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.