Computability, Logic and Mind

Addendum:   Husserl's Philosophy of Arithmetic which is actually a treatise on some of the highest categories and operations of the understanding, in particular the act of combination or synthesis.

We begin with the psychological characterization of that abstraction which leads to the (authentic) concept of the multiplicity, and subsequently to the number concepts. We have already indicated the concreta on which the abstracting activity is based. They are totalities of determinate objects. We now add: "completely arbitrary" objects. For the formation of concrete totalities there actually are no restrictions at all with respect to the particular contents to be embraced. Any imaginable object, whether physical or psychical, abstract or concrete, whether given through sensation or phantasy, can be united with any and arbitrarily many others to form a totality, and accordingly can also be counted. For example, certain trees, the Sun, the Moon, Earth and Mars; or a feeling, an angel, the Moon, and Italy, etc. In these examples we can always speak of a totality, a multiplicity, and of a determinate number. The nature of the particular contents therefore makes no difference at all. (Husserl, PA, p. 17 Dillard tr.)

Thus we must distinguish between concrete multiplicities and abstract multiplicities.  Husserl explores the the aspects of combination and synthesis involved, including syntheses of syntheses and so forth.

A Kantian question about computability and the mind

 
On order for the human mind to act in a way equivalent to a universal Turing machine, what is the minimal logic that needs to be supposed ? The statement of this question lacks clarity and precision - to furnish it will be our task in the future. Another form of the question: what is the minimal logic required by the human mind to be able to follow rules of any complexity (discarding limitation of space and time) ?

It is interesting to consider monadic second-order logic (MSOL) (because of its well-known connection to automata theory and recursion theory), but specially when considered as an extension of monadic first-order logic only (and not full first-order logic as is more common), as in J. R. Büchi - Weak Second-Order Arithmetic and Finite Automata. Mathematical Logic Quarterly, 6(1-6):66–92, 1960.

https://arxiv.org/abs/2301.10519

Algorithms to search for and check  proofs in axiomatic-deductive systems can all be implemented in MSOL. So in a sense MSOL is transcendental logic and the other logics are rule-based games (a posteriori). Thus multiple generality and non-monadic relations need not be considered philosophically fundamental; thus many arguments for the superiority of modern logic over ancient logic fail. Also it is very evident that Aristotle's metalogic in the Organon is second-order or even higher.

Kant and Computability Theory

It is strange that few have noticed that it can be strongly argued that the abstract concept of computability and its allied notions are a candidate for being part of the pure a priori necessary concepts for all our cognition and experience (Husserl seems to have anticipated some recursion theory in his Philosophy of Arithmetic).

We have the intimately connected triad of formal logic, arithmetic-combinatorics and computability theory.  To write and check a formal proof we already are deploying computability concepts. But to investigate computability notions we need formal logic and arithmetic. Computation, proof  and the sequence of the natural numbers share the ordered directed time-like quality (linear with branching possibilities). Note: we are not suggesting that computability exhausts human cognition. Also by computability we include all classes in the arithmetical and analytic hierarchies, etc. In a future post we will critique the denigratory use of the term  'mechanistic'  showing it does not hold water when confronted by a serious mathematical and philosophical analysis of the use of differential equations in science.  Computability theory seems very close to Kant's notion of rule and of an architectonic of reason. Church's thesis is a transcendental condition for the possibility of knowledge.

Computability has to do with prescriptive normativity (method) rather than mere general normativity (rules).

We wonder if Kant's realm of pure synthetic a priori intuition of space does not actually correspond to graph theory and combinatorics - and whether category theory, and specially higher category theory  are not best viewed from this perspective (cf. simplicial sets and cubical sets). Category theoretic diagrams have a a kind of dynamic nature - at least in the way they are commonly used and visualized - which recall Kant's schematism.
 

