Not sure what the concept stands for in Carnap, but I'm SURE 'category' is a BASIC category in Grice!
Title:Carnap, Completeness, and Categoricity: The 'Gabelbarkeitssatz' of 1928
Authors:Awodey, S
Carus, A W
Source:Erkenntnis: An International Journal of Analytic Philosophy, 54(2), 145-172. 28 p. 2001.
Document Type:Journal Article
Subjects:AXIOMATICS
CATEGORICITY
COMPLETENESS
LOGIC
LOGICISM
Persons as Subjects:CARNAP
GÖDEL, KURT
Abstract:In 1929 Carnap gave a paper in Prague on "Investigations in General Axiomatics"; a brief summary was published soon after. Its subject looks something like early model theory, and the main result, called the 'Gabelbarkeitssatz', appears to claim that a consistent set of axioms is complete just if it is categorical. This, of course, casts doubt on the entire project. Though there is no further mention of this theorem in Carnap's published writings, his 'Nachlass' includes a large typescript on the subject, 'Investigations in General Axiomatics'. We examine this work here, showing that it provides important insights into Carnap's development during this critical period.
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The term categorical in this context means "having only one model" or perhaps only one of each infinite cardinality. "up to isomorphism", i.e. the many different models all have the same structure. This has no connection with Aristotle's categories or with categorical syllogisms or with category theory.
ReplyDeleteAs to why this is thought to cast doubt on the project, that is a bit mysterious, since the property cited holds of the first order predicate logic (for example). Since I have not seen the work being discussed and no details are supplied I can't comment on whether the system actually has the property, but I would be inclined to accept Carnap's opinion (which he presumably substantiated).
I would guess that the supposed problem is that the condition cited is equivalent to "completeness" and hence cannot be had by any language in which it is possible to define (categorically) the natural numbers because of Godel's incompleteness results. Though I beieve Carnap probably did work with deductive systems which acheived this effect by means which excluded them from the scope of Godel's results (e.g. infinitary logics).
Looking at the date (which precedes the incompleteness results) it is more doubtful that Carnap could have proven the result in question. It is a completeness result and the completeness of predicate logic was not proven until 1930, also by Godel in his doctoral dissertation.
It is probable that Carnap knew of the completeness result before Godel's PhD dissertation was complete, and conceivable but not likely that he might have made use of the result in the way suggested.
Another possibility is that, since he was influenced by Hilbert in a way which was shown by his impending "syntactic" philosophy, he might simply have accepted Hilbert's opinion that all mathematical problems are soluble and hence assumed the completeness of the predicate calculus.
I'm afraid without sight of the details this is rather speculative.
RBJ
Thanks for the informed response.
ReplyDeleteWhat a pleasure to converse with a Carnap expert!
I suppose I will elaborate in due time. At least I'm relieved that there is no Aristotelian safe reference here!
In a few cases, we do have Grice wondering about why a word with philosophical 'pedigree' or something is used by logicians disparately. My favourite must be when he discusses 'protothetic' logic, I think it was. He has this: "This belongs to a branch of logic as it is or was practiced, called, I think, protothetic (why?)".
So, I wouldn't be surprised if he found Goedel's use of 'category' eschatological, or something!
Looking for an explanation of the use of "categorial" in mathematical logic I found this nice page on "categorical propositions":
ReplyDeletehttp://www.philosophypages.com/lg/e07a.htm
which tells me, incidentally, that in my earlier search for a name for the kind of language in which deduction is possible, (which I ended up with "propositional"), I might have been better tuned with history if I had chosen "categorical", which I might well do, for the sake of connecting myself with this excellent bit of Aristotle rather than making an appearance of saying something new.
Anyway, my guess is that the story for the mathematical usage runs like this.
The usage in "categorical proposition" is partly to do with definiteness of the predicate.
For a proposition to be categorical it has to be a subject-predicate proposition and the predicate must cleave the universe into two parts one of which has and the other does not the relevant characteristic. It must be definite (even if we don't know the answer) for any particular, whether or not it satisfies the predicate.
Thus we have something which suits Grice, for it is the root of the law of the excluded middle, which we can now use Aristotelian language to say must hold of any categorical proposition, and hence hold in natural language (contra Strawson) insofar as it is categorical (though Strawson could counter with the observation that not all of natural English is categorical, and that definite descriptions may lead you outside the realm of categoricity).
