Critical analysis of Robert Hanna's Cognition, Content, and the A Priori - Chapters 4 and 5 (continuously updated)

Chapter 4: Truth in Virtue of Intentionality, Or, The Return of the Analytic-Synthetic Distinction

The classical Kantian distinction between analytic and synthetic judgments is clearly very important in Hanna's broadly Kantian philosophy and so-called weak transcendental idealism. Our aim here is to examine in great detail Hanna's arguments upholding the above distinction against its foes and to determine i) whether the arguments could still remain valid from the perspective of philosophical suppositions distinct from Hanna's and ii) whether additional arguments can be adduced in favor of the analytic-synthetic distinction.

First of all does the analytic-synthetic distinction make any sense to us ? And why should it matter ? Understanding Kant's precise notion of this distinction in the CPR is no easy matter. While the distinction between a priori and a posteriori knowledge is both classical and more-or-less clear, the analytic-synthetic distinction is difficult...it seems to hinge upon the question of what constitutes purely logical knowledge, purely logical propositions...which in the light of the diversity of modern systems of formal logic would appear to be no easy question. And yet this very diversity hides behind a single underlying question: what is recursivity and how does it relate to logic and to the mind ? In order for any human mind to be able to understand a recursive axiomatic-deductive system it must possess a logic $L$ of sufficient strength so as to contain a fragment $L'$ capable of representing and expressing recursion theory - furthermore for any other human mind with logic $G$ and corresponding fragment $G'$ it must be that $L'$ and $G'$ are fundamentally equivalent. The equivalence class of these fragments is what constitutes the core analytic logic which guarantees that the human mind can understand and carry out rules in a self-reflecting way (and this is an important characteristic). This has a close connection to Church's thesis. Put in another way: core analytic logic cannot have been learnt through rules and procedures because rule and procedure cognition and competence presupposes precisely such a logic. Core analytic logic can be expressed (but not identified uniquely with) in turn in a minimal formal system, for instance monadic second order logic (and in which the first-order aspect is also monadic except for some basic binary relations). Monadic logic (and Hanna seems to want to identify its first-order variant with analytic logic) is certainly close to some of Kant's definitions of analyticity in terms of  'containment' though of course it will be very important to elucidate if such containment is intensional or extensional (or both).

New logical investigations

Let us face it. We know and understand very little about the 'meaning' of such homely terms as 'water' (mass noun). Meaning is not 'inscrutable' just very complex and has not been investigated with complete candor or penetrating enough insight.

A linguistic segment may acquire individual additions or variations of meaning depending on linguistic context  (there is no water-tight segmentation) and yet still contain a certain invariant meaning in all these cases - all of which cannot be brushed away under the term 'connotation'.  For instance compare the expressions 'water is wet', 'add a little water' and 'the meaning of the term 'water''. 

This is clearly related to psychologism and its problems and the inter-subjective invariance of meaning.

In literary criticism there is actually much more linguistic-philosophical acumen, for example in asking 'what does the term X mean for the poet' or 'explain the intention behind the poet's use of the term X'.

Let us face it. Counterfactuals and 'possible worlds' if they are no make any sense at all demand vastly more research and a more sophisticated conceptual framework. We do not know if there could be any world alternative (in any degree of detail) to the present one.  The only cogent notion of 'possible world' is a mathematical one or one based on mathematical physics. There is at present no valid metaphysical or 'natural' one - or one not tied to consciousness and the problem of free-will. 

Given a feature of the world we cannot say a priori that this feature could be varied in isolation in the context of some other possible world. For instance imagining an alternative universe exactly like this one except that the formula for water is not H2O is not only incredibly naive but downright absurd.

Just as it is highly problematic that individual features of the world could vary in isolation in the realm of possibility so too is it highly problematic that we can understand the 'meaning' of terms in isolation from the 'meaning' of the world as a whole.

There is no reason not to consider that there is a super-individual self (Husserl's transcendental ego or Kant's transcendental unity of apperception ) as well as a natural ego in the world.  What do we really know about the 'I', the 'self' , all its layers and possibilities ? The statement 'I exist'  is typically semantically complex and highly ambiguous. But it has at least one sense in which it cannot be 'contingent'. Also considerations from genetic epistemology can lead to doubt that it is a priori.  

