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Relativism, truth, and Protagoras


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Relativism, truth, and Protagoras

Timothy Chappell, University of Bath, August 3 2006

t.chappell@open.ac.uk
(For something like this argument see Chappell, “Reading the Peritrope”, Phronesis 2006; also on my OU homepage

http://www.open.ac.uk/Arts/philos/chappell.htm.)
Today people often say things like “All perspectives are equally valid”, or “This is my truth, tell me yours” (© The Manic Street Preachers), or “This is true for me, and that is true for you”.
They’ve been saying that sort of thing at least since Plato’s time (425-345 BC).
Call such views examples of relativism about truth (here just ‘relativism’ for short).
In the first half of this talk, I’m going to look at how Plato, in his dialogue the Theaetetus (c.380 BC), argues against relativism.

The exact shape of Plato’s argument is disputed. I’ll criticise two interpretations. Then I’ll offer a third interpretation (my own). This turns out, magically, to be correct.



Half-time entertainment: an anti-relativist argument that Plato never quite offers—but maybe should.

That will lead us, in the second half, to look at how we today might argue against relativism.

The talk will go to extra time, where I’ll look at moral relativism, and argue that the important non-realist view expressivism is closer to moral relativism than expressivists would like it to be.

First half: Plato’s argument against Protagoras’s relativism
First, an outline of Protagoras’ position (Plato, Theaetetus 152a2-10):
Socrates: Protagoras, or so I’ve heard, held that man was the measure of all things: of those that are, that they are; of those that are not, that they are not…and isn’t this roughly what he means: “Particular things are to me just as they appear to me, and are to you just as they appear to you”?

Theaetetus. Yes, that’s what he means.
First interpretation (Bostock, McDowell, Chappell CQ 1995)
Protagoras’ relativistic thesis is that all beliefs are true. In logicianese:
(R1) For any proposition p, if p is believed, then p is true.
And Socrates’ objection to (R1) at Theaetetus 170-171 is that some people don’t believe it (e.g. Socrates doesn’t): some people believe that not all beliefs are true.
So we can argue against (R1) by showing that if it’s true, then a contradiction is true:


  1. (R1) For any proposition p, if p is believed, then p is true. (Assumption for reductio)

  2. Not-(R1) is believed. (Premiss)

  3. So not-(R1) is true. (a, b)

  4. But a contradicts c! Reductio.

The key move here is (b), which slots in Not-(R1) as a value for p.

The argument applies (R1) to itself.

In Greek, this is called the peritropê (the turning-round of the argument).

Protagoras could just deny (b), and tell Socrates that in fact he does believe (R1); but that seems a bit high-handed and unconvincing.
Problem with the first interpretation:
The text of the Theaetetus makes it obvious how Protagoras can reply.

He can say that Socrates doesn’t understand his view properly.

Protagoras’ real view (he can say) is not that all beliefs are true, but that all beliefs are true for someone (= the believer). In logicianese:
(R2) For any proposition p and any believer x, if x believes p, then p is true for x.
Now we can’t run an argument like (a-d) against (R2). Try it and see:


  1. (R2) For any proposition p and any believer x, if x believes p, then p is true for x. (Assumption for reductio)

  2. Not-(R2) is believed.

  3. So it’s true that, if x believes Not-(R2), then Not-(R2) is true for x.

  4. But g doesn’t contradict e! The reductio fails.

So, if we want Socrates’ argument against Protagoras to work, maybe we should read Protagoras differently?


Second Interpretation (Burnyeat, Denyer, Sedley)
Protagoras’ point is not (R1). (R2) is closer to the truth, but still not the best expression of what Protagoras is really getting at. For that, try (R3):
(R3) p is true iff there is someone to whom p appears.
And Socrates’ objection is that there is no one to whom (R3) appears.
So we get this argument against Protagoras:


  1. (R3) p is true iff there is someone to whom p appears. (Assumption for reductio)

  2. But there is no one to whom (R3) appears. (Premiss)

  3. So (R3) is not true. (i, j)

  4. But k contradicts i! Reductio.

The key move here is (j), which slots in (R3) as a value for p. The argument is another peritropê.


Problem with the second interpretation
Why should we accept (j)? Doesn’t (R3) appear to Protagoras (i.e. Protagoras believes (R3))? He certainly says he does!
Of course we can reject his claim to believe (R3), just as (as we saw) Protagoras can deny Socrates’ claim (b) in the first argument. But this is just a dialectical stalemate.
Third interpretation (Chappell Phronesis 2006)
Protagoras is proposing two things: an account of truth as truth-for, and an account of belief as truth-for.

