Finally added table of categories.

]]>For what it’s worth: in Kant’s *Prolegomena* (§ 20, p.84) one finds in a footnote the suggestion to replace the term *particular judgment* with *plural* judgment (*judicia plurativa*). In another footnote (§ 39, p.122) he briefly mentions the idea to reorder the categories of quality to reality-limitation-negation yielding a progression from something to nothing.

Concerning the terminology “Vielheit”: In Karl Leonhard Reinhold in his “Versuch einer neuen Theorie des menschlichen Vorstellungsvermögen” notes (vol.2 p.412 of the Meiner edition of 2010, 3. Buch - Theorie des Verstandes)

Vielheit soll hier nur soviel als Mannigfaltigkeit überhaupt bezeichnen, in wieferne sie der Einheit entgegengesetzt ist.

This is still open for ambiguity. Mysteriously to me the editor Onnasch annotates this with

Mit dieser Bermerkung will Reinhold offenbar klarstellen, daß der hier verwendete Begriff nicht zu verwechseln sei mit dem der Kategorie “Vielheit”.

Reinhold’s table of judgments and categories at least relieves me of the troubles concerning the order of categories: he has generalities and modalities where they should be!

Be that as it may, there is a very nice paper by Onnasch on the early discussion around Kant’s table: Herleitung der Urteilsformen in Reinhold.

Concerning the rectification: *all students sing = all students are singing students*. This property, $Q(A)(B)=Q(A)(A\cap B)$, is a famous property of natural language determiners, called *conservativity* , that was introduced in a landmark paper of NL semantics by Barwise and Cooper as semantic universal for NL determiners. It means that the value of the generalized quantifier Q(A) that maps predicates to truth values, on the predicate B of type <e,t> does merely depend on their intersection whereas e.g. in *only students sing* it does not suffice to consider the set of singing students but you have to take also its complement into account (Hence, a linguist believing in conservativity as universal talks herself out of this counterexample by pointing out that *only* is a polymorphic quantificational adjective like e.g. *and* is a polymorphic conjunction).

Conservativity has an interesting *completeness* property due to Keenan and Stavi: conservative GQ over a finite universe E of entities coincide precisely with GQ that can be built from $\forall$ and $\exists$ by using the Boolean connectives $\vee,\wedge, \neg$.

This is interesting and mildly embarrassing from a Kantian perspective since singular quantity drops out of the picture: the individual *c* turns via the predicate $\{c\}$, *being c* , into the GQ *the c* denoting the principal filter generated by the individual whence subjects for singular judgments are obtained from application of $\exists$ or $\forall$ to “singular predicates”. A possible reply would be to say that a true singular judgement results from applying the predicate to the subject as argument in contrast to general and particular where the subject takes the predicate as an argument - hopefully one can convince oneself then that these predications are still needed for demonstrative anchoring in reality.

Another way out would be to insist that your negation is intuitionistic as suggested by non validity of the tertium-non-datur for infinite judgements and figure out what the notions of GQ and conservativity give on a topological algebra of open and closed sets with pseudocomplementation.

]]>My interpretation would be as follows: Yes, in the situation described the proposition “Some students sit on the ground” is true. But in view of the list of categories the question whether it is true or false is misleading. Actually, one has to ask, how can I make such judgments? And here here the $\exists$-intro rule seems to be a bit impure to Kant, as written in the Jäsche Logik (“von den besonderen Urtheilen ist zu merken, dass wenn sie durch die Vernunft sollen können eingesehen werden und also eine rationale, nicht blos intellectuale (abstrahierte) Form haben”). In the example: You see the student sitting, so first you have a singular judgment, “student X sits on the ground”, from which you use the $\exists$-intro rule to get the proposition. A category is a “pure concept of synthesis”. Therefore only the “rational” form of particular judgment can give rise to a category. And in this sense it is always a plurality, as it is at least contingent that there are many students sitting on the ground.

One could “rectify” the example to make the rational form of particular judgment applicable by saying “Some students are on the ground sitting students”. Here it becomes apparent that judgments are a connection of concepts. Thinking in a possible world semantic there are always infinitely many instances which fulfill a certain condition, i.e. to which concepts can relate.

]]>@24. As always I am at loss with following Kant in the quoted passages. But in order bring out my problem with the “plurality” of particular judgments let’s imagine a situation with a group of three students in front of townhall with exactly one of them sitting on the ground: is the particular judgment “Some students sit on the ground”, nevermind if the language is awkward, true or false in this situation? Now, usually modern logicians would say, that statements with Some have existential import, hence this is true, but since Kant brings up the category name “Vielheit-plurality” and no sitting plurality is present, one might suspect that for him the statement is false in this situation.

As addendum to the order of categories: In his remarks on the dialectics of categories (§11 of CPR), Kant says that the third category results from the combination of the first two and then goes on to call totality nothing but plurality conisidered as unity. Whence he commits himself explicitly to the order of quantative categories with totality as third as in the table of categories whereas general judgments come first in the table of judgments.

]]>On 25. and 26.: it seems that the introduction rule for existential quantification is what Kant means by “intellectual (abstracting) form [of a judgement]”.

]]>OK, I think now I understood the difference:

particular judgement from reason: Some $T_0$ spaces are $T_2$. Because be can prove $T_2 \implies T_0$.

contingent particular judgement: Some metrisable spaces are Hausdorff. Because we can have the cognition of a concrete particular space, let’s say $[0,1]$ which is both, i.e. we can

*abstract*to both properties. This fits well with Kant’s wording: “eine rationale, nicht blos intellectuale (abstrahierte) Form”.

