Prop $, number $Nat : Type$ and the interval $\mathbf{I}$ for path spaces or cubical sets in homotopy type theory. Paths represent change, the interval represents temporality. Plato speaks of equality being or not being equal to something, just as Voevodsky speaks of equality being equivalent to equivalence.

Formalism is not clarity

Ich setze also voraus, daß man sich nicht damit begnügen will, die reine Logik in der bloßen Art unserer mathematischen Disziplinen als ein in naiv-sachlicher Geltung erwachsendes Sätze system auszubilden, sondern daß man in eins damit philosophische Klarheit in betreff dieser Sätze anstrebt (...) - Husserl I Log. Unt.

On a surface level Aristotle's Organon and Physics are formally impressive and from a contemporary mathematical point of view quite suggestive.  Yet, if we analyze things very carefully we find that at a deeper level we are in the presence of a big step backwards from Plato which also cannot really be compared to the sophistication and brilliance of the Stoics. For in Aristotle the key fundamental terms and concepts ("term", "concept", "predication", "essential predication",  "proposition",  "huparkhein", "quality", etc.) are never defined, elucidated, clarified  and perhaps not even used consistently. There is also a serious lack of grammatical and linguistic analysis . To study Aristotle it is not enough either to engage in traditional "classicist"  or commentary-based methods of exhaustive textual analysis and nitpicking or to think that somehow modern mathematical or symbolic logic in itself is sufficient tool to clarify all problems.  Rather we must deploy what is scientific and sophisticated in modern philosophy to bring to light what lies beneath the surface of the Aristotelic texts. Fortunately we do have a kind of philosophical Principia Mathematica, and this is Husserl's Logical Investigations and other subsequently published and equally important texts complementing and developing this work.

Recall Husserl's distinction between judgments of existence and judgment of essence.  Can this help us understand the universal quantifier ? Consider:

1.All ducks can swim.

2.All people in this room are under 30.

3. All prime numbers greater than 2 are odd.

 What does 1) mean, what do we mean by 1).  That swimming is part of the definition of duck, that being able to swim is a logical consequence of the definition of duck - and here we are assuming an artificial consensus - or that everything belonging to the extension of duck (for instance, we could just take a heap of things and label it "duck") happens to have the property of being able to swim (this is unlikely or at most genetic, plausible) ?  For 2) we cannot state that being under 30 somehow is a logical consequence of the concept of being in this room.  2) is definitely a Husserlian 'judgment of existence'.  3) can be given an extensional reading but it also could be given a logical reading in the sense that being odd follows from the definition of being prime and the condition of being greater than 2.  Thus 3) differs from 1) and 2) by allowing both interpretations. 3) can also mean: there is an algorithm which takes as input a prime number and a proof that this number is greater than 2 and yields as output a proof that it is odd.

But consider a model-theoretic approach.  For a model $M$, representing the current world, or current global state-of-affairs, we may well have that $M \Vdash \forall x. \phi(x) \rightarrow \psi(x)$ without it being the case that for our theory $T$ we have  that $T \vdash \forall x. \phi(x) \rightarrow \psi(x)$. But the statement $M \Vdash \forall x. \phi(x) \rightarrow \psi(x)$ must itself be proven in some metatheory $T'$ and is thus again purely logical.

The extensional interpretation of 1) can be: i) that things in the extension of "duck" have the property of being able to swim. ii) that the extension of "duck" is contained as a set in the extension of "being able to swim". 

More profound is the dependent-type theoretic interpretation $\vdash p: \Pi_{ x : D} S(x)$ which reads: there is a function $p$ which takes as input a duck and yields a proof that that particular duck can swim. Compare this to Bobzien and Shogry's interpretation of Stoic quantification:

If something is a duck then that duck can swim. 

How far we are from understanding quantifiers, concepts, extensions and predication in general !

Radical mathematical logicism is the position that logic (or pure rationality) only exists fully in mathematics (and mathematical models in science).  Natural language can only attain an approximate rationality via a mathematical pragmatics (as in computer science).

