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 ?

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.

Consider an alien race that only understood first-order logic. How would we explain the rules of Chess, Go or Backgammon ? And how do we humans understand and learn the rules of these games when their expression in first-order logic is so cumbersome and convoluted and extensive ?  Expressing them in a programming language is much simpler...perhaps we need higher-level languages which are still formal and can be reduced to lower-languages as occasion demands. How do we express natural language high-level game concepts, tactics and strategy, in terms of low-level logic ?

Strange indeed to think that merely recursively enumerable systems of signs can represent or express all of reality...how can uncountable reality ever be capture with at most countable languages (cf Löwenheim-Skolem theorems, the problems with categoricity, non-standard analysis, etc.) ? 

All mathematical logic - in particular model theory - seems to be itself to presuppose that it is formalizable within ZF(C). Is this not circular ?  Dare to criticize standard foundations, dare to propose dependent type theory, homotopy type theory, higher topos theory as alternative foundations. 

The Löwenheim-Skolem theorems cannot be used to argue for the uncertainty or imprecision of formal systems because, for instance (i) these results are focused on first-order logic and the situation for second and higher-order logic is radically different (for instance with regard to categoricity). (ii) according to the formal verification principle these metatheorems themselves have to be provable in principle in a formal metasystem. If we do not attach precise meaning to the symbols and certainty to the deductive conclusion in the metasystem what right have we to attach any definite meaning or certainty to the Löweinhem-Skolem theorems themselves ?  

But of course the formal verification principle needs to formulated with more precision for obviously given any sentence in a language we can always think of a trivial recursive axiomatic-deductive system in which this sentence can be derived.  The axiomatic-deductive systems has to satisfy properties such as axiomatic-deductive minimality and optimality and type-completeness, i.e., it must capture a significantly large quantity of true statements of the same type - the same 'regional ontology'. Also the axioms and primitive terms must exhibit a degree of direct, intuitive obviousness and plausibility. And the system must ideally be strong enough to express the 'core analytic' logic.

The formal mathematics project might well be the future of mathematics itself.

The problems of knowledge: either we go back to first principles and concepts, the seeds, but loose the actual effective development, unfolding, richness, life - and also having to bear in mind that the very choice of principles might have to change according to goals and circumstance -  or else we delve into the unfolding richness of science but become lost in the alleys of specialization and limited, partial views.  Either we are too far away to see detail and life or we are too close to see anything but a small part and miss the big picture. Also when we are born into the world 'knowledge' is first forced onto us, there is both contingency and necessity. It is only later that we review what we learnt.  A great step is when we step back to survey knowledge itself, attempt to obtain knowledge about knowledge, to criticize knowledge. Transcendental knowledge is not the same as the ancient project of 'first philosophy'.

If we take natural deduction for first-order logic and assume the classical expression of $\exists$ in terms of $\forall$, then we do not need the natural deduction rules for $\exists$ at all. This can be used as part of my argument related to ancient quantifier logic.  Aristotle's metalogic in the Organon is second-order or even third-order.

Overcoming the categories and semantics - or rather showing their independence and holism. With this theme we can unite such disparate thinkers as Sextus, Nâgârjuna, Hume and Hegel - and others to a lesser extent (for instance Kant). Notice the similarity between the discussion of cause in Sextus, Nâgârjuna and Hegel. The difference is that Sextus aims for equipollence, Nâgârjuna to reject all the possibilities of the tetralemma while Hegel continuously subsumes the contradictions into more contentful concepts hoping thereby to ladder his way up to the absolute. And yet how pitiful is the state of logic as a science....once we move away from classical mathematics and computer science.  The idea of a formal mathematical logic (or even grammar) adequate for other domains of thought, remains elusive ! 

We can certainly completely separate the content and value of Aristotle's Organon and Physics from Aristotle's politics and general world-view. Can we do this for Plato too ? 

Cause-and-effect. The discrete case. Let $Q$ denote the set of possible states of the universe at a given time and denote the state at time $t$ by $q(t)$. Then this will depend on the set of previous values of $t$. Thus determinism is expressed by  functions $f_t: \Pi_{t' < t} Q \rightarrow Q$. Now suppose that $Q$ can be decomposed as $S^B$ where $B$ represents a kind of proto-space and $S$ local states for each element of $b\in B$ (compare the situation in which an elementary topos turns out to be a Grothendieck topos).  Now we can ask about the immediate cause of the states of certain subsets of $B$ at a time $t$ - that is the subset of $B$ who variation of state would change the present state.  But a more thorough investigation of causality must involve continuity and differentiability in an essential way. Determinism, cause-and-effect depend on the remarkable order property of the real line and indeed on the whole problem of infinitesimals...

The problem with modern physics is that it lacks a convincing ontology. Up to now we have none except the division into regions of space-time and their field-properties. Physics should be intuitively simple. But all ontologies are approximative only and ultimately confusing.

Does Lawvere's theory of quantifiers as adjoints allow us to view logic as geometry ? $\exists$ corresponds to projection and $\forall$ to containment of fibers. Let $\pi: X \times Y \rightarrow X$ be the canonical projection and let a geometric object $P \subset X\times Y$ represent a binary predicate. Then $\exists y P(x,y)$ is represented by the predicate $\pi(P) \subset X$ and $\forall y P(x,y)$ is represented by $\{x \in X: \pi^{-1}(x) \subset P\}$. For monadic predicates we use $\pi: X \rightarrow \{\star\}$ so that for $P \subset X$ we have that $\exists x P(x) = \{\star\}$ corresponds to $P$ being non-empty and $\forall x P(x) = \{\star\}$ corresponds to $P = X$. Combining this we see that $\forall x \exists y P(x,y)$ corresponds to $\pi(P) = X$ and $\exists x \forall y P(x,y)$ corresponds to $P$ containing a fiber $\pi^{-1}(x)$. Exercise: interpret the classical expression of $\forall$ as $\neg\exists\neg$ geometrically.  Conjunction is intersection, disjunction is union. What is the geometrical significance of classical implication $P \rightarrow Q$ as $P^c \cup Q$ (for monadic predicates). This is only $X$ if $P \subset Q$. So it measures how far we are away from the situation of containment.

We have meaning M and project it to a formal expression E in a system S. Then we apply mechanical rules to E to obtain another formal expression E'. Now somehow must be able to extract meaning again from E' to obtain a meaning M'.  But how is this possible ? Reason, argument, logic, language - it is all very much like a board-game. The foundations of mathematics: this is the queen of philosophy.

Jan Westerhoff's book on the Madhyamaka, p. 96.  I fail to see how the property "Northern England" can depend existentially on the property "Southern England".   Because conceptual dependency only makes sense relative to a formal system.  I grant  B may be a defined property and A's definition may explicitly use B. But why can't we just expand out B in terms of the primitives of the formal system in use ? And what does it even mean for two concepts to be equal ? What are we doing when replacing a concept by its definition (and Frege's puzzle, etc.) ?  

A must read: Hayes' essay on Nâgârjuna. Indeed svabhava is both being-in-self and being-for-itself !

T.H. Green on Hume is just as good as anything Husserl or Frege wrote against psychologism or empiricism.

