Language Games

The case of infinitely many propositions following from a single one

Is it impossible that infinitely many propositions should follow from a single one-in the sense, that is, that we might go on ad infinitum constructing new propositions from a single one according to a rule?[1]

Modal interpretation:

Is the following a valid argument, or rather, is it a possible argument:

◊ (p→q /\ (q→r) /\ (r →s)…n)))

Another interpretation:

◊ (p→(q /\ r /\ s…n))

In English: “it is possible that p implies q and q implies r and r implies s ad infinitum”

Now, the justification for the ‘ad infinitum’ is the idea that each the first proposition, p, is the first in a series of logical implications.

I think an example of this idea might be the following:

1. In order to move to the door of this room I need to travel the distance between this chair and the exit for this room.

1. The distance between this chair and the exit is roughly 6 feet.

1. It is necessary that in order to travel from this chair to the room exit, I must also travel from this chair to half of the distance to the exit, which is roughly 3 feet.

1. It is necessary that in order to travel from this chair to ‘half of the distance to the exit’, I must also travel to ‘half of the half of the distance to the exit’, which is roughly 1.5 feet.

You get the picture: a single proposition functions as a rule for the infinite regress that follows.

Wittgenstein’s response to this language game:

Suppose that we wrote the first thousand propositions of the series in conjunction.  Wouldn’t the sense of this product necessarily approximate more closely to the sense of our first proposition than the product of the first hundred propositions? Wouldn’t we obtain an ever closer approximation to the first proposition the further we extended the product? And wouldn’t that show that it can’t be the case that from one proposition infinitely many others follow, since I can’t understand even the product with 10¹⁰ terms?

We imagine, perhaps, that the general proposition is an abbreviated expression of the product.  But what is there in the product to abbreviate? It doesn’t contain anything superfluous?[2]

In a nutshell, Wittgenstein opposes the idea that any proposition p can serve as a rule for an infinite regress (or a series of infinitely many propositions following from p) from that proposition.

The assumption in the line of thought Wittgenstein opposes here is that any proposition p is inherently indeterminate; more carefully constructed, the idea is that it’s always possible to infer q or r or s, for instance, from any p.

For convenience I’ll call this view propositional indeterminism.

The logic of propositional indeterminism

Our idea that a proposition is indeterminate and thus may always be inferred from springs from our idea of language as a calculus.

Consider

P: (10+4)=Q: 14

And

P: (10+4)=Q: 14 or R: (7*2)

And

P: 10+4= Q: 14 or R(7*2) or S(5*2+4)

Let’s translate this into a logical argument.  To make it easier, I treated both addition and multiplication as logical conjunctions.  I also added inferences to make the argument valid.

1. If p, then q
2. If q, then (r and s)
3. If q then (t and u)[3]
4. If q then (r and s) or (t and u)

Wittgenstein, in effect, is saying that the expression ‘10+4′ is not coextensive with respect to the expression ‘14′.  It is in some circumstances, but not in the context of using their coextensionality as a foundation for an infinite regress. By itself, the proposition which says that ‘if 10+4′ then ‘14′ is a valid construction via the rules of arithmetic.  However, the expression doesn’t also say that “14 and 7 multiplied by 2 are identical”.  That they are identical can be expressed in another proposition, but it is not guaranteed or logically necessitated by the distinct expression that 10+4 also means 14.

[2] Wittgenstein, PG 250

[3] I realize that treating both multiplication and addition as a logical conjunction is problematic but that it is problematic is not relevant to this discussion

Simon van Rysewyk, author of the Wittgenstein Forum–a great blog, by the way—has a most helpful link to a Wittgenstein database called The Cambridge Wittgenstein Archive.

This database is for anyone ranging from the interested amateur to the most hardcore of fans/stalkers.   Its most noteworthy feature for me was the ability to search for Wittgenstein’s more personal works, such as his personal letters, entire notebooks, and notes Wittgenstein had taken (he was quite a note taker, from what I understand)

I’m not very familiar with online resources for Wittgenstein-related primary sources, but from what I can tell this is one of the most comprehensive available online.  Certainly it is more user-friendly than the Stanford Encyclopedia of Philosophy entry on Wittgenstein, for my money.

Does Brentano’s definition of intentionality as intentional inexistence presuppose a subject-object grammar?

I’m not sure, but let’s look at a few philosophical exercises I’ve concocted in thinking about this.

1. The ball appears red to me.
2. The ball is red to me
3. Ben thinks that the ball is red.
4. I assumed you meant that I hadn’t heard you completely.

1 and 3 are classical examples of “intentional propositions”.  4 is a bit trickier in a most fortunate way–it is fortunate because it is a counterfactual and thus specific to the English language (Mandarin, for example, “has no formal [that is, corresponding] translation for counterfactual expressions).  It is also intentionally rich—the expressions “I assumed” and “you meant” are clearly intentional phrases in the sense that ‘intentional’ means ‘to seem as if’–think of a simile.

