On Van Lambalgen et al.'s formalization of Kant

The paper by Van Lambalgen and Pinosio 'The logic and topology of Kant's temporal continuum' (which is just one of a series of papers by Van Lambalgen on Kant)  opens with a nice discussion and careful justification of the general idea of the formalization of philosophical systems. The coined expression 'virtuous circle'  is particularly fortunate. In this post, which will be continuously updated, we will critically explore the above paper and make some connections with our own work on Aristotle's theory of the continuum.

The primitives are called 'events', self-affectations of the mind, which must be brought into order by fixed rules.  The authors work over finite sets of events which is justified by textual evidence from the CPR (we will return to this later).  Their task is to formalize relations between events - and to thus develop a point-free theory of the linear temporal continuum.

We find that that their notation could be improved and the axioms better justified. Instead of the confusingly asymmetric (all for the sake of the substitution principle, I suppose, or for the transitivity axiom) $aR_- b$ and $cR_+ d$  let us write $a{}_\bullet \leq b$ and $d\leq_\bullet c$. Instead of $a\oplus b$ we write $a\leftarrow b$ and insead of $a\ominus b$ we write $a\rightarrow b$.

The basic idea is that : $x{}_\bullet\leq y$ does not need to imply that $x\leq_\bullet y$ or vice-versa.

Kant's concept of causality implies that in order for a part $x$ of $a$ to influence $b$ we must have $x{}_\bullet\leq b$.  Thus the following axiom is expected

\[  a\ominus b{}_\bullet\leq b\]

But let us look at axiom 4 for event structures (in our notation):

\[ cOb\,\&\, a\leq_\bullet c \,\&\, b{}_\bullet \leq a \Rightarrow aOb \]

Our task is to make sense of this by offering a more satisfactory account of the primitive relations. Let us consider the set of connected (hence simply connected) subsets of the real line $\mathbb{R}$ and the interpretations:

\[ a{}_\bullet\leq b \equiv \forall x \in a. \exists y\in b. x\leq y  \]

\[ a \leq_\bullet b \equiv \forall x \in b. \exists x\in a. x\leq y  \]

But this does not work for  $a{}_\bullet\leq b \Rightarrow a\leq_\bullet b$. But let us take our events to be bounded open intervals $(a,b)$ and consider

\[ (a,b){}_\bullet\leq (c,d) \equiv  b < d  \]

\[ (a,b) \leq_\bullet (c,d) \equiv a < c \]

\[(a_1,a_2)O(b_1,b_2) \equiv a_2 > b_1\,\&\, a_1 < b_2\]

Then if we consider $(0,1)$ and $(0,2)$ we have that $(0,1){}_\bullet\leq (0,2)$ but not $(0,1)\leq_\bullet (0,2)$. The inequalities must be strict for allowing  $(a,b){}_\bullet\leq (a,b)$ is absurd, for then we could not associate any clear or definite Kantian philosophical concept with the relation.

Now let us look at axiom 4:

\[ (c_1,c_2)O(b_1,b_2)\,\&\, (a_1,a_2)\leq_\bullet (c_1,c_2) \,\&\, (b_1,b_2){}_\bullet \leq (a_1,a_2) \Rightarrow (a_1,a_2)O(b_1,b_2) \] which becomes

\[ c_2 > b_1\,\&\,  c_1 < b_2   \,\&\,a_1< c_1\,\&\, b_2 < a_2 \Rightarrow a_2 > b_1\,\&\, a_1 < b_2\]

But this follows immediately, using in addition the fact that $b_2 > b_1$. The condition $c_2 > b_1$ appears not to be needed.

We could try defining $(a_1,a_2)\rightarrow (b_1,b_2) := (a_1,b_2)$ when $a_1 < b_2$ and $(a_1,a_2)\leftarrow (b_1,b_2) :=  (b_1,a_2)$ when $b_1 < a_2$.

