[FOM] Do the British not accept such principles as "p is true iff p"?

Dean Buckner Dean.Buckner at btopenworld.com
Fri Oct 17 15:53:52 EDT 2003

... asks Tennant.  I was taught by the late C.J.F.Williams, who was a
British logician of some authority (as a member of the editorial board of
Analysis he may have refereed Tennant's papers submitted to that journal).
He certainly would have questioned this principle, being a disciple of

On the principle itself, which I've questioned on FOM before, without any
sensible answer, the OBVIOUS objection is that the letter "p" in Tennant's
formulation occurs twice, but appears to be doing very different work.  You
can read it as a variable on the first occurrence, meaning "for any sentence
p", i.e. "p" is standing in for that sentence.  But I don't know how we are
meant to read the second occurrence.  Substituting "the sentence p" for "p",
to make it absolutely clear, gives

 For any sentence p, the sentence p is true iff the sentence p ...

The sentence p what?  "The sentence p" is not a sentence, though it refers
to one.  The problem is that in referring to a sentence we are using a noun,
which is not a sentence.  To convert it back to a sentence, we need to add a
verb, the obvious candidate being "is a fact" or "expresses what is the
case" &c.  Thus

 The sentence p is true iff the sentence p expresses a fact.

which is obviously  circular.  Friedman has mentioned Russell with approval,
so let me quote Russell.  Russell writes

"There appears to be an ultimate notion of assertion, given by the verb,
which is lost as soon as we substitute a verbal noun, and is lost when the
proposition in question is made the subject of some other proposition. ."
[as in "p is true iff p" for example]  "This difficulty, which seems to be
inherent in the very nature of truth and falsehood, is one with which I do
not know how to deal with satisfactorily.  . I therefore leave this question
to the logicians with the above brief indication of a difficulty." (In
sections § 52  § 54 and passim of Principles of Mathematics, also § 478
where he discusses the same question in the context of Frege).

Who are the "logicans" Russell is referring to?  Are they mathematical
logicans?  Does mathematical logic deal with this problem?  It is central to
Prior's work, and is an important part of the book by Strawson I mentioned
(which Friedman claims to have read).

As Slater has written

"In the equivalence between 'that p is true' and 'p', we find the connection
between reference to and expression of propositions: what is in one place
referred to with the nominal phrase 'that p' is expressed and asserted in
other places, with 'p'. So, given that it operates on a sentence, 'that' is
a nominaliser: it operates just like a demonstrative to produce, with the
sentence then immediately used, a linguistic phrase which refers to what is
then expressed.*"

Mathematicians have this habit of turning everything into nouns.
For example, defining a relation to be some sort of object, even
though grammatically relations are much more like verbs. So use of the word
"likes" commits you to the existence of the set of all ordered pairs (x,y)
such that
x likes y.


* See: Kneale, W. 1972, 'Propositions and Truth in Natural Languages', Mind,
81, pp 239f.
Slater, B.H. 2001, 'Prior's Analytic Revised', Analysis.

Dean Buckner,    : If one person can see it as a paradise
London,              : of mathematicians, why should not
ENGLAND        : another see it as a joke?

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