Intentional inexistence says that all intentional acts obtain independent of the truth of their object(s).  We can have thoughts without our thoughts “actually referring to” something.

The question is, does this sort of characterization presuppose a function of language, namely, “the subject-object” function?

How can one think what is not the case? That’s a great example of a philosophical question and it is one that Wittgenstein explicitly addresses in The Blue Book, page 31 (among other places, I am sure)

“How can one think what is not the case? If I think that King’s College is on fire when it is not on fire, the fact of its being on fire does not exist.  Then how can I think it? How can we hang a thief who doesn’t exist?” Our answer could be put in this form: “I can’t hang him when he doesn’t exist; but I can look for him w hen he doesn’t exist.

Wittgenstein here is commenting on the fact that the object of a thought not existing shouldn’t be as philosophically perplexing as we allow it.  Once we get outside the “rules of a term’s use” we realize that, in this case, we can have the thought that p, where p is “King’s College is on fire” without p being the case in the same sense that I can look for a person that doesn’t ‘exist’ in the strict (and local) sense of the word.

What about the thought that

r: if I think something is the case then q: that thing must actually be the case

This is really the same dillema.  The problem is the confusion over the word “to think” and “what the thought denotes”.  Consider this case:

r1: If something is the case then that thing must actually be the case

R1 is a tautology.  It says that the truth of itself must obtain.  Realize that many times when we say “I think” we are not assigning an intentional state (in the philosophical sense) but rather drawing attention to the up-and-coming assertion.

w: I think that Wittgenstein is saying here that the meaning of a term is how the term is used in a social context.

What would it take for that sentence to make sense, let alone be true?

response: But the assertion w deals with a matter of interpretation because we can’t actually know what Wittgenstein meant.

I agree that it is a matter of interpretation, albeit in an unimportant way.  Consider your use of “what Wittgenstein meant” and realize that my use of “Wittgenstein is saying here that…” may not coincide in the way you want.  Maybe that part of the expression can be interpretted only as saying “the best possible interpretation for Wittgenstein’s writings here is…”

Now consider the difference between

1. “Wittgenstein is saying here” means “what Wittgenstein meant here”
2. “Wittgenstein is saying here that” means “the best possible interpretation for Wittgenstein’s writing here is…”

Which did I mean when I said it?  Who knows–but realize this:  when we ask questions like “well what did you mean when you said that…” we presume that the speaker actually knew the exact meaning of the expression in question when he uttered it.  That is, we think that the speaker ought to know what he or she meant when he or she used it.  Consider the case in which S says that p but realizes that p might imply r to T and then revises p in light of this newfound suspicion.  What did S mean when S said that p?  It depends on the time at which S said that p.  It also depends on the questioner and what he or she knows about the use of that particular term.

As Wittgenstein notes in The Blue Book, page 31:

We are  here misled by the substantives “object of thought” and “fact” and by the different meanings of the word “exist”. Supposing we asked: “How can one imagine wh at does not exist?” The answer seems to be: “If we do, we imagine non-existent combinations of existing elements” [this is also Russell's answer as a representative of logical monism].

We’re misled by the quality of a thought with its facticity as an occurent event. If I have something I call a thought, then what the thought has as its content need not be true.  Just like I can be “looking for x” without knowing that “x exists”.

My thoughts say things that may not be true.  But their failure to be true [in the correspondence sort of way] need not mean I can’t have them.  Or that they are meaningless.

I look for things that may not exist. Often times my act of “looking for a particular baseball card” may defy the fact t hat “in all probability, no one has access to that particular card” for example.

As the student who was tasered and arrested for asking John Kerry ‘too many questions,’ gained national media exposure, the AP demonized him as “a known prankster” and the author of a “disorganized diatribe”– though the basis for the labels bears no relation to the incident. Somehow, Meyer’s personality justifies excessive force by police?

President Bush is a known alcoholic, but apparently nobody’s around to demonize him when he is confronted with hard questions. Regardless of the intent of the student, this situation was pretty horrifying.  I’m finishing up my response to it now and will have it available shortly.

read more | digg story

Ok, I published the FAQ page. Regardless of how unlike it is from a normal FAQ page, at least I can say that my blog on Ludwig Wittgenstein’s philosophy HAS a FAQ page.

That’s right: my blog is the first blog to both be about Wittgenstein and have an FAQ page.  Apparently I didn’t get the memo that philosophers never write FAQ pages, since they’re too busy thinking and all.

I don’t know that much about Whorf’s theory regarding the relationship between language and thought.  I have no experience with anything by him and I haven’t so much as seen a publication he’s authored.