There is, at first sight at least,  a huge chasm between our mathematics and the complex organic self-directed concreteness of living systems and consciousness. But this chasm can be bridged if we study mathematical theories qua theories, their diachronic and synchronic systemic articulation and organicity seen as an abstract version of consciousness and life. 

If in mathematics both formal and conceptual clarity are of great importance, in philosophy they are even more so.  While agreeing with the quote of Husserl we do not undermine the greatness of formal clarity and the huge progress in philosophy that, in the scale of things, would be achieved by a formal philosophy even if this not mean the ultimate clarity and the highest development of the philosophical project.

Words of Pawel Tichý from Foundations of Frege' logic (1988)

Fate has not been kind to Gottlob Frege and his work. His logical achievement, which dwarfed anything done by logicians over the preceding two thousand years, remained all but ignored by his contemporaries. He liberated logic from the straight-jacket of psychologism only to see others claim credit for it. He expounded his theory in a monumental two-volume work, only to find an insidious error in the very foundations of the system. He successfully challenged the rise of Hilbert-style formalism in logic only to see everybody follow in the footsteps of those who had lost the argument. Ideas can live with lack of recognition. Even ignored and rejected, they are still there ready to engage the minds of those who find their own way to them. They are in danger of obliteration, however, if they are enlisted to serve conceptions and purposes incompatible with them. This is what has been happening to Frege's theoretical bequest in recent decades. Frege has become, belatedly, something of a philosophical hero. But those who have elevated him to this status are the intellectual heirs of Frege's Hilbertian adversaries, hostile to all the main principles underlying Frege's philosophy. They are hostile to Frege's platonism, the view that over and above material objects, there are also functions, concepts, truth-values, and thoughts. They are hostile to Frege's realism, the idea that thoughts are independent of their expression in any language and that each of them is true or false in its own right. They are hostile to the view that logic, just like arithmetic and geometry, treats of a specific range of extra-linguistic entities given prior to any axiomatization, and that of two alternative logics—as of two alternative geometries—only one can be correct. And they are no less hostile to Frege's view that the purpose of inference is to enhance our knowledge and that it therefore makes little sense to infer conclusions from premises which are not known to be true. We thus see Frege lionized by exponents of a directly opposing theoretical outlook. 

(...)

To the most advanced among the exponents of the New Age logic even this is not enough. Why, they ask, cling dogmatically to consistency ? Why not jettison the law of non-contradiction (...) Men of action (the Lenins and Hitlers of this world) have long been familiar with the advantages of embracing contradictions. They know that it not only neatly solves all problems in logic proper, but provides an intellectual key to 'final solutions' in other fields of human endeavour.

What is a term for Aristotle ?

One of the uniquely interesting features of the Topics is that it offers a wealth of examples illustrating the fine logical-grammatical structure of those linguistic expressions considered terms for Aristotle.  The common view that in Aristotle we have essentially a term-logic while in the Stoics we have a proposition- or sentence-logic is untenable. Not only have Bobzien and Shogry shown that the Stoics had a highly developed theory of sub-propositional elements but the evidence points to the Aristotelic term being what we now call a noun-phrase which can include embedded sentences, in particular that-clauses. Thus this opposition is merely apparent

In Plato’s Sophist there is a discussion concerning ’sentences’, ’nouns’ and ’verbs’, sentences being combinations of nouns and verbs, and all sentences must have a ’subject’. Plato gives the example ’Theaetetus, with whom I am now speaking, is flying’. This illustrates how Plato’s concept of ’noun’ corresponds to our ’noun phrase’ and very likely to the Aristotelic term or ’oros’.

1. We must be careful not to confuse Aristotle's convertability and co-extensionality with our modern notions, in particular, first-order notion. Aristotle's convertability  is more of a co-predicability. Thus the Aristotelic 'extent' of a term is not framed using individuals but other terms in general. The extent of a term B is the collection of all terms C for which B can be said of C.  Thus subordinate species and the subordinate species of theses species are all part of the 'extent' of a genus.