There are dumb fallacies which mix up logic and psychology, ignore one of them, artificially separate them or ignore obvious semantic distinctions. And above all the sin of confusing the deceptively simple surface syntax of natural language with authentic logical-semantic structure ! For instance: 'Susan is looking for the Loch Ness Monster' and 'Susan is looking for her cat'.  It is beyond obvious that the first sentence directly expresses something that merely involves Susan's intentions and expectations whilst the second sentence's most typical interpretation involves direct reference to an actual cat. The two sentences are of different types.

We live in the age of computers and algorithms.  Nobody in their right mind would wish to identify a 'function' with its 'graph' except in the special field of mathematics or closely connected areas. If we wish to take concepts as functions (or take functions from possible worlds to truth values) then obviously their intensional computational structure matters as much as their graphs. Hence we bid fair-well to the pseudo-problems of non-denoting terms.

Proper names are like titles for books we are continuously writing during our life - and in some rare cases we stop writing and discard the book. And one book can be split into two or two books merged into one.

It is very naive to think that in all sentences which contain so-called 'definite descriptions'  a single logical-semantic function can be abstracted.  We must do away with this crude naive abstractionism and attend to the semantic and functional richness of what is actually meant without falling into the opposite error of meaning-as-use, etc.

For instance 'X is the Y' can occur in the context of learning: a fact about X is being taught and incorporated into somebody's concept of X. Or it can be an expression of learned knowledge about of X: 'I have been taught or learned that X is the Y' or it can be an expression of the result of an inference : 'it turns out that it is X that is the Y'. Why must all of this correspond to the same 'proposition' or Sinn ?

Abstract nouns are usually learnt in one go, as part of linguistic competence, while proper names reflect as evolving, continuous, even revisable learning process. Hence these two classes have different logical laws.

The meaning of the expression 'to be called 'Mary'' must contain the expression 'Mary'. So we know something about meanings ! 

How can natural language statements involving dates be put into relationship to a events in a mathematical-scientific 'objective' world (which has no time or dynamics) when such dates are defined and meaningful only relative to human experience ? What magically fixes such a correspondence ? This goes for the here and now in general ? What makes our internal experience of a certain chair correspond to a well-defined portion of timeless spatial-temporal objectivity ?

On Schmanalyticity and A Reappraisal of the Synthetic A Priori

 https://kritik.substack.com/p/if-theyre-wrong-theyre-wrong

Logic, mathematics, language and thought

Update 13/9/2024: The part of about Zalta's Object Theory and topos theory has now been given a more accurate and extensive development in a paper.

What if most if not all modern mathematical logic could be shown to be totally inadequate for human thought in general and in particular philosophical thought and the analysis of natural language ? What if modern mathematical logic were shown to be only of interest to mathematics itself and to some applied areas such as computer science ? 

By modern mathematical logic we mean certain classes of symbolic-computational systems starting with Frege but also including all recent developments. All these classes share or move within a limited domain of ontological, epistemic and semantic presuppositions and postulates.

What if an entirely different kinds of symbolic-computational systems are called for to furnish an adequate tool for philosophical logic, for philosophy, for the analysis of language and human thought in general ? New kinds of symbolic-computational systems based on entirely different ontological, epistemic and semantic postulates ? 

The 'symbols' used must 'mean' something, whatever we mean by 'meaning'. But what, exactly ? Herein lies the real difficulty. See the books of Claire Ortiz Hill.  It is our hunch that forcing techniques and topos semantics will be very relevant.

However there remains the problem of infinite regress: no matter how we effect an analysis in the web of ontology, epistemology and semantics this will always involve elements into which the analysis is carried out. These elements in turn fall again directly into the scope of the original ontological, epistemology and semantic problems. 

If mathematics, logic and philosophy have important and deep connections in was perhaps the way that these connections were conceived that were mistaken. Maybe it is geometry rather than classical mathematical logic that is more directly relevant to philosophy.