I am, however, being exposed to secondary literature in one of my cognitive psychology graduate seminars.  Within cognitive science, Whorf’s claim that language affects thought in profound ways and that two individuals who speak different languages will think differently is apparently some revolutionary stuff.  Revolutionary in the sense that its initial reception was not a welcoming one: most language theorists in the field of cognitive science and psycholinguistics poo-pooed the idea that native Hopi speakers, for instance, experienced and thought about the world in incommensurate ways as compared to English speakers.

In reading reviews of Whorf’s hypothesis concerning different linguistic communities I found it ironic that, on the one hand, most theorists seemed perfectly content with distinguishing between “language general” and ‘language specific” effects, or between “language effects” and “culture effects”, but not at all OK with the distinction between grammar and syntax, as it pertains to effects on “thought”.

One reviewer suggested that the distinction between grammar and semantics wasn’t very useful because there are many instances when the two become indistinct where “the effect on thought” can be explained equally well with recourse to grammar and/or to semantics.  Moreover, the one influences the other.

I grant that.  But in rejecting that distinction and accepting the equally vague, broad, and misleading “language vs culture” distinction, mainstream cognitive theory in this area seems to be forgetting the intimate bond “language” and “culture” share.  I want to say that where there is a difference in culture there is a difference in language.  Of course, being an avid reader of our pal Wittgenstein, my perspective on “language” is very peculiar and probably inappropriate for the sort of empirical ambitions cognitive scientists have.

I guess for psycholinguists and others in the field, it simply isn’t significant to consider the possibility that members of the same “linguistic community” (i.e. native English users) “have different thoughts” in virtue of their responses to questions regarding “thought” and/or “what they were thinking when x occurred.”

I like Whorf’s hypothesis, from what I hear of it.  I’d also like to say that a theoretical physicist, in virtue of his linguistic repertoire, has a very different “idea of” and therefore “experience of”, for instance, “spatial relations” or “the concept of time”.

That is, if we’re still thinking that there is such a thing as the “concept of time” that represents what all our instances of “understanding time” share.  Then again, maybe cognitive scientists are just playing another language game and ought not be ridiculed or questioned from non-players such as “people that try too hard to emulate Wittgenstein”.

“For instance, the sign for negation:

P

T | F

F | W

is worth no more and no less than any other negation sign; it is a complex of lines just like the expression “not-p” and it is only made into a sign for negation by the way it works-I mean, the way it is used in the game.  (The same goes for the T-F schemata for tautology and contradiction). What I want to say is that to be a sign a thing must be dynamic, not static.”

Several interesting things to note about this passage–which was taken from Philosophical Grammar, page p. 55, comment 17–but of striking importance is Wittgenstein’s claim that both “not-p” and the T-F (truth-false tables characteristic of first-order propositional logic) schemata are meaningful in virtue of the work they do–nothing about their meaning is inherent or intrinsic to their nature.  The meaning of these signs is something “dynamic”–or so “[Wittgenstein] wants to say”–they do not have meaning absent what use we get from them in this or that game.

But what is the difference between “not-p” and the negation sign in logic?  The question is misleading: according to Wittgenstein’s later philosophy, I don’t think he would grant that the question made much sense since it presupposes a strict “logical negation” in the sense of a logical contradiction of an assertion, p.  We ought not be confused by questions such as “what is the semantic difference, if any, between “the negation of something” and the “logical sign of negation”.  The question is asking us to compare the functions of two distinct operations occuring in very different language games.  And we can’t think of “negation” absent “what is being negated”.  Here, there is no strict sense of “not being”.

I’m adding an FAQ page to this site because I think it warrants one.

Let me know what you think once it’s completed!

Thanks,

David

My interpretation of Plato’s doctrine of negative assertions (or “not-being) in the Sophist at (roughly) 257e

Plato’s doctrine of negative assertions:

Stranger: May we not say that the existence of the not-beautiful is constituted by its being marked off from a single definite kind among existing things and again set in contrast with something that exists?

Theaetetus: Yes.

Stranger: So it appears that the not-beautiful is an instance of something that exists being set in contrast to something that exists.

Theaetetus: Perfectly.

Stranger: What then? On this showing has the not-beautiful any less claim than the beautiful to be a thing that exists?

Theaetetus: Just as much

Stranger: And we must also put the not-just on the same footing as the just with respect to the fact that the one exists no less than the other.

The existence of the not-A (the form not-A?) is constituted

(a) by its being separated from a single (definite) Form among existing things (particulars)

AND

(b) by its being separated from something that exists (form or particular??)

If and only if not-A then (p) A is distinguishable from a single definite form F among existing F-particulars and (q) A is distinguishable from other non-F particulars and/or non-F forms