The Church-Turing Thesis, Kripke and Kant

If we consider the abundance of hypothetical and counterfactual elements embedded irrevocably in our linguistic discourse, then a possible worlds semantician might be inclined to view the existence of Kripke's rigid designators as the transcendental conditions for the consistency and intelligibility of our discourse about the world.  But here we wish to discuss a Kantian turn in a different domain. What is it exactly that it means to follow a rule, a set of rules, to play a game,  learn how to use a language, carry out a logical debate, or in general to engage in the world ? For a subject  or mind to do this, it must be computationally competent, in other words, (at least) Turing-complete.  Secondly, it must be able to do this cross-platform, in an indefinite number of domains.  Thus the Church-Turing thesis, like Kripke rigid designators, appears as a transcendental condition for the possibility of our engaging in the world. It also suggests the a priori nature of a basic but fair portion of arithmetic, combinatorics and recursion theory.

Algorithms and Numbers

When we investigate the concept of computability we necessarily require arithmetic. When we investigate arithmetic from a logical point of view it is inevitable that we consider computability or are lead to it. Arithmetic and computability are inseparable notions and it is likely that the Turing-Church thesis is tied to the categoricity of the theory of natural numbers N. When one defines a natural number object in a topos the universal property gives us automatically the notion of a primitive recursive function. The fundamental concept in recursion theory is that of partial recursive function, which can be embodied or implemented in a variety of abstract machine models such as Turing machines. Partial recursive functions, which are partial functions (partial because the computation need not stop) from natural numbers to natural numbers can themselves be codified by natural numbers. This allows us to have a concept of constructive functional, extending the notion of recursivity to all finite types over the natural numbers. There are several possible constructions such as HRO and HEO. This entire process can be generalised to the algebraic setting of partial combinatoric algebras. Intuitionistic arithmetic in all finite types HA, the terms of which form Gödel's system T, is an alternative to set theory, category theory and type theory for doing mathematics. It has a model in ZFC given by interpreting terms of a given finite type by means set-theoretic maps and the natural numbers. This model is extremely complex (it can represent the real numbers) and very little can be said about it in general. What is amazing is that models of HA like HRO and HEO pack this entire structure into the microcosm of N ! This gives us a vision of the mathematical universe in which, as for the ancient Pythagoreans, all things are number.
Tichý has written a monumental work on the philosophy of logic and mathematics in which computationally inspired formal semantics is given allowing a consistent rectification, interpretation and extension of Frege's Begriffschrift and Russell's Ramified Theory of Types. In this system (which we call T) the structure of an expression is isomorphic to the computational process it signifies. Thus an expression in T is like a syntactically correct program in some (typed) programming language. Its meaning is the process of execution (evaluation) of such a program, or rather, a certain abstraction of this. For instance we can give a description of how a certain function (in the programming sense) is executed for an undefined value of its argument. But it is clear that a lambda-calculus type of formalism (or functional programming in general) is the most adequate for this approach and it is not difficult to see why Russell's and Church's systems should figure prominently. We could attempt to formalise in turn such descriptions of computational processes, but this will just evade the problem again. We must stop at an extra-formal foundation in terms of definite mental-cognitive processes and acts. Tichý in his anti-formalist argumentation underestimates just how much mathematics has to be used and assumed to treat combinations of symbols in a relevant and interesting way (for instance proving soundness, completeness and consistency). Hilbertian metamathematics is not consistent with relativism. 

What is Kant's logic ?

In this note we inquire into the nature of Kant's logic in the CPR. More specifically, we investigate the formal syntax and expressive capacity of Kant's conception of 'general logic' and 'transcendental logic'. Clues are furnished by concrete examples given by Kant, by the expressions occurring in the numerous proofs throughout the CPR and above all by the famous and perplexing table of the forms of judgment A70-76 (B95-101). We also ask if there be an overall consistency in the structure of the logic. Maybe the logic introduced via the table is weaker than the logic implicitly deployed in other places of the CPR, that is, the table-logic cannot cope with the expression of multiple generality found in the formulation, for instance, of the analogies of experience.
The original version of the second analogy in A was: for everything that happens there is something which succeeds it, according to a rule. Alles, was geschiet (anhebt zu sein) setzt etwas voraus, worauf es nach einer Regel folgt.  This expression has a logical form  which exceed the capacities of the classical theory of judgment if this last be identified, as is usual, with monadic first-order logic.

