How Does Plato Think That The Measure Doctrine Is Self-Refuting?

Protagoras’ Measure Doctrine (M) claims, “Man is the measure of all things” (152a). According to Plato, as he wrote in Theaetetus, Protagoras meant, “Everything is, for me, they way it appears to me, and is, for you, the way it appears to you” (152a). Essentially, truth is not independent of the individual, but truth is dependent on the perceptions of an individual; something is true because an individual believes it to be true.

(2) Why does Plato think that the Measure Doctrine is self-refuting?

Plato’s charge of self-refutation is as follows:

(M) Measure Doctrine holds that all beliefs are true. (170d)

Some people believe (M) is false.

The belief that (M) is false is true. (From (1) and (2))

Therefore, according to (M), (M) is false.

He claims that Protagoras himself would have to succumb to admitting the falsehood of his own doctrine. Plato goes specifically into Protagoras’ belief that his Measure Doctrine is true. He points out “Protagoras agrees that everyone has in his judgement the things which are.” (171a) Meaning, everyone has beliefs about what is true. As (1) states, Protagoras hold that all the beliefs of individuals are true. (170d) Protagoras assuredly would concur, according to (2), “on every occasion there are countless people who make judgements opposed to yours and contend against you, in the belief that what you decide and think is false[.]” (170d) Protagoras believes the Measure Doctrine to be true, but there are also “countless thousands” (170d) who believe the Measure Doctrine to be false. According to (1) and (2), Protagoras must agree that is his belief that ‘the Measure Doctrine is true’ is false. As Plato writes, “[T]he opinion of those who make opposing judgements about his own opinion—that is, their opinion that what he thinks is false—is true.” (170a) From (1) and (2), Plato can conclude (3). This offers self-refutation, as (3) contradicts (1). If someone states that p is true and then NOT p is also true, there exists a logical contradiction in the argument. It is the same for Protagoras’ argument, (1) states ‘(M) is true’ and (3) states ‘NOT (M) is true’. Plato rationalizes in (4), “So his theory will