René Thom: quantum mechanics is the intellectual scandal of the 20th century. An incomplete and bad theory  that includes the absolutely scientifically unacceptable nonsense of the 'collapse of the wave-function'. 

Bring genetic epistemology (child cognitive development) into the foreground of philosophy. Modify Husserl's method into a kind of phenomenological regression.

When we say 'we' do we mean I +  he/she or they  - or something different ?

On Schmanalyticity and A Reappraisal of the Synthetic A Priori

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

The meaning of quantifiers

Quantifiers very likely are equivocal. These differences in meaning are revealed when they occur within an intensional context. There are universal quantifiers which express the sum total of the instantiations (which must pertain to a finite or constructible domain) and those that do not (for instance mass nouns). There are quantifiers that are subordinate to relations of intension. Also in our previous post we argued that quantifiers express the ability to understand and follow rules, to do computation. All rules can be expressed via universal monadic second and first-order quantification (which is Aristotle's implicit metalogic in the Topics). There are also quantifiers that express generic properties.

It seems very plausible that all quantifiers are implicitly limited to some domain (even for 'everything'), that unbounded quantifiers are meaningless.  We should focus specifically on nested quantifiers $\forall (x:A) \exists (y : B). P(x,y)$ and $\exists (x: A)\forall (y :B). P(x,y)$. The first has meaning: every A has a P, and plausibly that P can be found through a computatable function.The second has meaning: there is an A which is the P of every B, which plausibly can be checked.

Perhaps Aristotle took pairs (combinations) of natural deduction rules for quantifiers as primitive rules.  Notice how $\exists E$ and $\exists I$ can combine, the latter being 'in the middle' of the former.

It is plausible that the Mitchell-Peirce system of quantifier logic (1885) is not only technically and philosophically superior to Frege's Begriffsschrift (1879) but was developed earlier (by Peirce's student O. Mitchell).  Modern dependent type theory uses Peirce's symbols $\Pi$ and $\Sigma$. Or even the Boole-De Morgan relation calculus approach can make this claim. For instance $\forall x \exists y R(x,y)$ can be formulated as $RU \cap \Delta = \Delta$ where $U$ is the universal relation and $\Delta$ is the diagonal relation. $\exists x\forall yR(x,y)$ can be expressed as $UR = \Delta$. $R(a,b)$ is expressed as $(a,b) \in R$. So we have an open theory expressing first-order logic for monadic and binary predicates. $R$ being transitive is simply expressed as $RR \subset R$. But what about when we leave relations as in $S(x,y,z) = R(x,y) \& R(y,z)$ ? Although obviously we cannot define such relations in our calculus it would appear (check this) that every first-order sentence or monadic or binary relation over binary predicates can be defined in the calculus (sentences via equations). Also how would we write $\forall x(G(x,x) \rightarrow \exists yH(y,x))$ which is equivalent to $\forall x\exists y(\neg G(x,x)\vee H(y,x))$ ? Maybe $((G\cap\Delta)^c \cup H^{-1})U \cap \Delta = \Delta$ ?

In Leibniz's Non inelegans specimen there are certainly definitions which employ $\exists$. The containment relation (let us write it $A\leq B$) is defined (Def. 3) as 

\[ A \leq B \equiv \exists N(B = A\oplus N)\]

But consider a general situation in which we have a defined predicate $P(x_1,...,x_n) \equiv \exists y_1...y_n\phi(x_1,...,x_n,y_1,...,y_n)$. Then there are situations in which  a $\psi$ involving occurrences of $P$ can be transformed into an equivalent (or rather stronger) sentence of the form

\[\forall z_1.....z_m\psi'(z_1,...,z_m)\]

in which $P$ does not occur, being replaced by $\phi(x_1,....x_n,y_1,....y_n)$ where the $y_i$ occur among the $z_i$.  For instance using Skolemization or constructivist techniques. Perhaps this fragment of Leibniz could be expressed in open logic. Transitivity of inclusions might be expressed as

\[\forall N,M((B = A \oplus N) \& (C = B \oplus M) \rightarrow C = A \oplus N \oplus M) \] Maybe Aristotle's ekthesis had to do with producing universal constructive sentences which are stronger than the sentence to be proven. Open formulas (unsaturated lekta) are problematic (and they not just universally quantified in disguise ?). An important task is to reformulate the axioms and rules of modern logic to deal only in closed formulas (sentences).

As for metalogic or metamathematics we must never forget that all results are themselves necessarily formalizable within certain recursive-axiomatic systems (including the very same system as the object system). Thus metalogic and metamathematics is just the study of formal reflections of systems into each other as well as self-reflection.  However knowledge and computation in recursive-axiomatic systems presupposes always a logical system at least as strong as monadic second-order logic. 

Hume's Treatise contains the following logical observations (when discussing causality): from the fact that every husband has a wife it does not follow that every man has a wife. Let $M(x,y)$ be the marriage (husband-wife) relation between a man $x$ and woman $y$. Then $Husband(a) \equiv \exists x M(a,x)$ and $Wife(a) \equiv \exists y M(y,a)$. Certainly $\forall x Husband(x) \rightarrow \exists y M(x,y)$ and $\forall x Wife(x) \rightarrow \exists y M(y,x)$. But it does not follow that $\forall x \exists y M(x,y)$.

Addendum on Aristotle:

We should examine the logic in Book V of Euclid which is supposed to be a presentation of Eudoxus' general theory of analogy. There are quaternary predicates involved $A(a,b,c,d)$  (expressing that $a$ is to $b$ as $c$ is to $d$) which have purely universal definitions. How much of Book V could be derived without $\exists$-rules ? 

What about monadic second-order logic ? Is this Aristotle's metalogic in the Analytics ? For instance what is the enunciation of Barbara (ignoring the existence condition) but

\[  \forall P,Q,R (\forall x(Px \rightarrow Qx) \& \forall x(Qx \rightarrow Rx) \rightarrow \forall x(Px \rightarrow Rx))  \]

How can this logic represent the logic of relations and multiple generality (we already know Prawitz's answer for full-second order logic representing first-order existential quantification) ?  That is, could we argue that ancient logic dealt with multiple generality through some kind of second-order monadic logic (perhaps with some basic binary relations) ? 

Or else Aristotle could have worked with the classical equivalence between $\exists$ and $\sim\forall\sim$.  

That is with natural deduction for classical predicate logic but without the rules for $\exists$. Instead use is made of the derivable Hilbertian axiom-schemes together with modus ponens (that is the simplest way to transform a proof using $\exists$-rules into one without them: $\phi(a) \rightarrow \exists x\phi(x)$ and \[\forall x(\phi(x) \rightarrow A) \rightarrow (\exists x\phi(x) \rightarrow A)\] where $x$ is not free in $A$. To prove this one can use the contrapositive of \[(\forall x\phi(x) \vee A) \rightarrow \forall x(\phi(x) \vee A)\]

For instance from the definition $Gxy \equiv \exists z(Rxz \& Rzy)$ and axiom $\forall x\exists y Rxy$ to derive $\forall x\exists y Gxy$, that is, show that \[ \forall x \sim \forall y \sim Gxy \] With Galen's hypothetical syllogism and the logic of the Topics and the proper theory of negation and conversion Aristotle had the full power of classical first-order logic - but we must deal with the problem of nested quantifiers vs.  prenex normal form.

