Meaning and Rule-Following
The Naïve Theory of Meaning
- Each word is associated with an idea.
- Ideas are “mental pictures”.
- Ideas represent in virtue of “matching” or “resembling” things in the world.
Examples: ‘green’, ‘square’, ‘horse’. For extended critical discussion, see the Wittgenstein (1960).
Wittgenstein on Representation, Resemblance, and Rules
139(b). I see a picture; it represents an old man walking up a steep path leaning on a stick.—How? Might it not have looked just the same if he had been sliding downhill in that position? (Wittgenstein 1973, 139(b))
What needs to be added to the account of representation to allow for this distinction?
- Add more to the picture? (motion? arrows?)
- Then what sort of mistake did I make; was it what we should like to express by saying: I thought the picture forced a particular use on me? How could I think that? What did I think? Is there such a thing as a picture, or something like a picture, that forces a particular application on us; so that my mistake lies in confusing one picture with another?—For we might also be inclined to express ourselves like this: we are at most under a psychological, not a logical, compulsion. […] So our ‘belief that the picture forced a particular application on us’ consisted in the fact that only the one case and no other occurred to us. (Wittgenstein 1973, 54–55)
- Resemblance is not enough to force one “application” rather than another.
- What we need is a rule that tells us how to interpret the picture.
Augustine on Meaning and Ostension
Augustine: Now do this: tell me—if I were completely ignorant of the meaning of the word ‘walking’ and were to ask you what walking is while you were walking, how would you teach me?
Adeodatus: I would do it a little more quickly, so that after your question you would be prompted by something novel [in my behavior], and yet nothing would take place other than what was to be shown.
Augustine: Don’t you know that walking is one thing and hurrying another? A person who is walking doesn’t necessarily hurry, and a person who is hurrying doesn’t necessarily walk. We speak of ‘hurrying’ in writing and reading and in countless other matters. Hence given that after my question you kept on doing what you were doing, [only] faster, I might have thought walking as precisely hurrying—for you added that as something new—and for that reason I would have been misled. (Augustine 1995, 3.6; see also W. V. Quine 1950, 622ff.)
- Resemblance is not enough to force one interpretation of the demonstration rather than another.
- What we need is a rule that tells us how to interpret the demonstration.
The “Rule” Theory of Meaning
- Each word is associated with a rule.
- To know the meaning of a word is to grasp this rule.
Examples: ‘green’, ‘square’, ‘horse’.
The Rule Following Paradox
But switching to rules doesn’t make the basic problem go away:
No course of action could be determined by a rule, because every course of action can be made out to accord with the rule (Wittgenstein 1973, 201).
There is no unique correct answer to this “fill in the blanks” question: \(2,4,6,8,\underline{\ \ \ },\underline{\ \ \ }\).
Suppose you use the word ‘bos’ in the past, and you have applied it to all and only cows. And suppose you have never before encountered a purple cow, until now. Does your word ‘bos’ apply to this cow? Does your word ‘bos’ mean “cow” or “purple cow”? Notice that the history of your usage of the word ‘bos’ does not answer this question. What does? What facts about you determine the meaning of your word, which rule you are following?
Kripke on plus and quus.
I […] use the word ‘plus’ and the symbol ‘\(+\)’ to denote a well-known mathematical function, addition. The function is defined for all pairs of positive integers. By means of my external symbolic representation and my internal mental representation, I ‘grasp’ the rule for addition. […] Although I myself have computed only finitely many sums in the past, the rule determines my answer for indefinitely many new sums that I have never previously considered. This is the whole point of the notion that in learning to add I grasp a rule: my past intentions regarding addition determine a unique answer for indefinitely many new cases in the future (Kripke 1982, 7–8).
You grasp the rule for addition, and you use ‘plus’ and ‘\(+\)’ to denote it. But for some \(n\), you have never attempted to add numbers larger than \(n\). Suppose \(n\) is 57. So what is:
\[68 + 57 =\ \line(1,0){40}\]
Note the analogy with ‘bos’. Just like the purple cow was a case you had never encountered, this sum is a case you have never encountered. This raises what Kripke calls a “metalinguistic” question: how do you know what you mean by ‘\(+\)’?
This sceptic questions my certainty about my answer, in what I just called the ‘metalinguistic’ sense. Perhaps, he suggests, as I used the term ‘plus’ in the past, the answer I intended for ‘\(68 + 57\)’ should have been ‘5’! […] perhaps I used ‘plus’ and ‘\(+\)’ to denote a function which I will call ‘quus’ and symbolize by ‘\(\oplus\)’ (Kripke 1982, 8–9).
\[x\oplus y = \begin{cases} x + y & \mbox{if } x,y < 57 \\ 5 & \mbox{otherwise} \end{cases}\]
What facts about you determine which rule you are following, and so what you mean by ‘plus’?
Dispositions?
Proposal: You mean \(+\) not \(\oplus\) by ‘plus’ because you are disposed, when asked to perform sums, to perform addition, not quaddition.
Three objections:
- Your dispositions only determine a finite number of cases, and so cannot uniquely determine the function \(+\) as opposed to some \(\oplus\)-like alternative.
- For very large numbers, the idea that you have any disposition to respond one way or another is nonsense.
- You can be disposed to make mistakes. But that means that the rule you are following is not determined exactly by your dispositions.
Goodman on Green and Grue (Goodman 1983, 72ff):
- Grue
- \(x\) is grue if and only \(x\) is green and first observed before 12/3/2014, or \(x\) is blue and not first observed before 12/3/2014.
You are disposed to accept the following inference:
- Every emerald we have observed has been green.
- So probably the next emerald we observe will be green.
You are not disposed to accept the following inference:
- Every emerald we have observed has been grue.
- So probably the next emerald we observe will be grue.
But
- What makes the first inference good and the second bad?
- Can you be sure that when you use the word ‘green’ you mean green and not grue?