So we have this connection bewteen "categorical" and "definite" which is then elaborated through history. For example, in Kant's categorical imperi
ative the word means "unconditional" which is another kind of definiteness.
The usage in mathematical logic is just that.
A set of axioms are categorical if they unambiguously characterise a particulat mathematical structure, i.e. if when construed as an implicit definition, they do that, they are definite enough to define a single structure.
Now we can fix the impression I had that Aristotle's use in categorical syllogism and in category is disparate (which I suspect only shows how ignorant I am of Aristotle), for the idea that predicates cleave the universe into two halves is an oversimplification.
Normally they have a domain in which they do this, and in the rest of the universe they just don't make sense.
These domains in which predicates can be said to be definite are the categories.
Categories are therefore part of a refinement of syllogistic logic recognising that few predicates really are universally definite, and admitting that they will normally only be definite when applied to the right kind of object.
Hence, we have the law of the exluded middle only if the proposition does not make any of Ryle's category errors.
In mathematical logic the category idea leads us into type theories and mathematics into category theory.
RBJ
This is good.
ReplyDeleteThe online Lewis/Short Latin dictionary has as transliteration for
κατηγορία
"In logic, a predicament, category or class of predicables (pure Lat. praedicamenta): Aristotelicae, Isid. Orig. 2, 26, 1; Sid. Ep. 4, 1: Aristotelica quaedam, quas appellat decem categorias, Aug. Conf. 4, 16; Serg. Expl. in Art. Don. p. 487, 25 Keil."
Oddly, that is Sidonius, Epistola, where, as Lewis & Short also tell us, we first encounter 'implicatura' in Latin.
The Liddell/Scott Greek Lexicon online provides:
κατηγορία.
in Logic, predication:
Cites:
Arist. Metaph.1007a35, etc.:
in plural, Id.APo.84a1;
especially affirmative predication, opposite to στέρησις (steresis) Id.APr.52a15; ἄπορον ἐν κ. Stoic.2.93.
But also:
"predicate,":
Arist. Metaph.1004a29, 1028a28, al., Epicur.Ep.1p.23U., etc.
More frequently: category qua head of predicables:
Arist. Topica 103b20 (ten), APo.83b16, Ph.225b5 (eight), Metaph.1068a8 (seven), cf.EN 1096a29.
-----
It's interesting how one would NOT think of 'quality', for example, as a 'category', but one should. I.e. students of philosophy go directly to define the categories of qualitas, quantitas, relatio and modus (in Kant -- and their base in Aristotle) without analysing per se the abstract notion itself.
There is also a Stoic use.
It should also be pointed out that the ORIGINAL use was 'accusation', and that Aristotle is giving a 'sophistic' term, as it were, a new 'philosophical' usage:
Thus reads the online etymological data bank at:
http://www.etymonline.com/index.php?term=category
"category: 1580s, from M.Fr. catégorie, from Late Latin categoria, from Gk. kategoria, from kategorein "to speak against; to accuse, assert, predicate," from kata "down to" (or perhaps "against") + agoreuein "to declaim (in the assembly)," from agora "public assembly," from PIE base *ger- "to gather" (see gregarious)."
"Original sense of "accuse" weakened to "assert, name" by the time Aristotle applied kategoria to his 10 classes of things that can be named."
"category should be used by no-one who is not prepared to state (1) that he does not mean class, & (2) that he knows the difference between the two ...." [Fowler]
---
Grice loved to use the term 'transcategorial' which is the best way to describe what Ryle was up to with his 'category' mistakes, of course.
Oddly, Thomason has an online essay on 'sortal' incorrectness which may have a Carno-Gricean ring to it.
Suppose I say, "My neighbour's three year old is an adult" (G&S, In Defense of a dogma -- example of analytically false). The point of it being analytically false is that it is transcategorial. It misdefines, or misattributes, or misaccuses. One difference between Carnap and Grice here would seem to apply to changes in belief. It would seem that for Carnap, a change in a meaning postulate means a change in language. In Grice, one may be more tolerant as to how a 'belief' can be changed without change in the linguistic framework. But I'm speaking vaguely, and only remotely connecting myself with the categoricity of Gooedel that started the thread.
But if 'category' is ALSO 'predicate', there is this further link that the "predicate" calculus is a 'category' calculus that lacks categoricity!