This models should be introduced right at the start of the paper to motivate the the definition of event structure. Notice that the set of events is here identified with the (infinite) subset $E \subset \mathbb{R}\times\mathbb{R} = \{(x,y): x < y\}$ but we could take only a finite subset.

We must check the axioms for event-structures for our model and also give a geometrical interpretation of the relations and operations above in terms of the identification of $E$ as a subset of the plane above.

A filosofia de Ludwig Wittgenstein à luz do diagnóstico de autismo

 https://philarchive.org/rec/SILAFD-6

Philosophy of Mind

Consciousness is.  But we must reflect and become aware first of all that there is consciousness. Make consciousness itself an object for consciousness.  But in this awareness of consciousness itself we must make sure that we see consciousness in its purity and totality. That all experience, past, present, future, in whatever mode, the world itself, is immanent in consciousness.  Thus does consciousness stand outside itself and perceive itself in itself by itself - the primordial stream which has sometimes been called the stream of thought, although to be accurate we must include all modalities of consciousness experience under the heading of cogitata.  Temporality is unveiled in its primordial role. This shift of awareness* can have a sobering and even liberating effect, like an exit from Plato's cavern.  We are left with an agent-self standing both inside and outside consciousness, the universal sphere of consciousness as an object, and the relation between the agent-self and the universal sphere.

But here we take a practical turn which will hopefully lead us to truth and allow us to avoid many errors. 

There is the primordial transcendental knowledge that there is something seriously wrong with our consciousness that needs to be corrected, there is something that needs to be overcome.  The agent-self must somehow act on consciousness and hence ultimately act on itself to remedy this situation.  Part of such primordial wrong is the powerlessness and passivity of the self-agent, specially in its naive unreflected self in which it inhabits naturalistically and in oblivion  a world unknowingly constituted by consciousness. And no, we are not confusing merely 'psychological' problems with philosophical ones. The divorce of psychology and philosophy is itself philosophically questionable.

The transcendental ego in the Husserlian sense is merely passive, shallow and theoretical (in the sense of being talk about knowledge rather than knowledge, the menu rather the the actual dish).  Because it lacks power and it cannot act, it cannot attain to knowledge, indeed despite some rupture it does not leave the western conceit that genuine knowledge can be divorced from a vital transformation of the subject.

How can the transcendental ego gain a grasp or foot-holding on consciousness itself so that it does not remain a vain chimera ? The answer involves the theory of embodiment, but in a way radically different from any naturalism or existentialism. To us the theory of embodiment is the theory of the body from a first-person perspective, the phenomenology of the body as experienced exclusively from the inside (the external theory being developed from and based on the internal theory). We claim that the phenomenology of  inner-body consciousness is necessary and of immense importance to any correct philosophy of consciousness.  The transcendental ego must first of all focus, concentrate and investigate inner-body consciousness.

Focus on the inner-body consciousness is the solution, the vital foot-holding, for  the transcendental ego becoming active and gaining power over the total immanent sphere of consciousness and thus to at last access the long sought-for clarity in knowledge.  Inner-body consciousness is a kind of organic crystallization of consciousness which reveals itself to be very deep with roots in manifold aspects of ordinary human psychological  experience - even neurological ones fall within the scope of a strictly internal point of view. It is the necessary path to pass through in order to be able to  grasp and bring order to other spheres of more subtle conscious modalities. That is: we must bring to the light of consciousness what we are in fact deeply identified with without knowing in order to achieve dis-identification from it (we will not go into here the philosophy of self and its modalities of unification, synthesis, negativity and subsumption).

The body is thus the initial and essential tool and means by which the transcendental ego attains the correct relationship to its own total immanent sphere of consciousness. It is also, as we shall see in a future essay, the key to a Kantian-Schopenhauerian ethics of universal compassion, which involves the overcoming of solipsism. It is also the basis for  a new philosophical top-down organicist architectonic of the sciences which is not without strong connections to Goethe's,  Schopenhauer's and von Uexküll's theories of natural science.