That is, he’s claiming


(R4) p is true = p is true for someone, and x believes p = p is true for x.
So whenever I say that something is true, I can equally say that it’s true for someone (me).
And whenever I say that something is true for someone (me, or you, or whoever), I can equally say that it’s true.
And whenever we say that someone believes something, we can equally say that that something is true for him (and vice versa).
(R4) yields a contradiction as follows:
m. (R4) p is true = p is true for someone, and x believes p = p is true for x. (Assumption for reductio)

n. Protagoras believes (R4). (Premiss)

o. Socrates does not believe (R4). (Premiss)

p. So (R4) is true for Protagoras. (m, n)

q. But (R4) is false for Socrates. (m, o)

r. So (R4) is true. (m, p)

s. And (R4) is false. (m, q)

t. But s contradicts r! Reductio.


This is a peritropê too: it feeds in Socrates’ and Protagoras’ beliefs as instances of p, and Socrates and Protagoras as instances of x.

Half-time entertainment: the anti-relativist argument that Plato never (quite) presents
u. (R5) All truth is relative (to some believer or some context of belief).

v. (R5), if true, is either a relative truth or a non-relative truth.

w. If (R5) is a relative truth then the non-relative truth “Not all truth is relative” is consistent with (R5).

x. If (R5) is a non-relative truth then (R5) is a counter-example to itself, and so false.

y. Either way, the opponent of relativism has nothing to fear from (R5).
Another peritropê…look, in particular, at (v).
Does Plato run this argument against relativism about truth?
I don’t think so.
He comes close to running it, especially according to the First Interpretation.
Perhaps, on his own principles, he should run this argument.
But he never quite does.
Second half: how we might argue against relativism
First Argument


  1. Relativism says that we should talk, not about “truth”, but about “truth for people”. We should say, for any proposition p that we agree with, not “p is true”, but “p is true for me”.

  2. But this loses the normativity of truth.

If I say “p is true”, I give you reason to believe p. If I say “p is true for me”, you can just reply “how interesting” or “so what?”.

“p is true for me” implies nothing about anyone’s reasons to believe (not even mine).

So “true for someone” isn’t an adequate stand-in for “true”.

So relativism about truth won’t do.


A worry about this argument
Does “p is true” give me reason to believe p? What if p = “the 34562nd phone number in the current edition of the Paris telephone directory is 76-34-90”? Why should the truth of that proposition give me any reason to believe it?
Possible answer
The truth of p at least gives me reason (a) not to believe not-p, and (b) to believe p if I consider the question whether p.
To put it another way: saying that

we have reason to believe what’s true

doesn’t have to mean saying that

we have reason to believe everything that’s true:


no more than
saying that we have reason to do what’s good

has to mean

saying that we have reason to do everything that’s good.
So truth is still normative (just in a modester way than we first proposed).

And truth-for still isn’t normative.

So the argument still works.

Second anti-relativist argument (or train of thought)
What could “a relative truth” possibly be? Let’s look at precise examples.
(RD) “Derek is to the left”
is a relative truth, indeed a doubly relative one.
But that doesn’t mean that (RD) is some sort of sinister threat to non-relative1 truth!
It just means that (RD) is semantically incomplete.

What semantic (in)completeness is
If I say

(BD) “Derek is taller”,


you’ll say “Taller than who?”; and I might reply “Taller than Bashful”.
Likewise if I say

(RD) “Derek is to the left”,


that just prompts the questions (a) “To the left of what?” and (slightly less obviously) (b) “From what viewpoint?”
And we can answer these questions, e.g. by
(NRD) “Derek is to the left (a) of Grumpy (b) from where Sneezy is”.
So when you fill out (RD), you get (NRD). But (NRD) is a non-relative truth!
So semantic completion gives us a smooth and easy road from (RD)’s sort of relative truth to (NRD)’s sort of non-relative truth.
It shows how relative truth is a much more boring phenomenon than the relativist hoped.
Objection from the relativist
‘You’re completely missing the point. I’m not talking about the relativity of left or taller than. I’m talking about the relativity of truth.

I am saying that “is true” is like “is taller than”.

My point is that “is true” is semantically incomplete as it stands.

You need to add a reference to who the truth is true for.

“Is true” always needs to be spelt out as “is true for [someone]”.
My reply to this objection

‘I see what you mean; but there’s a point that you’re missing.

When we semantically completed (BD) “Derek is taller” by turning it into “Derek is taller than Bashful”, what we ended up with was a non-relative truth.

And when we semantically completed the relative truth (RD) by filling in who Derek was to the left of, and from whose viewpoint, we got the non-relative truth (NRD).

The point generalises: whenever you fill in these semantic completions, you end up with a non-relative truth.

So when you fill out the relative truth


(T1) “p is true”
as, say,
(T2) “p is true for Protagoras”,
(T2) will be a non-relative truth.
But relativism about truth is the claim that all truths are relative!

So (T2) is a counter-example to relativism about truth.’