Fowler is often compared with Grice & Co. So it's interesting he says, as per above, that a category is NOT a class. Russell would NOT agree!
The origins you indicate are odd, but language evolves in strange ways.
ReplyDeleteThat "quality" is a category seems OK to me, of course qualities are not themselves categories. This squares well with the connection with type theories, in which qualities have a type which relates to the type of thing of which they are qualities.
(this is more elaborate than Aristotle).
I can't say I concur with your analysis of the neighbour's three year old, and the idea that it is a category mistake surely contradicts the idea that it is analytically false. Category errors surely yield nonsense rather than falsehoods. Otherwise if you negated a category error you would end up with a truth.
Carnap's meaning postulates are possibly the weakest point in his mature account of semantics.
They must be understood as implicit definitions, and it is therefore inappropriate (IMO) to talk of changing ones mind about whether they are true, one can only change ones mind about whether they correctly characterise the intended concept (and hence whether the language thus defined is the one the designer had in mind) or about whether that concept is the right one to have in the language. "Believing in the truth" of a meaning postulate suggests a confusion as to its status.
Perhaps not JUST a class, a special kind.
In a type theory, if one regards the types as complex analogues of categories, then if one has classes as well they are likely to be either virtual ones, extensional versions of propositional functions or else typed ones. In either case there corresponds to each type a universal set or class (which may or may not be a fiction) and the categories are then just the universal classes.
The connection with predicates is then seen to be reasonable. A predicate denotes a propositional function whose extension is a class, and in the special cases where the predicate is tautologous all these might possibly be thought of as categories.
RBJ
Excellent points. I would think I would take issue, if that's the expression, but friendlily so, with your view of a negation of a sortal incorrectness as 'nonsense'. I do hold them as true. In fact, you may recall that I have this poll in "The Grice Club":
ReplyDelete"Is Grice the greatest philosopher that ever lived?"
where one is allowed to vote:
"(1) Yes.
(2) No.
(3) Truth-Value Gap."
-- a joke of course, seeing that Grice hated them -- 'truth-value gaps'.
So, one has to be careful.
With the tree-year-old child NOT being an adult, the point is, as you say: trite: it is TRUE.
Grice does not consider other scenarios, well trodden by Carnap:
"Caesar is a prime number"
---- "I don't understand what you mean".
-- So, we don't want to GO there -- although perhaps we should.
But back to the three three-year-old child not being an adult. Why? Well, I would think there is a 'meaning postulate' here:
(x)( Cx --> -Ax )
(No child is an adult.)
equivalent to Carnap's "No bachelor is married"
(x) (Bx --> - Mx)
(if I have the symbols right).
"No child is an adult". What about "That's not so general: this one child IS an adult". -- which is what A is informing B about A's neighbour's three-year-old child.
Perhaps the point is comparable to what you say about 'accuse', to 'low down' ("kata") to the public market place ("agora") -- "kat-agorein" -- category. Some odd origin for such a basic philosophical notion.
I once concocted a scenario, seeing that in English, 'child' involves TWO different predicates, in, say: Italian: 'age' thing and 'kinship' thing. "The children of the sun", for example. Or Einstein's children. Suppose they are 87 and 91 years old, respectively, and their names are Jack and Jill.
A: I saw two children in the park today.
It turns out that they are 'the children of Einstein'. So, in this case, a 'child' IS older than 3-years, and is STILL a child. In fact, Einstein's children ARE well grown ups into adults.
The "Age" category features large in a lot of lexicalisation. As when we say "girl", or "boy". Predicates are best seen as binary. So that
GIRL -- for example -- is defined as: FEMALE -ADULT. While woman would be FEMALE+ADULT.
So in Grice's and Strawson's example:
x is a three-year old.
x is an adult.
Where IS the problem. The thing seems to be fuzzy, and they are careful to choose a pretty early age: three years. For surely depending on what country we are uttering that, the 'legality' about what consitutes a 'minor' may vary (14, 16, 18). So that, someone who is 15 years old may be an ADULT in some cultures. The way we are going, as a psychologist friend of mine tells me: some girls reach puberty at 12, which makes them ADULTS. She told me some reach puberty at 9, which makes them ALSO adults, then.
---- So one needs a criterion, etc. A 'type' as you say, extensionally defined, perhaps, or something.
There may be other examples. Etc.