Our use of  'body' is perhaps misleading as it is used in a very technical sense: that (Kant's noumenal X) which enters into a particular kind of relationship to consciousness which characterizes inner, first-person body-experience and body-consciousness. Consider the section dedicated to the so-called 'refutation of idealism' in Kant's CPR. Our goal is to show in an analogous (or transposed) way that the certainty of our own consciousness and its reflected embodied experience - in particular the reflected embodied experience of pain - necessarily presupposes the existence of other similarly embodied consciousnesses external to us, when we consider ourselves as particular embodied consciousnesses. The moral law we wish to establish and develop (which has both Kantian and Schopenhauerian foundations) has two inseparable and necessary components: compassion for self and compassion for others, the duty to alleviate one's own suffering and to alleviate the sufferings of others, or, in the words of the Gospels, to 'love thy neighbor as thyself', supposing, of course, that by 'neighbor' is meant every human (or in general sentient) being.

*The path to  transcendental consciousness involves the detachment-inducing awareness of temporality (including past and futurity), of the transience, change, frailty and compositeness of all domains of our embodiment and experience (psycho-physicality) as well as scrutiny of the now, the actual, what in fact is really there in the midst and in light of this transience. This should become ever more subtle, anchoring in the realm of pure thought.

We can also say that transcendental consciousness is attained by a decision to examine what is really present before us, what is positively pristinely there rather than a projection, anticipation, interpretation and extension conditioned by volition, etc. This leads to a conversion inward, a unification which is also an inversion and shift of the entire domain of consciousness. 

Transcendental consciousness involves the process by which we come face to face - but in a detached way - with the current of our own thoughts (possibly entwined with empirical-sensual content). This is what Husserl describes in the first chapters of the Cartesian Meditations.

Or rather transcendental consciousness is a freely flowing unified state beyond the stopping (or self-limitation) of consciousness in intentional acts.

See also:  Scarfe, A. (2006). Hegelian “Absolute Idealism” with Yogācāra Buddhism on Consciousness, Concept (Begriff), and Co-dependent Origination (Pratītyasamutpāda). Contemporary Buddhism, 7(1), 47–73. doi:10.1080/14639940600877994

Serious philosophy might be said to be simply studying Kant and finding  ways or reconciling Fichte and Schelling, Schopenhauer, Frege and Husserl...except that i) modern mathematics has opened the way to the development of the genuine philosophical logic which few bother to learn and understand: categorical logic and type theory and ii) we now have access to immense Sanskrit libraries of Buddhist philosophy which few bother to learn and understand.

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.

Theory of mind

The fate of modern western philosophy involves, it seems, the conflict (and attempt at harmonization) between psychology, logic and the perceived 'laws of nature'.  It has always had strong tendency to dualisms, meta-isms (confusing going beyond something with overcoming it),  reductionisms (in the materialist or idealist directions) or ad hoc amalgmations, constructions, neologisms (intensionality, schematism, etc). There has always been a deep confusion and misguided mixture between the mental and the logical which seems hardwired into the terms themselves like 'concept' and 'idea'.  Schopenhauer accuses Kant of a heilosen Vermischung der intuitiven und abstrakten Erkenntnis. Frege and Husserl both struggled with this. Earlier analytic philosophy likewise attempted to evade the psychological in ways less cogent. It seems that by trying to analyze the psychological and the logical the mind inevitably puts both into each one of them and also into their very relationship.

Let us drop  logicism, psychologism and meaning-as-use for the moment, and drop all reductionist dogma. Let us engage in the philosophy of mind, starting from basic, down-to-earth, common-sense considerations.  From the most basic and obvious and neglected it is possible to ascend to the most subtle, all-encompassing and unexpected.  An organic system-theoretic thinking about the larger role of sensation and perception and its multilateral connection without any artificial assumptions. Bringing in the importance of the body and physiology to the mind without trying to restrict or reduce the mind.  For the mental has a continuous gradation (analogous to the gradations in energy and frequency in physics) of yet qualitatively distinct levels all bearing an organic inter-dependence.

