[FOM] Another easy solution does not work

[Andrew Boucher]

Todd Wilson wrote:

>Actually, under the Barwise/Etchemendy treatment of propositions (at
>least what they call "Russellian propositions"), each proposition in
>the above infinite list is again just the Liar, and thus is
>paradoxical.  The statements above are translated in the B/E formal
>language as:
>
>    (1)  - True[that_2]
>    (2)  - True[that_3]
>    (3)  - True[that_4]
>    ...
>
>whereupon each of the associated propositions is equal to
>
>    [Fa [Fa [Fa ...]]] ,
>
>and thus all equal to each other (and to the liar, f = [Fa f]).  Thus,
>the analysis that leads to the conclusion that the odd-numbered
>propositions must have the opposite truth value of the even-numbered
>propositions, under this account, is flawed because it assigns
>different truth values to what, after some investigation, are revealed
>to be the same propositions.
>  
>
It sounds like B/E's view evolved significantly from "The Liar."  On p. 
98 they admit that the identification of propositions (in the case of a 
finite cycle) could be an error ("things that should be distinct are 
here modelled by the same set").  And they propose adding indices to 
distinguish them.  Indeed, B/E were pretty severe with the Russellian 
analysis of propositions  ("very unsatisfactory", p. 105), and on my 
reading anyway tended to promote the Austinian account, where 
ascriptions of truth are supposed to have an indexical to a situation. 
 The last approach would favor the uniqueness of each statement.  

In any case one can embellish the infinitary example in any number of 
ways to promote uniqueness of the propositions (under both Russellian 
and Austinian analysis) and presumably therefore dismiss the specter of 
self-reference:

(1')  Statement (2') is not true & 2 = 2
(2')  Statement (3') is not true & 3 = 3
...

It does however look like these criticisms can be directed equally at the BEM analysis of Yablo's paradox, so the infinitary sequence is not in fact an improvement in the regard (thus, my error).  If so it looks like a draw between the two approaches:  Yablo's paradox involves explicit reference to an infinite set but produces a direct contradiction, while the infinitary sequence 

(1) Statement (2) is not true
(2)  Statement (3) is not true
...

has statements which refer singularly but only produce a contradiction if one makes some assumption which essentially implies that statements (1) and (2) should have the same truth value.   I'd have to think more about this.

But as I said in my last post, the debate between self-reference and (for lack of a better word) infinitary-reference may be moot, since anyone willing to ban self-reference would seem naturally of disposition to forbid infinitary-reference as well (not that I would propose this by any means, since it seems to avoid rather than explain the paradox, as well as cutting out a fair amount of legitimate discourse).


Thanks to Todd for his correction, for Richard's formalization - always instructive - and I look forward to reading Tennant's article (once I get my hands on it - anyone have an electronic copy they can send me?). 

Andrew Boucher
Unaffiliated