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.

The collected works of F.W. Lawvere

https://github.com/mattearnshaw/lawvere

On the Field-only Approach to Quantum Field Theory

This post consists in  only some incomplete sketches and is obviously very tentative.

René Thom called quantum mechanics 'the greatest intellectual scandal of the 20th century'. Maybe this was too harsh, but quantum theory was meant originally just as to be crude provisional proto-theory destined to give place to something to better (which has not...due to political, military, economic and industrial reasons ?). Consider the double-slit experiment. The 20th century was also the century of dynamical systems and chaos theory. It is clear to us that the random aspect of the double-slit experiment must be explained in light of chaos theory, thus of an underlying deterministic system. In a classical setting there will also be a pseudo-random aspect for particles traversing the two slits (but without the interference pattern). Nobody would think of interpreting this as a probabilistic collapse of a wave-function. In the non-classical situation it would occur to almost anyone to see the wave-function as a real physical field associated to particles (a "pilot-wave"). If we rule out local hidden variables (but do we really need to ?), then we are lead to non-local yet deterministic non-linear systems which generate the pseudo-random phenomena of quantum theory in the standard way of chaotic dynamics. Even numerous colliding perfectly elastic particles is a deterministic system which yields Brownian motion. To do: study the argument involving single photons and half-silvered and full-silvered mirrors described in Penrose's The Emperor' New Mind p.330 (1st edition).  Both the photon and the wave-function are real existing physical entities and the randomness of the reflection can be given an underlying deterministic explanation. Some wave-packets are empty of particles yet still have physical meaning. We could also consider space as being like a Poincaré section for some higher-dimensional continuous dynamic. Of course there is an easy objection to our proposal: what about maximally delocalised solutions for the free particle Schrödinger equation ? Due to many other difficulties we could also take A. Hobson's approach that  'there are no particles, only fields'.  Here small-scale irregularities, the fact that we are dealing with approximations, etc. could well explain the 'collapse of the wave function' - if we postulate that quantum fields are to have here an intrinsic holistic nature so that their localized interaction around the boundary of their support entails an immediate (or very fast) alteration of the entire field (Hobson gives the analogy to popping a balloon). In the double-slit experiment if we think of the wave-function as a single entity then in reality only one small portion of the wave-front will hit the screen first - which will be determined by sensitivity to initial conditions and many perturbations and irregularities in the instruments involved in the experiment. This, based on Hobson's own analysis, could furnish the missing piece to eliminate any appeals to probability, even in a field-only interpretation. The $|ready >$ state is itself complex and fluctuating (deterministically). Thus the pseudo-randomness of which $A_i$ region will effect the 'pop'. However there seems perhaps to be a difficulty in interpreting the apparently random aspect of the experiment above discussed by Penrose (it would suggest that the result of  'popping the balloon' must still be considered random).  But is this experiment really so different from the double-slit one ? We need to find the inner geometric deterministic dynamics of field interactions that could account for this behaviour. Maybe use the fact of the interference of the environment (and entanglement) in all experimental conditions. 

 

Hobson interprets $|\Psi|^2$ as the probability of interaction of the field. We need to add an extra dimension to $\Psi$  and an accompanying deterministic non-linear dynamic field-process (as in nonlinear PDEs) which explains the resulting interaction probabilities in a totally deterministic way. This is where chaos theory is the key to quantum theory. This applies to interaction and to spin-measurement. Consider the classical orbitals of the Hydrogen atom. Some have nodal points which seem to rule out a particle interpretation. Also spin basically involves extending the phase space of the original wave function, for instance for a single particle $L^2(X, \mu) \otimes \mathbb{C}^2$. Thus our proposal is not surprising. On the other hand if we consider the orbitals of the Hydrogen atom is seems natural that they should posses also some kind of dynamic nature in an extended dimension (analogous to spin) related to the amplitude of the original wave function.

In the Penrose experiment considered above consider the detectors in two distant locations in which each spin configuration has a 1/2 percent chance and in which the two measurements are always correlated. We view the electron wave as a single entity even if divided into two packets. The unity is expressed in the phase in the extended dimension which oscillates not as two independent oscillators (one for each packet) but as a single oscillator, thus guaranteeing the correlation of the measurements. 

A model: a packet could have a phase oscillating between UP and DOWN  in the extended dimension which determines the measurements (interaction probabilities). But two coupled packets would oscillate between UP x DOWN and DOWN x UP globally.

Consider two distinct localized wave packets  $P_1$ and $P_2$ centered around points $-x$ and $x$ for $x > 0$ larger than their wave-length. If $P_1$ moves forward and $P_2$ moves backwards so that they exchange places then the resulting quantum state and hence the physical state of the system will be exactly the same as it was initially. Thus the "indiscernibility of identicals" follows immediately from the field-only approach while it is problematic for the particle approach.

Miscellany of philosophical observations

1. Quantum theory gave us the idea of introducing negative probabilities, i.e. signed measures. 

2. Category theory is intensional (non-extensionalist) mathematics based on minimal logic, thus hyper-constructive.  We ask about a natural number object (the concept of an 'element'  is not taken as a primitive; rather we have only generalized elements $1 \rightarrow A$) in a given category, that is, about its universal property;  we construct concrete generalized element 'numbers' through composition of primitive morphisms $z : 1 \rightarrow N$ and $s : N\rightarrow N$. Recall how the concept of primitive recursive function emerges naturally from this definition...

4. There have always been different notions of 'quantification' (and the corresponding determiners) which were conflated by extensionalist logicians.  This is clear in the distinction between intensional, conceptual universal quantification and extensional quantification. Also such distinctions are brought to light by the behaviour of quantifiers in propositional attitudes.  Constructivism tried to bridge the gap between extension and intension via a kind schematism (see previous post). We must bring all the different kinds of quantification to light again. 'Some' seem to be even richer in nuances than 'for all'. The distinction between the classical and intuitionistic/constructivist 'some' is deeply rooted in and reflected in cognition and natural language semantics. For instance, the intuitionistic interpretation fails for existential formulas in the scope of propositional attitudes. I may believe that the money in a book in the library without there being a specific book in which I believe the money is in.

Are set-theoretic extensions are atomistic structureless heaps, like the extreme abstract atomic alienated negativity in certain stages of Hegel's phenomenology of spirit ? This is not really so, they can have a very definite tree-like structure. Groupoids have more organic unity. We must investigate what it means to quantify over groupoids.

5. Some people are scared of homotopy type theory, higher category theory or of Coq and Agda. I respect that.  I feel the same about fractal calculus. But perhaps fractal calculus has something to do with the following important question. Numerical, discrete, computational methods are routinely used to find approximate solutions of differential (and integral-differential) equations. But we also need in a turn a theory of how differential and smooth systems can be seen as approximations of non-differential and non-smooth systems. Is this not what we do when we apply the Navier-Stokes equations to model real fluids ? Recall how continuous functions with compact support are dense in the $L^p$ spaces of integrable measurable functions (but see also Lusin' theorem).  Can all this be given a Kantian interpretation ? An analogy of experience: how the very notion of measurable function supposes the standard topology and Borel structure on the real line $\mathbb{R}$.

