For the esoterically adventurous in the ontology project only, read on for a disquisition on the question of ontology without reference to existence involving Hilbert, Husserl and Heidegger leading to a syntactic and semantic approach for rigorous philosophical method.

In pursuit of a rigorous philosophical method that purges the folie of existence from ontology, I’d like to present a few thoughts that, to mix metaphors, may draw from my Rosetta Stone a sword that cuts through the semantic underbrush:

The syntactical development of the predicate calculus by the Hilbert ε-operator has resisted so far all attempts at semantic interpretation.  Briefly, the Hilbert ε-operator is a successful syntactic formalization of the notion “the individual variable x such that F(x) is true, if any,” where F(x) is a predicate.  One replaces the sentence (∃x)F(x) by the proposition F(ε(F(x))).  Likewise, before the work of Kripke and others, the semantics of intuitionistic and modal logics were not known.

Gian-Carlo Rota, section “Syntax and Semantics”, chapter “Syntax, Semantics and Identity of Mathematical Items”, part “II. Philosophy: A Minority View” of his book “Indiscrete Thoughts”.

See the Stanford Encyclopedia of Philosophy’s article for more background on Hilbert’s so-called “Epsilon Calculus”.

Notice that although Hilbert manages to eliminate the folie of the existential quantifier; his syntax “has resisted so far all attempts at semantic interpretation”.

Let me now return to my Rosetta Stone to introduce Husserl’s notion of Fundierung to this problem:

Husserl’s Third Logical Investigation, ostensibly dealing with the phenomenology of whole and parts, is meant to introduce the notion of Fundierung.  Fundierung is frequently used in phenomenological literature, although little has been written about it since Husserl introduced it.  Husserl himself, while using Fundierung extensively, never felt the need to reopen the discussion…

...The queen of hearts as (emphasis JAB) an item in a bridge game is a function related by a Fundierung relation to the queen of hearts as (emphasis JAB) a card pure and simple.  The other term of a Fundierung relation is called the facticity:...the queen of hearts as (emphasis JAB) a card in the deck…

...Like all facticities, they are indispensible in a pen’s function; this indispensability of facticities leads to the mistaken “identification” (emphasis JAB) of facticities with the function of pens.  The absurdity of this reduction can be realized by eidetic variations:  no amount of staring at this object as (emphasis JAB) an assemblage of plastic, metal, and ink will reveal that the object we are staring at “is” a pen, unless my previous familiarity lets me view the pen through the facticities upon which it is founded.

We consider next the controversial Fundierung relation between viewing and seeing.  This Fundierung relation is on par with those of the preceding examples; however, this time we find it hard to admit that we are dealing with the same kind of relation.  My viewing the pen is founded upon my seeing something;  my reading the content of a printed page is founded upon my seeing it.  I may see the printed page without viewing it as (emphasis JAB) reading material…

...Viewing, in manifold modes, is a function; seeing is the facticity that founds viewing.  Pretending to reduce “reading” to a series of psychological or physical processes, as Wittgenstein mockingly pretends to do, is to commit the same reductionist error a child makes when he dismantles a clock to investigate the nature of time.  The Fundierung relation separates seeing from viewing by an abyss, all the more insurmountable because it is a logical abyss.  Seeing may be a process taking place in time, one that founds my view.  But viewing has the same standing as the rules of the game of bridge, the third declension, or the cohomology of sheaves.  None of these items may be said to “exist .” (emphasis JAB)

Gian-Carlo Rota, chapter “Fundierung as a Logical Concept”, part “II. Philosophy: A Minority View” of his book “Indiscrete Thoughts”.

