"What the Tortoise Said to Achilles" is a brief dialog by Lewis Carroll which playfully problematizes the foundations of logic. The tortoise challenges Achilles to use the force of logic to make him accept a particular deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression.
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2 What's wrong here 3 Where to find the article 4 References |
The discussion begins by considering the following logical argument:
Summary of the dialogue
If we take A and B as the two indicated sides, we can formalize these statements in mathematical symbols as:
- (A): ∀x,y,c: (x=c and y=c) ⇒ x=y
- (B): ∃k: A=k and B=k
- (Z): A=B
The Tortoise is obviously a troublemaker, since (Z) follows necessarily from (A) and (B) given the standard laws of logic. Again using mathematical symbols, we can rigorously show this as follows:
- Let s be the "same" to which A and B are equal. (The second premise guarantees that there is such an s)
- A=s and B=s.
- (A=s and B=s) ⇒ A=B. (Specialization of (A))
- A=B. (Modus ponens)
- (C): (A) and (B) ⇒ (Z)
- (A): "Things that are equal to the same are equal to each other"
- (B): "The two sides of this triangle are things that are equal to the same."
- (C): (A) and (B) ⇒ (Z)
- therefore (Z): "The two sides of this triangle are equal to each other"
- (D): (A) and (B) and (C) ⇒ (Z)
- (A): "Things that are equal to the same are equal to each other"
- (B): "The two sides of this triangle are things that are equal to the same."
- (C): (A) and (B) ⇒ (Z)
- (D): (A) and (B) and (C) ⇒ (Z)
- ...
- (n): (A) and (B) and (C) and (D) and ... and (n) ⇒ (Z)
- therefore (Z): "The two sides of this triangle are equal to each other"
Several philosophers have tried to resolve the Carroll paradox. Isashiki Takahiro (1999) summarizes past attempts and concludes they all fail before beginning yet another.
What's wrong here
Where to find the article
References