My problem is with Kant’s intentions. Most interpreters with modern in logic in mind identify particular judgments with a term formation rule for existential quantification. A general problem if you start to throw modern gadgets at him is to still to keep the different cases independent and complete. For instance if you identify the predicate negation with classical negation you don’t need both of $\exists$ and $\forall$.

Kind of curious is his disjunction, this ressembles the requirements $\phi_i\wedge\phi_j\vdash \bot$ on admissible disjunctions $\vee\phi_i$ occurring in disjunctive logic with additional exhaustivity $\top\vdash\vee\phi_i$. (Sorry, this was thought as reply to David’s post. )

]]>@Thomas Holder:

Furthermore, by calling “Some A” of

pluralcategory, does this not suggest an interpretation as “Several A” hence stronger then existential ?

I don’t claim to give a formalization but in my impression the way Kant thinks about objects and predicates is not like sets but like predicates (i.e. properties). As far as we are speaking about objection of apperence we can index these objects (i.e. imagine them at different places in space). Therefor, if it’s possible that something exists, it automatically exists in many different ways (“numerically different”). Actually, this way of thinking is probably a reason why Kant came up with the singular judgement at all. Singular and universal judgement work in the same way, the only difference is the predicate.

In the Jaesche Logik (p.159ff) the following passage may illustrate Kant’s way of thinking:

- von den
besonderenUrtheilen ist zu merken, dass wenn sie durch die Bernunft sollen koennen eingesehen werden und also eine rationale, nicht blos intellectuale (abstrahierte) Form haben: so muss das Subject ein weiterer Begriff (? latior) als das Praedikat seyn.

Kant continues to say (by a drawing) that “$a$ is $b$” means $a \subset b$ in this case. But then there is also the case of the contingently particular judgement (“zufaelliger Weise partikulaer”) in which (according to the drawing) $a \cap b, a\setminus b, b\setminus a \neq \emptyset$. I have to admit I am still a bit confused by this (even grammatically by the wording).

Anyway, the page before the relation of singular and universal judgement is explained:

]]>

- Die einzelnen Urtheile sind der logischen Form nach, im Gebrauch den allgemeinen gleich zu schaetzen; den bey beyden gilt das Praedikat vom Subject ohne Ausnahme.

I guess “Some A” could be seen as dependent sum before truncation to the existential, or as a pullback as in Science of Logic, so potentially several.

From a different tradition, Kit Fine thought to define modalities, ’true in n worlds’, giving rise to a form of graded modality.

]]>For my remark 19 I simply read off the order of quantities of *judgment* from the table of *categories* supposing that of course the orders correspond to each. Now, I realize that the order of quantities of judgment are *general, particular, singular* whereas the categories of quantity are *unity, plurality, totality*. This still leaves the mismatch between the respective quantifiers involved in the judgments of quantities and the *modalities* and adds the puzzle why the categories are reordered respectively to the judgments in the quantity colummns ? Furthermore, by calling “Some A” of *plural* category, does this not suggest an interpretation as “Several A” hence stronger then existential ?

Added a criticism of Kant’s formation of the categories by Ryle.

]]>We have necessity and possibility arising from the base change adjunctions deriving from $W \to \ast$. We don’t seem to have written it down, but we could look at $a: \ast \to W$ as picking out the actual world and use it to generate a monad and comonad, such as something skyscraper sheaf-like.

]]>Assuming a possible world semantics for modality, the order of modalities in Kant’s table of judgements under 4, namely possibility, actuality, and necessity becomes puzzling on comparison with the quantities under 1, i.e. $dthat x$, $\exists x$ and $\forall x$ (purloining Kaplan’s demonstrative quantifier “dthat” as stand-in for singular judgments). Note that Kant gives some argument for his orders in § 11 of the B edition namely the dialectical relation between the categories corresponding to the judgement forms (Third category stems from the synthesis of the preceding two) but this fails to account for the relative ordering of the first two categories.

Interestingly, in the reflection of Kant’s table in the ordering of the respective paragraphs in the “subjective logic” of Hegel’s WdL we find the modalities reordered to actuality, possibility and necessity.

]]>Added table of judgements with remark on Jäsche Logik.

]]>I improved some of the formulations in the article concerning the B Deduction.

Lucas Immanuel Janz

]]>Put in links to wikisource (German edition).

]]>Provided a brief summary of “The Idea of a Transcendental Logic. (General Logic and Transcendental Logic)”

Lucas Immanuel Janz

]]>Added table of contents, made structure of sections closer to the way they are used in Science+of+Logic.

]]>Expanded the explanations on 1st and 2nd editions.

]]>Added reference to

- Bitbol, Michel, Kerszberg, Pierre, Petitot, Jean (Eds.),
*Constituting Objectivity: Transcendental Perspectives on Modern Physics*Springer (2009)- Jean Petitot,
*Noncommutative Geometry and Transcendental Physics*, pgs 415-455

- Jean Petitot,

which has already been linked at Immanuel Kant.

]]>added reference to Attempt to Introduce the Concept of Negative Quantities into Philosophy

]]>Added link to wikisource.

]]>Fixed now I believe. Apologies for the inconvenience.

Every time this kind of bug crops up, I think: if only we had a ’staging’ environment which we could run tests against! There are quite a few things to do before that can be put in place, though. But I’ll add it to the Technical TODO list (nlabmeta).

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