1. Rule-Following
Wittgenstein

2. Saul A. Kripke
Child prodigy, proved first completeness theorem for modal logic (Kripke semantics) at 17.
Wrote Naming and Necessity
Has a theory of truth designed to solve the paradoxes

3. The Wittgensteinian Paradox

4. Kripkenstein
In his book, “Wittgenstein on Rules and Private Language,” Kripke provides a “Wittgensteinian paradox,” which he then provides a “skeptical solution” to, as an interpretation of Wittgenstein.

8. “Pattern Problems”
Based-Problems.html%7Cpage1

9. Rule-Following
What makes it true that you are following one rule, rather than a different one that is also compatible with what you’ve done so far?

10. Rule-Following
Furthermore, how can we tell if you’ve made a mistake and are not following the rule you intended to?

11. Rule Following
(Sec. 201) “This was our paradox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule.”

12. Hume on Causation
Some event A happens. Then some event B happens. What makes it true (if it is) that A caused B, rather than something else that happened before B caused B?

13. Rule-Following
Q: What makes it true that you are following one rule, rather than a different one that is also compatible with what you’ve done so far?

14. Plus and Quus

15. Features of the Case I use the word ‘+’ to mean addition.
I have in my life computed finitely many sums.
There is a largest number that has ever been part of my past computations: suppose for example it is 57.
Although the sums I’ve computed are finite in number, the rule that I ‘grasp’ determines the sums of infinitely many pairs of numbers.

16. A New Problem
Now I am asked to add 57 and 68. Naturally, I respond ‘125.’ I think (a) the answer is correct and (b) the answer is in accord with my past use of ‘+’ and the rule I learned in school.

17. The Skeptic’s Challenge
The skeptic comes along and asks: How do you know that in the past, your uses of ‘+’ meant plus, rather than quus? x quus y = x plus y, if x, y < 57 = 5 otherwise

18. The Skeptic’s Challenge
NOTE: while the skeptic’s challenge is bizarre, the skeptic is not a skeptic about everything, just this. It would suffice to provide one fact about your past behavior/ mental states that justified you in now believing ‘+’ means plus and not quus.

19. The Skeptic’s Challenge
Of course, if there is no fact about your past usage that justifies you in one response over another, then the same would be true of your present usage.

20. All of Language
The worry here is a general one about rule following and hence all of language, not simply mathematics.

21. Definition of ‘Grue’
An object x is grue =df x is examined before t0 and green; otherwise x is blue

22. t0

23. The Algorithm Solution

24. Counting
Adding isn’t one of our “primitive functions”: usually we add by doing more basic procedures, like counting.

25. Quounting
But the skeptic can just push the problem further. How do you know that by ‘count’ you meant count and not quount? Quount: like count up to 57, always returns 5 thereafter.

26. Quum A similar response goes for abstract definitions of addition.
x + 0 = x x + s(y) = s(x + y)

27. The Dispositional Solution

28. Dispositions
The vase is fragile, the safe is not. But this isn’t a matter of how they are: neither IS broken. To be fragile is to be disposed to break, if struck.

29. Dispositions to Add
Clearly in the past, I have only computed the values of ‘x + y’ for finitely many numbers. But perhaps I had a disposition to give an infinite number of responses, for arbitrary x and y.

30. The Disposition Theory
Theory: the function I mean by ‘+’ is the one consisting of the answers I am disposed to give when queried with ‘x + y’.

31. The Problem of Finitude
“[S]ome pairs of numbers are simply too large for my mind… to grasp. When given such sums, I may shrug my shoulders for lack of comprehension; I may even… die of old age before the questioner completes his question.”

32. Ceteris Paribus?
Suppose we say: the answer to ‘x + y’ is the answer I am disposed to give, all things being equal.

33. If I did have the memory, and the time…
“How in the world can I tell what would happen if my brain were stuffed with extra brain matter or my life were prolonged by some magic elixir? … We have no idea what the results of such an experiment would be. They might lead me to go insane, even to behave according to a quus-like- rule.”

34. The Problem of Error
Some individuals commit systematic errors of addition. According to the disposition theory, these aren’t “errors,” rather, these individuals denote nonstandard arithmetic functions by ‘+’.

35. -Lesson: dispositions might (
-Lesson: dispositions might (!) solve the problem of finitude, but they don’t capture the normative force of meaning: uses of words are correct or incorrect.

36. CTM
One popular view among philosophers is the Computational Theory of Mind: the brain is a computer and the mind is its software.

37. Adding Machines
On this view, the brain when it adds is an adding machine. So if Kripkenstein’s argument works, it should work for adding machines too. But can the skeptic maintain that the calculator means ‘quus’?

38. Adding Machines
Problem of Finitude: calculators can only process numbers of a certain size. What fact determines how they add/ quadd after? Problem of Error: gears break, wires melt. What fact determines when the output of the calculator is wrong?

39. The “Special Quale” Solution

40. Qualia
Is there as special ‘what it’s like’ to mean addition?