My view is that Rota, himself, missed the significance of Hilbert’s elimination of the existential quantifier in his pursuit of introducing Fundierung as a logical concept.  But I’m getting ahead of myself…

Bringing now Heideggar to the table, you might want to, at your leisure, read my ancient post here at MR titled “The Primordial “As”: Gian-Carlo Rota” but let me reiterate simply this part:

From “Indiscrete Thoughts” by the late Gian-Carlo Rota, Chapter XVII “Three Senses of ‘A is B’ in Heidegger”:

The tradition of philosophy constrains us to use words like “problems,”  “solutions,” “arguments,” and “relationship.”  There is at present no alternative to this language.  Heidegger attempted to develop a language which he considered more appropriate, and time will tell whether his lead can be followed.

and

Heidegger shows that the condition of possibility of Fundierung can be found only if Fundierung is viewed as a special (ontic) (emphasis JAB) instance of some universal (ontological) (emphasis JAB) problem.  The Zusammenhang between A and B is a problem only when the phenomena of Fundierung and “A is B” are taken in isolation.  The problem is made to disappear by uncovering a universal notion of which both Fundierung as well as the “is” in “A is B” will be special instances.  Where shall we find the betokening of such universality?  We will find it when we place Fundierung and the “is” in parallel with other phenomena which will all be seen as instances of one and the same universal concept.

Descarte had seen the problem of the is in “A is B” as an impenetrable mystery.  Heidegger does not attempt to dispel the mystery.  Instead, he shows that Descartes’ mystery is “the same” mystery as several others, for example, the mystery of Fundierung.  He shows that “the same” mystery is found in all speech whatsoever.  You see the mystery in the is of “A is B,” you see another mystery in Fundierung, because of the prejudice that the two mysteries of the is and of the “Fundierungszusammenhang” are of a different kind.  But what if we realized that the same mystery is the condition of possibility of all “relationships?”  Then the mysteries would reduce to one single mystery, namely, the condition of possibility of “relationships.”  But a universal mystery is no different from a universal law.  Heidegger concludes with the discovery of the universal law of the as.

The universal as is given various names in Heidegger’s writings:  It will be the primordial Nicht between beings and Being, the ontological (emphasis JAB) difference, the Beyond, the primordial es gibt, the Ereignis.

The discovery of the universal “as” is Heidegger’s contribution to philosophy.

Heidegger’s later thought in no way alters this discovery.  He came to believe that the language of phenomenology, in which his middle writings are couched, was inadequate to his discovery.  The later Heidegger wanted to recast his discovery in a non-objectivistic language, since the universal as lies beyond objectivity.  The universal “as” is the surgence of sense in Man, the shepherd of Being.

The disclosure of the primordial “as” is the end of a search that began with Plato, followed a long route through Descartes, Leibniz, Kant, Vico, Hegel, Dilthey and Husserl.  This search comes to its conclusion with Heidegger.

IMHO, Rota did not exaggerate the significance of Heidegger’s discovery here.

Now let me try to unify the Epsilon Calculus and the universal “as” in an approach to a rigorous philosophical method for ontology without reference to the folie of “existence”, via this esoteric syntax and semantics.

We will interpret the syntax:

x(y=z)

to mean:

“y and z are identical as an x”

or, equivalently (in our first foray into this semantic minefield where knowledge of the positions of the mines are “imprecise”):

“y and z are identical as it matters to x”

For the time being, I’ll choose the later meaning for reasons that may become obvious presently:

Now let’s interpret the syntax:

x(y≠z)

Which is the same as the logical negation of the prior expression:

~x(y=z)

to mean:

“y differs from z as it matters to x”

Now we may define:

Existence: x(x≠y)

That is to say:

“x differs from y as it matters to x” or, in the, hopefully, now, obsolete language of the predicate calculus:

∃y

Where did “x” go? 

That is precisely the mystery!

Now, we proceed to drag Hilbert, kicking and screaming, into our semantic minefield by defining his Epsilon Calculus’s ε operator via its predicate calculus form:

(∃x)F(x)

in our new syntax:

i(i≠x)F(x)

The choice of “i” as the variable hints at a closure of identity within this new syntax and semantics.