2.  Relative idion is a trinary predicate: $RI (x,y,z)$ read: $z$ is the relative idion of $x$ relative to $y$.

4. It will be good to have a complete glossary of all the 'terms' mentioned in the Topics and the relationships between them, either rejected or accepted (and sometimes, unfortunately both) by Aristotle. Examples of terms in book V : bear a very close resemblance to the soul , the primary element wherein the soul is naturally found,  possession of sensation,  being natural sentient, the most rarefied and lightest body, the substance to which "man" belongs as a species, that which is made of body and soul. We at once notice some grammatical issues with this translation, not to mention logical, ontological issues (the flexibility is amazing, even the meta-level predicabilia themselves can be part of the post-determiner of the noun-phrase). How much better is the original Greek. For instance the first term is : to homoiotaton psukhê(i). This appears to be an ellipsis for 'the thing ousia most similar to the soul', that is if we postulate the general structure of the Aristotelic term to be

Genus G + post-modifer

In modern logic the Peano iota operator involves individuals. But for Aristotle it is natural to conceive Peano operators for species, for instance the accepted property of animal as to ek psukhês kai somatos sugkeimenon. But this can be read: the composite consisting of soul and body. Definitions and properties of mass-nouns are not uncommon in the Topics. 

One topic is the following:

forall C.  Subexp(C,B) and Species(C,A) => not Idion(A,B)

Wait a minute ! Are we talking about the linguistic expressions or what they mean ? How dare you put the predicates for subexpression and species and idion all in the same formula !  But what if we want to deal with globs which include both the syntactical, semantic and relational aspects at once ? Fans of ordinary language and ordinary usage,  what do you mean when you say "the word 'cat' " ?   The glob ! Only linguists can invent some fancy brackets to isolate the purely phonetic or orthographic aspect, like /cat/.  Like it or not, Aristotle deals with globs - or perhaps, writing elliptically,  he does not. Maybe our Subexp is wrong, and it should be replaced with semantic inclusion ?

What is the concept of man ?  Does it differ for different people of different cultures at different times and places ? Are we after the invariant in all these concepts ? How does a concept relate to other concepts, to the mind and to the concrete things to which it is predicated ? What is the genesis of concepts and how do they relate to perception ?  What about the different modalities and intentions in which a concept takes part in ? What is predication of essence  and how can one know a concept yet definitions be defeasible ? Surely rationality, bipedalism, etc. are simpler, clearer, concepts and perhaps thus more invariant. But is the genus simpler than the species ? Does predication of essence relate to the essence qua embodied in a concrete substance or to the concept ?

Being entirely in different things at once

Sextus Empiricus in his Outlines, dealing with genus and its species, rips off an argument of the Parmenides. If we interpret the genus-species relation artificially in terms of ordinary geometric inclusion then obviously there will be problems.  But there is nothing contradictory about several objects $a_1,a_2,...,a_n$ all sharing the same relation with the same object $b$, i.e. $R(a_1,c), R(a_2,c),...,R(a_n,c)$. In fact they all share the same property $\lambda x. R(x,c)$. 

But what if we question set theoretic equality and replace it with path-connectedness ? For instance $x \subset y$ means that for every $z\in x$ we have that there is a continuous path $p_{zw}$ from $z$ to $w$ for some $w \in y$.  Then we can well have that $x_1,x_2...,x_n \subset y$ and at the same time $y\subset x_1,...,y \subset x_n$.

Consider the following set in the plane with the relative topology:

 Then by our definition we have that  $\{x_i\}\subset \{y\}$ and $\{y\}\subset \{x_i\}$ for $i=1,...,5$.

Also, what is a set of elements ? The elements must have a distinguishing property or relation prior to being organized into the set (this discussion needs to be greatly expanded).