The basic unit of Kant's logic is the judgment. Our inquiry is aimed at  the general forms of judgments. Attending to form means abstracting from content. But is can also mean taking heed of the function of the understanding involved - this is what delimits the transcendental logic from the general logic. It is an appeal to the understanding which allows Kant to justify the separate modes of singular and infinite judgment.

Was Kant conservative or innovative  in his logic ? There appears to be conflicting evidence in the CPR.
In a footnote Kant considers categorical  judgments and the four syllogistic figures as a mere fragment of logic, a fragment that does not take into account  hypothetic and disjunctive judgments (on p.78 (6) Hanna seems to forget that not all judgments are directly about relations of concepts; hypothetical judgments are first of all relations of judgments).

A crucial question is how  we are to interpret the table of the forms of judgment.

The table has four 'Titeln' (which we translate as 'rubrics') each having three modes. Thus there are 12 entries in the table.  Did Kant intend that any judgment must fall within exactly one of the 12 entries ? Or possibly more than one ? Does the table have some (perhaps) limited similarity to the inductive definitions of the syntax of formal languages, for instance, first-order logic ? These are the questions we shall investigate.

It is clear that judgments can fall into modes of more than one rubric and but that they cannot fall into modes of all rubrics at once. Also that evidently they can fall into only one mode of each rubric (the third mode of each rubric is already conceived as a kind of synthesis between the first two). For instance 'necessarily all men are mortal' would seem to fall into definite modes  of the rubric of quantity and of the rubric of modality. We can wonder about how Kant conceived the negation mode of the rubric of quality. For instance "all men are not mortal"  can mean that all men are in fact not moral or that some men are not mortal. In other words, is Kant considering judgment negation and term negation ? Do we have in this example a combination of two modes of the rubrics of quantity and quality respectively ?


Take the hypothetical mode of the rubric of relation.  Kant's example is 'If perfect justice exists then the obstinately wicked will be punished'.  Such judgments would seem to be of the form 'if $J_1$ then $J_2$' where $J_1$ and $J_2$ are themselves judgments. There does not seem to be further evidence in the CPR about whether $J_1$ and $J_2$ can in fact be any judgments (and thus we would have an anticipation of modern inductive definitions) or if there is some kind of constraint.  Anyhow someone may want to express the judgment that 'it is not the case that if perfect justice exists then the obstinately wicked will be punished'. It is difficult to see how Kant would have rejected this judgment which fits both in the rubric of relation and that of quality under the negation mode.  There is also the important question of whether the disjunctive mode of relation corresponds to our modern propositional disjunctive connective.  It seems that the table presents an attempt (perhaps not entirely successful) at an amalgam of the Aristotelic syllogistic (in its full modal version) and the full propositional calculus (which already received a sophisticated development in the Stoics).

We have shown that judgments can fall into modes of more than one rubric.  But it also seems certain that  no judgments that fall under certain combinations of modes of all four rubrics. For instance: what mode of quantity should we assign to hypothetical judgments ?  But Kant's explanation in B100 seems to suggest that the modality modes apply only to assertoric judgments (he speaks of an affirmation or negation).

There is no doubt that Kant distinguished between judgment and proposition Satz. Judgments involve intentional attitudes to propositions. Let us see how B100/A75 can  refine our understanding of the hypothetical judgment. Kant writes that both the antecedent and consequent are problematic but the consequence relation itself is assertoric. Similar considerations seem to apply for the disjunctive judgment. This suggests that hypothetical judgments cannot themselves fall within the problematic mode and thus it is difficult to see how Kant would have interpreted Aristotle's modal syllogistic.