What the relativist might reply to this reply

‘No, (T2) isn’t a counter-example. (T2) is semantically incomplete too. It needs to be filled out as (T3) “It’s true for someone else that it’s true for Protagoras that p”. And (T3) too will be semantically incomplete…’


My reply to this reply to my reply

‘It’s just plain barmy to say that every time we have a truth we have an infinite regress of extra relativisations that we need to add to that truth to make it semantically complete.’


The relativist’s next move

Why is it barmy? You non-relativists like to say that, for any true p, you can assert that it’s true that it’s true that p, and it’s true that it’s true that it’s true that p, and so on ad infinitum. Why is my regress any more vicious than yours?’


My next (and winning?) move

‘There’s no parallel. I say that, for any true p, you can assert that it’s true that p, etc. You say that, for any true p, you must go on adding the relativisations, or you won’t have a semantically complete statement. That means you never have a semantically complete statement of sub-infinite length! And that’s a disastrous consequence, because semantic incompleteness means meaninglessness.’



Extra time: is expressivism a moral relativism?
Our argument about relativism can be modified a little to deal with the case of moral relativism.
We can say: the moral relativist thinks that moral utterances are semantically incomplete: e.g.
“Stealing is bad”,
to be semantically complete, needs to be spelled out as
“Stealing is bad from my perspective”,
or something like that.
But this analysis faces the objection about normativity that I offered to relativism about truth: “Stealing is bad” gives people reasons in a way that “Stealing is bad from my perspective” just doesn’t.
What if we try and reinstate the normativity by saying this?
It’s good for people to say things like “Stealing is bad from my perspective”.
The problem is obvious: moral relativism says that we have to treat this too as semantically incomplete, and turn it into
It’s good from my perspective for people to say things like “Stealing is bad from my perspective”.
In short, the moral relativist will have to choose between finite semantic completeness and normativity. (Cp. the relativist about truth, who could only get semantically complete assertions by making them of infinite length.)
And this looks like a hopeless position.
So moral relativism is bunk.

Now most modern non-realists in ethics are not moral relativists of the kind discussed here.


Many of them (e.g. Simon Blackburn, Allan Gibbard, Mark Timmons, Jimmy Lenman, Mike Ridge, and (so he tells me) Derek Matravers) are expressivists.
Expressivism: moral utterances aren’t (really) assertions: they’re expressions of our attitudes.

(E.g. “Stealing is bad” is (really) more like “Boo for stealing!” than it is like “Derek is short”.)


Question: Is expressivism really a form of moral relativism?
Official answer: Of course not. Moral relativism says that moral utterances are not truths2, but truths-for-someone.

But expressivism says that moral utterances aren’t truths at all: they’re expressions of attitudes.



A fortiori, then, expressivism doesn’t say that moral utterances are truths-for-someone.
However…
If the realist tells you that moral utterances are truths, then you won’t want to ask “Whose truths?” unless you’re attracted to relativism. (And you shouldn’t be: as above, relativism is bunk.)
But if the expressivist tells you that moral utterances are expressions of attitudes, then you surely will want to ask: “Whose attitudes?”
When we answer this question, we start relativising attitudes to their holders.
So then we have to rewrite
“Stealing is bad”,
not simply as
“Boo for stealing!”,
But as
“‘Boo for stealing!’ according to Derek.”
And now expressivism is in deep doo-doo.
For if we’re expressivists, all we can do to restore reason-giving force to “‘Boo for stealing!’ according to Derek” is to say that we approve of Derek’s attitude.
This is meant (on the official expressivist line) to come out as
Hurrah for (Boo for Stealing!)!
But once we ask the question “Whose attitudes?”, we see that actually it comes out as
(X) Hurrah according to us for (‘Boo for stealing!’ according to Derek)!
And (X) is no more genuinely reason-giving than the relativist’s
It’s good from my perspective for people to say things like “Stealing is bad from my perspective”.
For (X) to become reason-giving, we’d have to stick another “Hurrah!” on the front of it.

But then the same question about whose attitude this is would arise... and so on indefinitely.



Hence expressivism collapses into something very like moral relativism:
For expressivism faces just the same problem as moral relativism about keeping hold of both finite semantic completeness and normativity.
Therefore, and finally, a quick argument in modus ponens:


  1. If moral relativism is bunk then expressivism is bunk.

  2. Moral relativism is bunk.

  3. [Exercise to do in your own time: fill in the conclusion.]



1 “Non-relative”, not “absolute”, because it’s apt to sound like some sort of spectacular achievement for truths to be absolute—but it isn’t at all difficult for them to be non-relative. They are non-relative when there is no non-trivial way of specifying some believer or context of belief to which they are relative. I take it most truths are like this most of the time. If that’s what absolute truth is, then I guess I believe in absolute truth—but as before would prefer to skip the boggle-inducing rhetoric of the word “absolute”.

2 Or falsehoods, of course—but for simplicity I concentrate on “correct” moral utterances.





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