While we distinguish between formal and conceptual-philosophical clarity we hold that in philosophy both are even more indispensable and fundamental than in mathematics. Clarity and transparency are the highest virtues in philosophy. And, without implying completeness or achievement,  formal clarity in philosophy would be, in perspective, a huge step forward. We do not see modern philosophy as excelling ancient philosophy in clarity or rigour nor do we see "analytic" philosophy (excluding some notable exceptions) as exhibiting greater clarity (formal or conceptual) than "continental" philosophy. Lack of clarity, formal, conceptual, not to mention deductive, has been the original sin of modern philosophy.  The clearest philosophical "concept" is still murkier and vaguer than the most complex scientific or mathematical one.  There is hardly a single example of a definite philosophical proof. The idea of focusing on the analysis of language as a surrogate for philosophy is circular  (the metalanguage is not less complex than the object language) and leads to even greater confusion and ambiguity. This is in ironic and tragic contrast to the huge development of mathematics in the direction of formal and deductive clarity. 

We need a new language for philosophy. And we need a new methodology and a new form of intuition and insight.

If it is one of the tasks of philosophy to break the domination of the word over the human spirit by laying bare the misconceptions that through the use of language often almost unavoidably arise concerning the relations between concepts and by freeing thought from that with which only the means of expression of ordinary language, constituted as they are, saddle it, then my ideography, further developed for these purposes, can become a useful tool for the philosopher. -  Frege, Begriffsschrift, Preface.

There have been two really significant discoveries and projects in philosophy. The first, objective, corresponds to Leibniz's idea of a characteristica universalis, calculus ratiocinator and mathesis universalis. This objective philosophy was, although this is little known, given substantial and valuable development in the earlier work of Edmund Husserl (though mixed and alongside other concerns and investigations: the unfortunate mixture and confusion we discussed above). It is thanks to the pioneering work of Claire Ortiz-Hill that the purely objective Husserl has been unearthed and brought to light. This aspect of Husserl is not incompatible with formalization and with Leibniz's project, rather it amounts to a substantial contribution to it. It offers devastating arguments against Carnap and Quine within an illuminating historical perspective on Fregean extensionalism.  We equate this objective Husserl with the Leibnizean project of a formal philosophy and it is an urgent task to gather together all subsequent significant work which can be seen as contributing to this project (for instance, Zalta's project).

This is no vague utopian project.  A modal type theory with special distinguished metaphysical predicates (for platonic participation, abstract objects, representation, relativization, etc.) can already accomplish something. There is nothing wrong with initial modesty, with employing weaker or partial systems first to elaborate solid results in axiomatic philosophy. Type theory is a kind of paradise of thought and yields substantial results - this without claiming any type theory to be complete or absolute.

We posted previously about a Hegelian interpretation of the various interrelated systems of type theory. Zalta's book Axiomatic Metaphysics book can be given a Hegelian interpretation. Indeed three systems are presented in order of successive strength, each subsuming the previous one. These three systems are deployed to formalize three successively more complex ontologies which also reflect a historical progression: the Platonic theory of forms, Leibniz's theory of possible worlds and monads and Frege's theory of objects, concepts and senses. To do: embed all these systems in dependent type theory or more specifically in Coq.

The second really significant progress in philosophy was Husserl's discovery of transcendental consciousness, which we might conveniently called the 'subjective' Husserl.  It is our task to unearth to true meaning of Husserl's discovery and to guard it against subsequent misunderstandings and appropriations (existentialist, naturalist, etc.).  The objective and subjective Husserl do not contradict each other, they are complementary, though the subjective Husserl remains the most profound, all-encompassing and important perspective. The key to re-establishing the 'subjective' Husserl against modern philosophical aberrations lies in its connection to ancient philosophy, specially ancient Eastern philosophy.

