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I added a discussion of space in Kant’s Transcendental Aesthetics in Critique of Pure Reason.
By the way, the translation of the quote from Kant in the section “On Aristotelian logic” seem a bit strange: I think the original German sentence was “Begriffe aber beziehen sich als Prädikate möglicher Urtheile auf irgend einen noch unbestimmten Gegenstand” (“But conceptions, as predicates of possible judgements, relate to some representation of a yet undetermined object.”).
PS The automatic function to create this thread in the nforum did not word.
Did you tick the box above where you report your changes?
About the entry, we should convey the structure better. You’d never know that ’Transcendental Logic’ is such an important part. But it goes 5 layers deep, see here. Perhaps when I have time.
PS The automatic function to create this thread in the nforum did not word.
As David suggested, it seems from the logs that the box ’Indicate your changes at the nForum’ was not ticked.
There is a remote possibility that you did tick it and something when wrong between the user interface and the tool which handles the nForum announcing, but no bug of this nature has been reported previously. Indeed, it is quite a long time since any bug has been discovered in the announcement functionality.
OK, I think I forgot to click the button.
I still think that it’s a user interface mistake to allow the user to enter text but then silently discard that text if they forget to tick the checkbox.
Thanks for bringing this up Mike, I either overlooked this suggestion first time around or had since forgotten about it. I have tweaked the user interface now to remove the checkbox completely, and make the announcement purely based on whether there is anything in the checkbox; and have tweaked the instructions accordingly. Let me know if this is satisfactory; if not, let me know what would be better! I felt it best not to have both the checkbox and automatic announcing if something is in the box, as some people might assume that the checkbox will be respected.
This is great, I think it’s exactly right, thanks!
It appears to be impossible to rename a page at the moment – the software gives the “A change of page name must be indicated on the nForum” error. Probably it is looking for the checkbox that no longer exists.
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).
added reference to Attempt to Introduce the Concept of Negative Quantities into Philosophy
Added reference to
which has already been linked at Immanuel Kant.
Added table of contents, made structure of sections closer to the way they are used in Science+of+Logic.
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. , and (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.
We have necessity and possibility arising from the base change adjunctions deriving from . We don’t seem to have written it down, but we could look at as picking out the actual world and use it to generate a monad and comonad, such as something skyscraper sheaf-like.
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 ?
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.
@Thomas Holder:
Furthermore, by calling “Some A” of plural category, 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 besonderen Urtheilen 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 “ is ” means in this case. But then there is also the case of the contingently particular judgement (“zufaelliger Weise partikulaer”) in which (according to the drawing) . 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.
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 and .
Kind of curious is his disjunction, this ressembles the requirements on admissible disjunctions occurring in disjunctive logic with additional exhaustivity . (Sorry, this was thought as reply to David’s post. )
OK, I think now I understood the difference:
particular judgement from reason: Some spaces are . Because be can prove .
contingent particular judgement: Some metrisable spaces are Hausdorff. Because we can have the cognition of a concrete particular space, let’s say 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”.
On 25. and 26.: it seems that the introduction rule for existential quantification is what Kant means by “intellectual (abstracting) form [of a judgement]”.
@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.
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 -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 -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.
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, , 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 and by using the Boolean connectives .
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 , 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 or 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.
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.
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