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

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.