Kripke's Kwas math.

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Term Definition

x ⊕ y (quads) if x, y less than 57 → x + y<br> otherwise → 5

1. background

In "Wittgenstein's Language Game" (1982), Saul Kripke brought up the fictitious operation ⊕ (quus) to ask "what is meaning?" and "what is following rules?"

It looks like addition, but by including a preliminarily limited condition (bordering on 57), it makes it impossible to distinguish whether one was really adding or following the rules of quasi-arithmetic.

2. rule-following skepticism

1. we have calculated in the past that "68 + 57 = 125".

2. but could also be interpreted as "in fact, "+" in the past meant ⊕" (it is always 5 if it exceeds 57)

→ Which is correct cannot be determined from a finite record of actions.

This is the skepticism of "indeterminacy of meaning" = any behavior can be consistent with multiple rules.

Kripke's Conclusion (Skeptical Solution)

Internal "understanding" and "intention" are not enough to fix rules.

Instead, "community custom" (community consent) becomes the criterion for deeming a given usage "correct.

If everyone in a society normally adds up to 57 or more, the "+" would function as an addition.

If ⊕ is adopted by the culture, "+" should be interpreted as such.

Thus, "having meaning" is nothing more than following norms supported by communal practices.

4. point organization

Kwas arithmetic is a thought experiment: create an operation that is "almost identical" to addition but differs in one respect, showing that the identification of rules can be arbitrary.

Finite data problem: It is impossible to uniquely determine the "correct rule" based on a finite number of examples.

Wittgensteinian reading: meaning does not reside in the mind of the individual, but in public activity.

Kripki calls this a "skeptical solution."

5. impact

Developed into a discussion on the sociality (externalism) of the contents of the mind.

This is similar to the question of "which rules can be said to have been learned from training data" in AI and machine learning.

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*In short**.

The quasi-arithmetic is a trick to expose the conundrum: "When you say '+', is there any evidence that you really mean 'addition'?" Kripke concluded from this that "meaning is rooted in communal practices, not in the inner life of the individual.

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