What if a first step towards finding this new logic were the investigation of artificial ideal languages (where we take 'language' in the most general sense possible) and the analysis of the why and how they work as a means of communications.

In our work we discuss the importance of Cantor's set of parts-forming operator $P$ and in particular for classical philosophy. Now another fundamental operator is the extension-forming operator $\{x : \phi(x)\}$ which in set theory allows one to form the set of all elements satisfying a given property $\phi(x)$ - and this of course in its unrestricted form leads to Russell's paradox.   These two operators are the cornerstones of topos theory. Now what is Zalta's encoding relation but an intensional form of the extension-forming operator but acting on properties, i.e. an intensional second-order analogue of the above operator ? Given a property of properties $\Phi(p)$ we can form an abstract object $a$ such that $a$ encodes precisely those properties $\phi$ for which $\Phi(\phi)$.

In topos theoretic language (remember that second-order logic can be considered as a restriction of HOL): let $X$ be a object/type. Then there is a correspondence between morphisms $\phi: X \rightarrow \Omega$ (elements $1 \rightarrow PX$) and subobjects $j: \{\phi\} \rightarrow X$. There is a morphism $ \in : \Omega^X \times X \rightarrow \Omega$ which tells whether an $1 \rightarrow X$ belongs to a $\phi: 1 \rightarrow \Omega^X$ (a $\{\phi\}$ for $\phi: X \rightarrow \Omega$).  And likewise there is a correspondence between morphisms $\Phi: \Omega^X \rightarrow \Omega$ (elements $1 \rightarrow P\Omega^X$) and subobjects $J: [\Phi]\rightarrow \Omega^X$. Hence Zalta's encoding relation can be seen analogously as the canonical morphism $\in: \Omega^{\Omega^X} \times \Omega^{X} \rightarrow \Omega$, in other words it tells whether a $\phi: X\rightarrow \Omega$ belongs to (i.e. is encoded by) a $[\Phi]$. 

To be more faithful to Zalta's system we need an isomorphism $\rho: X \cong \Omega^{\Omega^X}$ ($X \rightarrow PPX$) so that all subobjects $[\Phi]$ can be seen as 'elements' of $X$. Cardinality considerations suggest that there a well-defined limits to the sets of properties that can be formed in this context - for instance we can consider a the effective topos, etc. Note that more accurately such an isomorphism should be restricted to the subobject of abstract objects $Y\hookrightarrow X$ so that $X \cong Y \oplus Y'$ and we have an isomorphism $\rho': Y \cong \Omega^{\Omega^X}$.  Zalta's encoding relation can be defined in terms of $\rho'$. This is highly significant because it allows a small window for polymorphism or at least some freedom from type restrictions. A $q :X\times X \rightarrow \Omega$  for instance can have implicitly arguments of type $\Omega^{\Omega^X}$ alongside those of $X$.

We can now convert our Coq formalizations of some fragments of Zalta's formal metaphysics into topos theory.

Now notice a further analogy with algebraic geometry. Let individuals be seen as points in a space $K^n$ and properties as polynomials in $K[x_1,...,x_n]$ (the property holds of an individual if it is a root of the polynomial). Then the identification of $K^n$ with the set of maximal ideals of $Spec\,K[x_1,...,x_n]$ is analogous to what we have seen above.  The maximality in this case expresses that if an object encodes a property it will encode all the logical consequences of that property. Of course in Zalta's system only abstract objects encode. Also Zalta's encoding relation for an object is quite distinct from gathering properties which hold of that object.

Can $\Omega$ besides being considered a space of generalized truth-values be a model for Stoic lekta, for Fregean Sinnen, for propositional meanings ? If $A$ represents state-of-affairs then for the Stoics we have a function  $T:A\times\Omega\rightarrow B$ where $B = \{ t, f \}$. But then in a topos we get our 20th century intensional logic perspective:

\[ \Omega \cong B^A \]

But at the same time meanings are very different entities from functions from states-of-affairs to truth-values. We can have different impossible concepts. So our 'functions' need to be intensional, for instance corresponding to different computations.