Pippin is right to criticize a reading of Kant’s Deduction that takes there to be a nonconceptual content that (1) is in and of itself objectively valid and (2) is built upon by means of subsequent acts of judgement (or understanding), such as for example Robert Hanna (2008) believes. Such readings do not make sense in the Kantian context, where it is precisely the goal of the Deduction to demonstrate how it is possible that we can determine, a priori , how thought content and sensory content hook up inwardly , which justifies our conceptual claims about empirical objects. If the contribution by sensible content, more precisely, ‘transcendental content’, were really supplied ‘from the outside’, one would be none the wiser from any argument in the Deduction that supposedly showed how we are justified in making claims about objects, how pure concepts are justifiably (necessarily) employed in any judgement that says that some a is F. If it were true that such content is supplied from the outside, Kant could not have shown the fundamental intimacy between the pure concepts and empirical knowledge of objects, precisely the goal of the Deduction. Apperception and Self-Consciousness (2021).

Also we need to take heed of Bolzano's and Brentano's criticism of Kant's notion of analyticity. True logics form a tightly-knit organic whole or family (there is nothing remotely arbitrary about when a formal system constitutes or not a genuine logic).  Thus analyticity = a certain type of logical consequence. There is minimal analyticity, intuitionistic analyticity, classical analyticity, linear analyticity, relevant analyticity, etc.  Kant's notion is mistaken in the literal sense (which seems to betray strictly monadic thinking). However 'containment' can be given an interpretation in terms of 'unpacking' definitions and constructive type theory. This will be elaborated in our theory of definition and proof. We will examine Robert Hanna's slightly pedantic and convoluted treatment of analyticity in 'Kant and the Foundations of Analytic Philosophy'.  Hanna is spot on about 'microstructure'.  However the framing of analyticity in terms of necessary extensional equivalence is at least questionable as is several features of what he says about formal logic in general.

We have finally the correct purification and separation between the objective and the subjective. But both belong intrinsically to our 'rationality'. But wait...what happened to 'formalism is not clarity' ?  This clarity cannot be sought for directly in the objective beyond the axiomatic-deductive method. The subjective Husserl gives the sought for clarity, that is, shows it, rather than expresses it.

Gödelianism (see M. van Atten and J. Kennedy, The Philosophical Development of Kurt Gödel) postulates that there can be more than one valid method for attaining philosophical truth, more specifically,  both Husserl's transcendental subjectivism and Leibnizean objectivism can be equally valid and complementary philosophical paths. For Gödel the Kantian project extends to all possible minds, not just to the conditions of the specifically human mind.

Category theoretic view of definition

 View the objects of a category as Aristotelic species/genera.  Why not compare the process of definition with canonical algebraic constructions ? That is, given $A$ seen as a genus, one of its species $B$ results from a particularization of $A$ through some condition. Can we view this as being expressed in the standard generators and relations presentation (or quotient construction) ?

\[ R \hookrightarrow A \mapsto B \]

As in our upcoming paper we propose that modern mathematical definitions (specially as practiced in proof assistants) go beyond orthodox Aristotelic ones in that they can consider products of genera -   new species are generated also from inducing conditions of products of genera and taking the quotient:

 \[ R \hookrightarrow \Pi_I A_i \mapsto B \]

Think of the different notions of 'generators' of a category.  Of course this works better for strong logical restrictions of the kinds of objects we are considering (consider the HSP theorem in Universal Algebra). This approach could also have strong roots in ancient philosophy, for instance seeing the generators as basic kinds of logoi in a Stoic (see post on Stoic categories) and neoplatonic sense. The above considerations are not meant to be rigorous but only a kind of logico-mathematical metaphor.  But we could be more precise in a topos-theoretic context.