6.  What are distributions ? They allow a mathematical treatment of the vague notion of particle. Indeed particles are just euphemisms for certain kinds of stable self-similar field-phenomena. The great geniuses in physics were those who helped build geometric physics (which is what is most developed and sophisticated in modern physics):  Leibniz, Lagrange, Euler, Hamilton, Gauss, Riemann, Poincaré, Minkowsky and many others.  But it is no use playing around with highly sophisticated geometric physics (which looses all connection to experiment)  if you haven't solved the problem of quantum theory first. 

Distributions are clearly in themselves meant to be idealizations and abstractions of actual functions with their ultimate aim being approximation results. What is a dirac function ? This will depend on the scale. Dirac functions in nature are only approximate.

7. Study differential geometry as type theory; dispel all difficulties in a general understanding of mathematics as a language. It is of utmost importance to give physics, specially quantum theory, great formal logical and mathematical and philosophical rigour. Outstanding example: Peter Bongaarts' book.

8. Many of our concepts have a tripartite nature $(A, A^\circ, \bar{A})$ expressing certain $A$, certain not $A$ and the grey neutral area $\bar{A}$. For instance: bald, not bald, sort of bald but not really bold. Each one in turn will depend on an individual and a possible situation of affairs. But this is not enough. In order to do any kind of 'logic' here we need some kind of quantified probability measure, for instance the ability to measure quantities of individuals and states of affairs. Then the sorites is resolved by presenting a tripartite distribution.  Thus it is interesting to have a logic which can express probability distributions.

9.  The goal is to pass from language-based philosophy to pure logic based philosophy. But this needs a mediator. The mediator can only be advanced, sophisticated, mathematical models, qualitative, essential, extending to all domains of reality (deformations, moduli are the right way to study possible worlds). All aspects of Kant and Husserl can be given their mathematical interpretation and from thence their logical-axiomatic interpretation. The same goes for naturphilosophie via René Thom and Stephen Smale. Theoretical platonism and idealism is not enough. We need this realized applied platonism. Mathematics furnishes a rigorous way of dealing with analogy and integrating analogy into philosophy. Also mathematics furnishes the deeper meaning and interpretation of Kant's theory of categories and schematism. Mathematics furnishes us with a way of studying concepts which is not divorced from the conceiving mind but at the same time is not psychologistic.

10. How do mathematicians think, actually prove theorems and have insight and intuition - all of which is very different from a low-level proof-search for some formal axiomatic-deductive system ? In particular how can formal logic and intuition agree ?  If logic is the science of valid thought, then it just cannot ignore this question.  We certainly think immediately using admissible rules.

Consider a formal logic $L$ in which we have the concept of atomic predicate, equivalence and equality. Let $T$ and $P$ be a countably infinite set of symbols no occurring in the language of $L$. By a prelogic we mean a finite set $(t,p)$ consisting of a finite sets $t,p$ of formulas in $L(T,P)$ of the form $q(x_1,...,x_n) \equiv ...$ and of the form $t(x_1,...,x_n) = ...$. We write $(t_1,p_1) \leq (t_2,p_2)$ iff the symbols in the left sides of $t_2,p_2$ all occur in $t_1$ and $p_1$ and furthermore if $t_1 \subset t_2$ and $p_1 \subset p_2$.  For each prelogic $(t,s)$ we further associate a set of intuitively valid sentences $ISen \subset Sen(t,s)$ and intuitively valid inferences  $IDed$ which are subsets of $Sen(t,s) \times Sen(t,s)$, where $Sen(t,s)$ denotes the set of sentence whose symbols occur all in $t,s$.

11. The problem of the denotation of the selection of one of two orientations of vector space or one of the square roots of $-1$. 

12. Some important authors to study: Albert Lautman and Jean Petitot. A synthesis of Kant and Husserl within the framework of an enlightened mathematical structuralism.

13. Determinism may be only local. Determinism (think analytic continuation) is like a covering space. Only one continuation and lifting of a path for a chosen point in the fiber. But we can have instead of a locally constant sheaf a constructible sheaf. There is a stratification in which non-deterministic switches or choices take place (although they can be perfectly continuous).

14. What is completeness for a logical-deductive system ? And relative to a class of models ? Take intuitionistic propositional logic.  The classical logical-deductive notion of completeness does not apply anymore. Only a model theoretic one.  And the model theoretic one needs to change to become multi-valued, i.e. as in topos theory or at least the Heyting algebra of truth-values. This was the insight behind Kant's transcendental dialectic: that $A \vee \neg A$ is not a universal law of reason.

Schopenhauer on the Content of Compassion

https://phil.washington.edu/research/publications/schopenhauer-content-compassion

Instead of simply saying that the compassionate person perceives no distinction between herself and the object of her compassion, we should say that she perceives them to not be distinct spatiotemporal individuals. That is, she perceives them to be distict only in the way that Platonic ideas are distinct. The latter distinctness is not sufficient for individuality in the normal sense, for Schopenhauer, since he calls space and time alone the principle of individuation (OBM 4:267, p. 250). The key difference is that, at the level of ideas, things metaphysically overlap with each other in ways that they do not at the spatiotemporal level. (p.7)

So we have here a holology which also suggests comparison with Plotinus' theory of how ideas are unified in the nous.

Hegel's phenomenology of spirit is the antithesis of both Kantian and Schopenhauerian ethics. It is essentially anti-transcendent, pragmatist, relativist, collectivist, deterministic and culminates in a totalitarian-statist mysticism (a fascism based on an esoteric Christianity which subsumes and buries the the possibility of the resurgence of the enlightenment).  This is seen in the treatment of the phase of practical reason and its transition into "spirit". Hegel turns Kant's noumenon into his secular pragmatic collectivist fatalistic thing itself Sache Selbst. From thenceforth it is no longer about the individual but only the drama of the collective. The individuality of the individual is allowed to manifest in its "negativity" only for the sake of, and in function of, the development of the power and self-transparency of the collective.  What corresponds to this 'spirit' is the third section of what is inappropriately called 'Begriff' in the Science of Logic, the weakest and most ad hoc part,  which appropriates the far deeper insights into the structure of consciousness found in Kant and Fichte. 

At the basis of ethics must be a consciousness which does not make a distinction between self and others or between today, yesterday and tomorrow. Ethics concerns the timeless individually directly cognizable universal ought which is completely independent from any hypothetical necessary developmental law or process just as it has nothing to do relativistically with arbitrary convention. At a social-cultural level there can indeed be ethical progress, but this should never be seen as the working of some kind of natural law or the result of necessity. Confusing the ideal of human ethical progress with speculation about evolution in natural science has been a very serious error.  Human ethical progress is a normative ideal never a law or natural necessity. A normative ideal that remains invariant throughout recorded historical humanity, even if tragically it seems to be less and less realized in the world today. There are non-human sentient beings which cannot be subject to the normativity of the moral law. But we could explore how there is an implicit, albeit imperfect, morality already at work in nature. What we must reject are arbitrary speculations attempting to link non-human and human beings whereby such a link serves as a foundation or explanation of morality.

