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.