Another question: how does logical equivalence enter into the classification of the table ?  For instance is 'John is not not-handsome' affirmative or negative in quantity ?

Kant uses 'extension' of a term in its traditional sense. And at least as far back as Porphyry's Eisagogê we a correlative to extension called 'comprehension', the accumulation of differences applied to a given genus defining a species. The greater the comprehension the smaller the extension and vice-versa. For Kant individual terms have no extension, they are Begriffe ohne Ausnahme.

Now comprehension determines extension, but this is a many-to-one relation. Thus a synthetic judgment, as explained in B11,  must be in a certain sense 'extensional', that is, equivalent to a relation between extensions such as: the extension of term A is contained in the extension of term B, even though this is not deducible from the comprehension of A and B as in 'a rational animal is an animal'.  Compare how property and inseparable accident differed from definition in Porphyry and Aristotle.


Consider a judgment such as 'if the world is infinite and the world is eternal then the world is eternal and the world is infinite'.

This is an 'analytic' judgment which does seem directly related to the comprehension of terms or conceptual 'containment' of terms.

In conclusion we find that the determination of the exact formal nature of Kant's logic in the CPR - specifically with regards to  expressive power and formal nature of the syntax - is a difficult open problem.

We propose that various interpretations be weighed by the aptness to express key judgments in the CPR (the work \cite{lam} would appear to be relevant to this project). Of course this is being  charitable to Kant as the insufficiency of his logic could be taken at face value as a simply a flaw, specially if we consider how the table of categories derived from the judgment table permeates and guides the entire architectonic of the CPR.


Consider the 'analogies of experience'.  The original version of the second analogy in A was: for everything that happens there is something which succeeds it, according to a rule - Alles, was geschiet (anhebt zu sein) setzt etwas voraus, worauf es nach einer Regel folgt,

\[\forall x (Hap(x) \rightarrow \exists y RegSuc(x,y))\]

At first sight this would seem to transcend the expressive capacities of the logic defined by the table of the forms of judgments.

But we note that according to Bobzien and Shogry Stoic logic could handle this. In their view the Stoics would have expressed it in a regimented expression employing anaphora thus

if something happens then something follows it according to a rule.

The section on Subjective Notion in Hegel's 'Encyclopedia Logic' has a structure which appears very 'conservative' in that is does not seem to differ much from the core conceptual architecture of Kant's logic (although of course Hegel's interpretation and agenda is vastly different). Perhaps Hegel's treatment could shed light on some details of Kant's logic in the CPR.

What would the table of the forms of judgment be in modern terms, if we considered modal monadic first-order logic ?

We could say that any sentence of the logic is in exactly one of the following forms:

(1)$\exists x \phi$, (2)$\forall x \phi$, (3)$A(c)$,  (4)$\neg \phi$, (5)$\phi_1\rightarrow \phi_2$, (6)$\phi_1 \vee \phi_2$,  (7)$\square \phi$, (8)$\lozenge \phi$.

where the $\phi$ and $\phi_i$ can in turn be any of the 8 types. Here $A$ is an atomic predicate and $c$ a constant. Evidently we can organize these in a sequence of pairs corresponding to Kant's 4 rubrics. In classical logic these can be reduced to 5 forms. If each of $\forall$ and $\exists$ can be defined in terms of the other using $\neg$ then which one is to be considered a 'primitive' concept of the understanding ? William Lawvere might have answered: none of them, but rather the concept of adjunction of which both are a case of. We retort: the concept of adjunction depends on both quantifiers for its intelligibility. Furthermore: this problem can be seen simply as a strong argument for the primordiality of intuitionistic logic (in which such a problem does not arise) over classical logic. A topos is naturally intuitionistic, the classical version is a special case. If Kant's antinomies raise many difficulties for a rigorous formal interpretation, perhaps the profound metalogical thought behind them was simply that $A \vee \neg A$ is not a law of reason in general. Thus there need not be a proof either than the world infinite or that the world is not infinite.

Find evidence for second-order monadic logic in Kant.