The correct theory of ethics is much like Frege's  philosophy of logic. Or, to paraphrase Husserl:

Whatever is a moral duty, is absolutely, intrinsically a moral duty: the moral law is one and the same for men or non-men, angels or gods. Moral laws speak of the ought in this ideal unity, set over against the real multiplicity of races, individuals and experiences, and it is of this ideal unity that we all speak when we are not confused by relativism.

The moral law implies that we should strengthen our historical organizations dedicated to the universal unconditional upholding of human rights and international law.

Analytic metaphilosophy

By analytic metaphilosophy we mean a methodology which aims to apply mathematical and formal logical rigor and the full force of linguistic analysis to philosophical texts in order to assess their argumentative and epistemic value. Analytic metaphilosophy has a strong affinity and connection to early classical analytic philosophy (though it has many brilliant precursors before that time) but no connection to subsequent post-classical analytic philosophy - indeed it can be conceived as the ultimate tool to thoroughly refute and show the nonsense and vacuity of its various currents as well as some of those that claim to break with it.  A novel aspect is that it attaches an enormous importance to the style of philosophical texts and aims to be far more wary and careful about the pitfalls, delusions and psychological foibles of the whole process of literary creation. It never forgets that the philosophical writer is never far from the precipice of literary fantasy often exhibiting egocentric and sycophantic qualities geared to socio-economic advantages rather than epistemic and ethical goals.  Analytic metaphilosophy studies in particular the sociology and psychology of sophistry and literary delusion. More Platonico we shamelessly proclaim that it is impossible to engage in analytic metaphilosophy without either a solid undergraduate background in mathematics and mathematical logic or at least a couple of years of experience working on a formal mathematics project employing a proof assistant.  In a nutshell: analytic metaphilosophy applies mathematical standards of logic and rigour to philosophical texts and refuses to be impressed by the mirages and artifices of language (though an ideal philosophical text can have also beauty, elegance and clarity of style as in the writings of Frege and Claire Ortiz Hill). Jargon-laden and convoluted texts rarely betray deep, complex, rigorous or valid thoughts. Such pseudo-difficulty is of an entirely different nature from the 'difficulty' of mathematical texts. Analytic metaphilosophy can also be seen as a kind of prolegomena and justification for the formal philosophy project we described in previous posts.

In that most rigorous, clear and certain of the sciences, mathematics,  mistakes and confusions still arise, there are gaps in proofs, there are unjustified assumptions, careless generalizations, confusions in terminology, silly oversights, circular reasoning, etc. As the length of the proof increases so does the probability of error, even for the best mathematicians and Fields Medalists like Vladimir Voevodsky. Careful checking by several experts is absolutely necessary. In some areas the length of the proofs become almost too long for this to be feasible, so proofs are formalized and run through specialized proof checking software.

Now consider philosophy, the least rigorous, clear and certain of human epistemic enterprises. Once a philosophical 'argument' becomes long, convoluted and (on the surface level) complex one can be almost certain that it is wrong or inconclusive. The same goes for texts with an elevated number of technical terms or  'jargon density', so to speak. 

Mathematicians have an artist's liberty to use and invent symbols for their primitive and defined notions and variables.  The philosopher, shackled by natural language and lack of mathematical training, is in a very dire situation, terms are pathologically and enormously semantically overloaded and the resulting terminology is opaque, ambiguous and stylistically repugnant. Perhaps this can be partially overcome by the construction of an artificial language for philosophical terms.

The majority of philosophical texts have nothing to do with the logical standards and rigour of mathematics or even the exact sciences. What reason is there to attribute to them any meaning or epistemic value at all ? Or even social value ?  And many of their authors are aware of this, don't care, or consider it a virtue.  They have their communities with their founding narratives and (non-reflectively) received doctrines and they happily engage in their 'language-games'  and strictly controlled boxed (bottled ?) 'innovations'. They have their own 'logic' and 'rules' and 'criteria' for parsing and deciding the value or legitimacy of textual-output - and this output is a torrent, an endless deluge and logorrhoea that seemingly might be generated by large language models. Ex falso quodlibet. 

Analytic metaphilosophy (which favours Martin-Löf type theory as an adequate intensional and modal logic) is entirely immune from objections culled from mathematical logic itself such as all-too-common misunderstandings and misappropriations of Gödel's incompleteness theorems.  Although mathematics can be conceived as a subset  of logic (by assuming special axioms such as univalence), logic can also be conceived as an application of mathematics. There is no concept of computability without involving arithmetic and no concept of arithmetic which does not involve some notion of computability.

Recall that we hold that logic is embodied in a closely unified (organic) family of formal systems which are related to each other by gradation or (mutual) embeddability and reflection. There is no trace of convention or arbitrariness.

And in nowise does our metaphilosophy claim to be philosophy itself or a substitute for it. However knowledge of applied mathematics at an advanced theoretical level has huge conceptual advantages for thinking about possible worlds, possibility, causality, identity and states of being which far surpass the crude models used in analytic philosophy.

Analytic metaphilosophy can be integrated seamlessly into the Gödelian and Husserlian frameworks as complementing and helping the metaphilosophy and methodology of ultimate evidences and intuitions as well as ethical metaphilosophy*. It works alongside it and provides powerful aid by refuting anti-idealist arguments.

Schopenhauer's detailed criticism of Kant in the WWR and T. H. Green's long introduction to Hume's Treatise are  good examples of pre-Fregean analytic metaphilosophy. The investigation of the expression of multiple generality and its associated reasoning in ancient philosophy is clearly a cornerstone to analytic metaphilosophy's  approach to ancient and early modern philosophy.

*ethical metaphilosophy focuses on the explicit and implicit content relating to human and animal rights, in particular the status, dignity and consciousness of animals,  in historical philosophy - and thereby comes to general conclusion about the value and merit of different philosophical systems.  Leaving aside ancient eastern philosophy, the cases of Porphyry, Schopenhauer and Husserl are enough to de-fang ethics-based anti-idealist arguments whilst arch-anti-idealist Nietzsche's raving praise of Descartes' view of animals delivers a fatal counter-blow.

Medieval philosophy has been criticized for being the handmaiden of theology and merely a tool for the apologetics of the dogma of organized religion.  If this is justified then philosophy also cannot be allowed to be the handmaiden for para-scientific ideology and dogma either, which is what we find predominantly in the so-called 'philosophy of mind'. Is there anything more silly than  a priori arguments for speculations  based on incomplete or faulty experimental science ? Image the money it saves on equipment and resources for the materialist neuroscientist.

