Is it a Contradiction?

London Ed writes,

I am interested in your logical or linguistic intuitions here. Consider

(*) There is someone called ‘Peter’, and Peter is a musician. There is another person called ‘Peter’, and Peter is not a musician.

Is this a contradiction?  Bear in mind that the whole conjunction contains the sentences “Peter is a musician” and “Peter is not a musician”. I am corresponding with a fairly eminent philosopher who insists it is contradictory.

Whether or not (*) is a contradiction depends on its logical form.  I say the logical form is as follows, where 'Fx' abbreviates 'x is called 'Peter'' and 'Mx' abbreviates 'x is a musician':

LF1. (∃x)(∃y)[Fx & Mx & Fy & ~My & ~(x =y)]

In 'canonical English':

CE. There is something x and something y such that x  is called 'Peter' and x is a musician and y is called 'Peter' and y is not a musician and it is not the case that x is identical to y.

There is no contradiction.  It is obviously logically possible -- and not just logically possible -- that there be two men, both named 'Peter,' one of whom is a musician and the other of whom is not.

I would guess that your correspondent takes the logical form to be

LF2. (∃x)(∃y)(Fx & Fy & ~(x = y)) & Mp & ~Mp

where 'p' is an individual constant abbreviating 'Peter.'

(LF2) is plainly a contradiction. 

My analysis assumes that in the original sentence(s) the first USE (not mention) of 'Peter' is replaceable salva significatione by 'he,' and that the antecedent of 'he' is the immediately preceding expression 'Peter.'  And the same for the second USE (not mention) of 'Peter.'

If I thought burden-of-proof considerations were relevant in philosophy, I'd say the burden of proving otherwise rests on your eminent interlocutor.

But I concede one could go outlandish and construe the original sentences -- which I am also assuming can be conjoined into one sentence -- as having (LF2). 

So it all depends on what you take to be the logical form of the original sentence(s).  And that depends on what proposition you take the original sentence(s) to be expressing.  The original sentences(s) are patient of both readings.

Now Ed, why are you vexing yourself over this bagatelle when the barbarians are at the gates of London?  And not just at them?

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Comments

[London Ed]

>>LF2. (∃x)Fx & Mp & (∃y)Fy & ~Mp

Actually I am not so sure of this as an interpretation, for it does not capture the 'another person called "Peter"' bit. In classical logic, the constants are not essentially ambiguous, as they are in English.

[BV]

Right. I made a change.

[London Ed]

>>Right. I made a change.

To this:

LF2. (∃x)(∃y)(Fx & Fy & ~(x = y)) & Mp & ~Mp

I still suspect this is incorrect, but I am not an expert. The reason is that in predicate logic (unlike ordinary language) there is no semantic connection between 'being called Peter' and the use of the logical constant for Peter. Thus your formalisation does not imply Ex Fx and Mx. I may be wrong (although pretty sure).

[BV]

Let's focus on (LF1). I assumed that 'x is called "Peter"' is interchangeable salva significatione with 'x = Peter.' That seems like a harmless assumption. Or can you show that it isn't?

Another assumption I clearly stated above:

>>My analysis assumes that in the original sentence(s) the first USE (not mention) of 'Peter' is replaceable salva significatione by 'he,' and that the antecedent of 'he' is the immediately preceding expression 'Peter.' And the same for the second USE (not mention) of 'Peter.'<<

That seems harmless as well.

No one who understands English will understand 'There is a man named "Peter" and Peter is a musician' as meaning that there are two men being referred to.

[london ed]

I understand your point, but my question is about classical predicate logic. I am 99% certain that classical logic is not allowed to 'see through' the predicate to the assignment of the constant. I agree we can do this in ordinary language. I.e. in English "'this person is called "Peter"' is interchangeable salva significatione with 'this person = Peter.' "

I don't think this holds in classical i.e. formal logic. Of course we can assign the constant 'Peter' to something, but the assignation is not itself a proposition of the calculus. I think Frege himself makes this point.

But this is not a speculative question. Only appeals to authority are valid here.

[BV]

Well, do the authorities (Frege, Russell, Quine, chiefly) address this question head on? If yes, please cite them.

Is it your point that in MPL (modern predicate logic) it is expressly denied that one can move from

x is called 'N'
to
x = N
?

Or is it your point that nothing in MPL sanctions this move and that the move needs sanction?

And why is this a logical question, strictly speaking, and not a question of the translation of O.L. into logical symbols?

[London Ed]

I believe, in the sense of being fairly certain, that it is ill-formed.

On the authorities, Frege says in S&R that if one asserts "Kepler died in misery" then the sense of the sentence does not contain the thought that the name 'Kepler' designates anything. Otherwise we would have to represent the negation not as "Kepler did not die in misery" but as "Kepler did not die in misery OR the name 'Kepler' has no reference".

He is arguing this on the general principle that logic cannot have propositions which assert things about the meaning of those propositions. Tarski made a similar argument. Also Buridan produced many paradoxes caused by doing the same kind of thing.

>>And why is this a logical question, strictly speaking, and not a question of the translation of O.L. into logical symbols?

It is simply about the well-formedness of the predicate calculus.