Nevertheless Kant's language is often indistinct, indefinite, inadequate, and sometimes obscure. Its obscurity, certainly, is partly excusable on account of the difficulty of the subject and the depth of the thought; but he who is himself clear to the bottom, and knows with perfect distinctness what he thinks and wishes, will never write indistinctly, will never set up wavering and indefinite conceptions, compose most difficult and complicated expressions from foreign languages to denote them, and use these expressions constantly afterwards, as Kant took words and formulas from earlier philosophy, especially Scholasticism, which he combined with each other to suit his purposes; as, for example, "transcendental synthetic unity of apperception," and in general "unity of synthesis" (_Einheit der Synthesis_), always used where "union" (_Vereinigung_) would be quite sufficient by itself. Moreover, a man who is himself quite clear will not be always explaining anew what has once been explained, as Kant does, for example, in the case of the understanding, the categories, experience, and other leading conceptions. In general, such a man will not incessantly repeat himself, and yet in every new exposition of the thought already expressed a hundred times leave it in just the same obscure condition, but he will express his meaning once distinctly, thoroughly, and exhaustively, and then let it alone. "_Quo enim melius rem aliquam concipimus eo magis determinati sumus ad eam unico modo exprimendam_," says Descartes in his fifth letter. But the most injurious result of Kant's occasionally obscure language is, that it acted as _exemplar vitiis imitabile_; indeed, it was misconstrued as a pernicious authorisation. The public was compelled to see that what is obscure is not always without significance; consequently, what was without significance took refuge behind obscure language. - Schopenhauer, WWR (II).

A filosofia de Ludwig Wittgenstein à luz do diagnóstico de autismo

 https://philarchive.org/rec/SILAFD-6

Meaning of the logical connectives

The meaning of the implication/conditional operator $A\rightarrow B$ is simply that of a relation of truth values (as Kant described the hypothetical judgment in the CPR). It has nothing to do with causality, inference or relevance.  If we take $0$ as false and $1$ as true then $A\rightarrow B$ is simply the proposition which states that the truth value of $A\leq $  truth value of $B$. What is paradoxical about the fact that for any proposition $A$ we have that 0 $\leq$ truth value of $A$ ? What is paradoxical about the fact that given any two propositions $A$ and $B$ we have that either truth value of $A\leq$ truth value of $B$ or truth value of $B\leq $ truth value of $A$ ? Relevancy is irrelevant in the face of propositions regarding the relationship of the truth-values of propositions - which are purely  mathematical. Logical connectives are in a way a reflection-into-self of logic, they are propositions - having truth values - about the truth values of propositions. This is clear even in the semantics of linear logic, interpreted as a many-valued logic.  And the many-valued truth value of $A\& \sim A$ can be seen for instance the the result of a voting process. There can be a draw between $A$ and $\sim A$ and this itself be a value.

In general implication means that there is some computable function that takes terms inhabiting in $A$ into terms inhabiting $B$.  That is, we can compute $B$s in terms of $A$s.  Connectives are semantically truth-value based or in general witness based. Their legitimacy and value is untouched.  We can however think of an additional, alternative theory of intensional connectives, relevance, inference and causality. Notice that if an effect is unique to its cause then classical logical connectives cannot capture causality.

Category theory has since decades developed a useful tool for dealing with contextualism and pragmatics: fibered category theory.

Even Girard's linear logic can be understood in terms of phase semantics; as an algebraic many-valued logic.  $\multimap$ is interpreted much like in realizability or dependent type theory.

In case you didn't know

Subjective idealism or the idealism in general found in Kant (correctly interpreted), Schopenhauer and Husserl has nothing to do with relativism or psychologism and is immune to all anti-psychologist arguments (including Moore's arguments against Berkeley-style 'idealism'). Nor can objectivity be founded on any simplistic and confused empiricism; nor a logic based on nominalist, conventionalist  of socio-pragmatic premises - which ignores the irreducible reality of intensional entities - even deserve that name. Sophisticated formal languages such as dependent type theory are meaningful and epistemically adequate in their own right, existing alongside natural language.  There is nothing cognitively or ontologically normative about natural language, let alone the English language.  One of the most unjustifiable and harmful  errors is stating that mathematics is justified solely by its applications in natural science or its indispensability in the language of natural science. This thesis is laughable to anybody with any serious knowledge of the history of mathematics and theoretical physics.

As in Kant, philosophy may form a tightly-knit organic whole. There is no good reason why ethics and the theory of knowledge might not exhibit vital connections to each other. Departments do not have to mean water-tight compartments. Divisions need a justification from a higher perspective, just as the species of a genus share both community and difference. Ethics rests on theory. The activity of theory can itself have a deeply ethical significance: sapere aude !

Historical progress will always be only an ought rooted in individual freedom and endeavor, never an automatic necessity.

Conceptual engineering is just the sociological version of the old psychologism. As sociology it is interesting and has its merits. But it does not contribute anything of philosophical value per se, although it can suggest problems such as the critical analysis of sociologically accepted fallacies and contradictions implicit in popularly used jargon and terminology.

Socrates and Husserl suffered similar fates: they were 'killed' by their times. The message and spirit of Husserl's philosophy (after Husserl himself was banned by the Nazis) was killed and then appropriated by existentialism, naturalism and Catholic neoscholasticism.

Philosophy has its stand-up comedians such as those that argue for logical nihilism.

The stronger one's scientism the greater the probability of having a very little knowledge either of scientific theories or the scientific method(s). Philosophy, logic, mathematics and ethics are epistemically (cognitively), semantically, ontologically (topically) robustly independent, preeminent (i.e. a priori) valid disciplines in their own right and do not depend on nor are subservient in any way on natural science - rather they furnish vital elements necessary for the progress of natural science (and this includes ethics, of course).

There is one name for the synthesis of many of the worst philosophical errors of the past: (neo)pragmatism, linguistic pragmatism, meaning-as-use - Pittsburgh School conceptualism and inferentialism.

To the most advanced among the exponents of the New Age logic even this is not enough. Why, they ask, cling dogmatically to consistency ? Why not jettison the law of non-contradiction (...) Men of action (the Lenins and Hitlers of this world) have long been familiar with the advantages of embracing contradictions. They know that it not only neatly solves all problems in logic proper, but provides an intellectual key to 'final solutions' in other fields of human endeavour. (Pawel Tichý)

Just because Rorty was 'canceled' by the reigning philosophy does not make him ipso facto into some kind of hero of truth valiantly defending a radical alternative; on the contrary it can well be that his program was actually worse that the status quo and just represented in a more thorough way its ultimate consequences or original motivations. Rorty was the Trotsky of analytic philosophy.

https://chryssipus.blogspot.com/2023/10/pyrrhonian-strategy-in-rortys-mirror-of.html

Regarding Rorty let us quote from J.N. Mohanty's The possibility of transcendental philosophy (1985) p.59 :

Impressive as he is in his scholarship, he has given very few arguments of his own. He uses Sellars' arguments against the given and Quine's against meaning, as though they cannot be answered, but he has done little to show they cannot be. He plays one philosopher against the other, and would have one or both dismissed, according as it suits his predelineated moral. These are rhetorically effective but argumentatively poor techniques. What does it matter if Sellars rejects the concept of the given - one may equally rhetorically ask - if there are other good philosophers who accept the viability of that concept? There is also an implied historicist, argument that has little cutting edge. If the Cartesian concept of the mental had a historical genesis (who in fact ever wanted to say that any philosophical concept or philosophy itself did not have one?) whatever and however that origin may be, that fact is taken to imply  that there is something wrong about the concept.

The Young Carnap's Unknown Master

https://www.routledge.com/The-Young-Carnaps-Unknown-Master-Husserls-Influence-on-Der-Raum-and-Der-logische-Aufbau-der-Welt/Haddock/p/book/9780754661580

Examining the scholarly interest of the last two decades in the origins of logical empiricism, and especially the roots of Rudolf Carnap’s Der logische Aufbau der Welt (The Logical Structure of the World), Rosado Haddock challenges the received view, according to which that book should be inserted in the empiricist tradition. In The Young Carnap's Unknown Master Rosado Haddock, builds on the interpretations of Aufbau propounded by Verena Mayer and of Carnap's earlier thesis Der Raum propounded by Sahotra Sarkar and offers instead the most detailed and complete argument on behalf of an Husserlian interpretation of both of these early works of Carnap, as well as offering a refutation of the rival Machian, Kantian, Neo-Kantian, and other more eclectic interpretations of the influences on the work of the young Carnap. The book concludes with an assessment of Quine's critique of Carnap's 'analytic-synthetic' distinction and a criticism of the direction that analytic philosophy has taken in following in the footsteps of Quine's views.

Stephen Hicks in Explaining postmodernism

Showing that a movement leads to nihilism is an important part of understanding it, as is showing how a failing and nihilistic movement can still be dangerous. Tracing postmodernism’s roots (...) explains how all of its elements came to be woven together. Yet identifying postmodernism’s roots and connecting them to contemporary bad consequences does not refute postmodernism.

What is still needed is a refutation of those historical premises, and an identification and defense of the alternatives to them. The Enlightenment was based on premises opposite to those of postmodernism, but while the Enlightenment was able to create a magnificent world on the basis of those premises, it articulated and defended them only incompletely. That weakness is the sole source of postmodernism’s power against it. Completing the articulation and defense of those premises is therefore essential to maintaining the forward progress of the Enlightenment vision and shielding it against postmodern strategies.

The names of the postmodern vanguard are now familiar: Michel Foucault, Jacques Derrida, Jean-François Lyotard, and Richard Rorty. They are its leading strategists.

Members of this elite group set the direction and tone for the postmodern intellectual world.

Michel Foucault has identified the major targets: “All my analyses are against the idea of universal necessities in human existence.” Such necessities must be swept aside as baggage from the past: “It is meaningless to speak in the name of—or against—Reason, Truth, or Knowledge.”

Richard Rorty has elaborated on that theme, explaining that that is not to say that postmodernism is true or that it offers knowledge. Such assertions would be self-contradictory, so postmodernists must use language “ironically.”

Against this Kantian ethics postulates:

1. Moral dignitarianism, the anti-egoistic, anti-utilitarian, and anti-relativistic universalist ethical idea that every rational human animal possesses dignity, i.e., an absolute, non-denumerably infinite, intrinsic, objective value or worth, beyond every merely hedonistic, self-interested, instrumental, economic, or utilitarian value, which entails that we always and everywhere ought to treat everyone as persons and never as mere means or mere things, and therefore always and everywhere with sufficient respect for their dignity, no matter what merely prudential reasons there are to do otherwise.

2.  Political dignitarianism, the anti-despotic, anti-totalitarian, and anti-Hobbesian- liberal yet also liberationist, radically enlightened idea that all social institutions based on coercion and authoritarianism, whether democratic or not-so- democratic, are rationally unjustified and immoral, and that in resisting, devolving, and/or transforming all such social institutions, we ought to create and sustain a worldwide or cosmopolitan ethical community beyond all borders and nation-States, consisting of people who who think, care, and act for themselves and also mutually sufficiently respect the dignity of others and themselves, no matter what their race, sex, ethnicity, language, age, economic status, or abilities.

Husserl:

 Whatever is true, is absolutely, intrinsically true: truth is one and the same whether men or non-men, angels or gods apprehend and judge it. Logical laws speak of truth in this ideal unity, set over against the real multiplicity of races, individuals and experiences, and it is of this ideal unity that we all speak when we are not confused by relativism.  

P. Tichý (Foundations of Frege's Logic):

Fate has not been kind to Gottlob Frege and his work. His logical achievement, which dwarfed anything done by logicians over the preceding two thousand years, remained all but ignored by his contemporaries. He liberated logic from the straight-jacket of psychologism only to see others claim credit for it. He expounded his theory in a monumental two-volume work, only to find an insidious error in the very foundations of the system. He successfully challenged the rise of Hilbert-style formalism in logic only to see everybody follow in the footsteps of those who had lost the argument. Ideas can live with lack of recognition. Even ignored and rejected, they are still there ready to engage the minds of those who find their own way to them. They are in danger of obliteration, however, if they are enlisted to serve conceptions and purposes incompatible with them. This is what has been happening to Frege's theoretical bequest in recent decades. Frege has become, belatedly, something of a philosophical hero. But those who have elevated him to this status are the intellectual heirs of Frege's Hilbertian adversaries, hostile to all the main principles underlying Frege's philosophy. They are hostile to Frege's platonism, the view that over and above material objects, there are also functions, concepts, truth-values, and thoughts. They are hostile to Frege's realism, the idea that thoughts are independent of their expression in any language and that each of them is true or false in its own right. They are hostile to the view that logic, just like arithmetic and geometry, treats of a specific range of extra-linguistic entities given prior to any axiomatization, and that of two alternative logics—as of two alternative geometries—only one can be correct. And they are no less hostile to Frege's view that the purpose of inference is to enhance our knowledge and that it therefore makes little sense to infer conclusions from premises which are not known to be true. We thus see Frege lionized by exponents of a directly opposing theoretical outlook.

Schopenhauer vs. Schopenhauer

The questions Shapshay asks in her book and her theory of an internal contradiction or tension in Schopenhauer regarding compassion vs. renunciation are very incisive and relevant to our thesis, which is as follows.

1. Schopenhauer had an imperfect grasp of ancient Indian philosophical and spiritual traditions.

2. Schopenhauer's theory of renunciation was largely colored by Christian mysticism (from the middle ages to the 17th-century) and in particular by Eckhart and Luther.

3. This lead to a miscomprehension  and misreprentation of ancient Indian traditions due to a falsely postulating their essential unity with Christian mysticism in so far as being brought together under the heading of the common phenomenon of renunciation and negation of the will.

4. Christian mysticism and many important ancient Indian traditions (in particular original Pali Buddhism, Samkhya and the Yoga of Patanjali) are mutually antagonistic and irreconcilable. For the practice promoted by such traditions (called in Pali bhâvanâ) can be seen as the consequence to one the two fundamental sides of the positive ethics of compassion : compassion for others and compassion for self. For instance, the corner-stone of original Buddhism is the rejection of causing suffering to others and practices involving self-torment or causing suffering to self. We have the duty both to cultivate the alleviation of the suffering of others and the suffering of our own self (to do: investigate Kantian aspect).  This is radically opposite to Christian mysticism. For suffering (of the agent) is an instrumental, circumstantial, empirical cause for practicing compassion and self-development but never an essential or sufficient cause; there is no ethical or social value in suffering per se be it voluntary or involuntary.  This completely rules out the legitimacy of the concepts of vindictive (as opposed to preventive and corrective) justice as well as vicarious atonement and of course all misguided forms of asceticism based on mental or physical self-harm.

5. Such Indian traditions completely evade the important objections raised by Shapshay and are fully compatible with the ethics of compassion and hope.  It is the theory of art in WWR3 (rather than the theory of renunciation in the fourth book) that offers a far more accurate philosophical interpretation of the effects and ultimate aim of self-development.

Project

1. Natural deduction and quantifier logic in ancient philosophy.

2. What was Kant's logic in the CPR ? Was it adequate even to express the transcendental analytic ?

3. Claire Ortiz-Hill's analysis of equality and identity in Frege and Husserl in the light of dependent type theory and in particular homotopy type theory. How Gentzen and Martin-Löf show us the most promising path in philosophy.

https://chryssipus.blogspot.com/2023/11/equality-and-sameness-from-frege-to.html

Our considerations on 'holology' and higher category theory - are in fact extremely relevant to the philosophy of concepts, objects, extensions, abstractions and intensions all concerning which ancient philosophy has many important things to say. Why should the 'object' that is an 'extension' of a 'concept' be a set rather than an $n$-groupoid ? How do we account for 'some' in mass-nouns and propositional attitudes ?

4. (Book) Kant, Schopenhauer, Husserl (both of the earlier and later phase) and certain ancient eastern traditions: logic, epistemology and ethics with reference to the interpretations of Hanna, Schulting and Shapshay.

Kant and Husserl in their 'cognitive semantics' agree remarkably well  with the basic architecture of the mind (or consciousness) layed out in the Pali suttas. But in some fundamental points, in which he differs or corrects Kant, Husserl is closer and in other points (logical, dialectical) Kant is. As for ethics, we might consider a synthesis of Kant and Schopenhauer.

4a. Original Buddhism was neither empiricist (in modern terms) nor relativist. And neither were Pyrrho and Sextus. 

5. All philosophers at the table: what is axiomatic philosophy, why it matters and how it is possible. 

5a. Computer assisted axiomatic philosophy using dependent type theory.

6. Theory of theory, theory of proof and genealogy of the theory of definition.

7. The Hegelian Kant, Husserl and category theory as universal ontology.

8. In defense of analyticity and refutation of inferentialism/proof-theoretic semantics and of pragmatic, social, relativist and coherentist accounts of truth, meaning and inference.

9. Formalize Porphyry's Eisagoge (done) and pin-point difficult questions and uncertain aspects.

(...)

Leibniz's dream is more than a dream

Leibniz's mathesis universalis, characteristica universalis and calculus ratiocinator are more than dreams or utopias. Nor is talk of formal philosophy mere metaphilosophical speculation and wishful thinking.

Zalta's Object Logic in its three degrees of unfolding (each subsuming the previous one) offers a non-trivial axiomatization and formal proofs of some interesting aspects of three great systems of ontology: Plato, Leibniz and Frege. Both the series of Object Logics and the series of these three ontologies can be given a Hegelian interpretation.

Furthermore this axiomatic metaphysics can be embedded and expressed in dependent-type theory. Here are some examples in Coq.

Plato: 

https://github.com/owl77/CoqFormalisations/blob/main/zalta2.v

Leibniz:

https://github.com/owl77/CoqFormalisations/blob/main/zalta3.v

Frege (for now just an embedding of modal typed object logic)

https://github.com/owl77/CoqFormalisations/blob/main/zalta4.v 

Kant, Husserl and beyond

We recommend Corijn van Mazik's paper Husserl’s covert critique of Kant in the sixth book of Logical
Investigations (2018).  Also Robert Hanna's  Kant and the Foundations of Analytic Philosophy (2001).  

Both Kant and Husserl (of the 5th and 6th Logical Investigations) lay out a theory of the structure of the human mind, of consciousness, cognition and experience which does not forget ontology nor (bodily) sensation.  Both Kant and Husserl offer a logic and a theory of objectivity, perhaps both platonic and constructivist (using dependent type theory shown to express (a substantial) part of universally valid laws of reason for rational agents).  There is a Leibnizean mathesis (universal ontology) project looming in both Kant and Husserl - and Hegel's Logic combined with modern category theory (and model theory) seems a promising way of realizing it.

We find that both architectures, while insightful and brilliant, are yet radically insufficient, both from below and from above.  

From below because they neglect, unlike classical philosophers like Aristotle (see our paper on De Anima), to take into account many crucial psychological and physiological elements in the structure of consciousness, such as a the key role of feeling, desire and habit in cognition and mental experience, as well as the psycho-physiological act of perception.

From above: this radical insufficiency was overcome by Husserl, following the supremely important critique of Kant implicit in the Logical Investigations, in his famous so-called transcendental turn. But in Husserl's discovery there still lurked the danger of not understanding how transcendental subjectivism is at the same time objective absolutism. Also there are some dangerous equivocations surrounding the term ego in 'transcendental ego'.  It would have been better and safer to adopt a purely negative approach and to speak of a transcendental consciousness not conditioned by an ego. Also missing in Husserl is the crucial connection between the awareness of universal temporality/temporalization and the transcendentality of transcendental consciousness. Missing is the description of how the worldly ego is constituted.

However there are also many important aspects in Kant's transcendental dialectic which were more or less overlooked or not given prominence by Husserl: the notions of inconsistency and incompleteness (undecidability) of conceptual and logical systems viewed formally. Kant's transcendental dialectic is in fact a more developed and consequential version of certain elements of Pyrrhonism. How can Kant's critical (transcendental) knowledge escape the bounds set by his theory of knowledge ?

Hanna talks a lot of 'embodiment' and throws the charge of solipsism at Husserl.  But the correct unfoldment of transcendental subjectivism involves apodeictic realism regarding other consciousnesses, human and otherwise,  together with the reality of the first-person (human or otherwise) experience of the body, of embodiment. Can we reconcile Kantian ethics and Schopenhauer's ethics of universal compassion ?

This is how Robert Hanna articulates Kantian morality and its political implications:

1. Moral dignitarianism, the anti-egoistic, anti-utilitarian, and anti-relativistic universalist ethical idea that every rational human animal possesses dignity, i.e., an absolute, non-denumerably infinite, intrinsic, objective value or worth, beyond every merely hedonistic, self-interested, instrumental, economic, or utilitarian value, which entails that we always and everywhere ought to treat everyone as persons and never as mere means or mere things, and therefore always and everywhere with sufficient respect for their dignity, no matter what merely prudential reasons there are to do otherwise.

2.  Political dignitarianism, the anti-despotic, anti-totalitarian, and anti-Hobbesian- liberal yet also liberationist, radically enlightened idea that all social institutions based on coercion and authoritarianism, whether democratic or not-so- democratic, are rationally unjustified and immoral, and that in resisting, devolving, and/or transforming all such social institutions, we ought to create and sustain a worldwide or cosmopolitan ethical community beyond all borders and nation-States, consisting of people who who think, care, and act for themselves and also mutually sufficiently respect the dignity of others and themselves, no matter what their race, sex, ethnicity, language, age, economic status, or abilities.

Finally we must give an account of aesthetics (including platonic and neoplatonic theories) and how it positively harmonizes with philosophy and ethics. We must value beauty greatly in itself but be realistic about its context and the way it manifests in the process of human life.