In this paper I argue that Frege’s concept horse paradox is not easily avoided. I do so without appealing to Wright’s Reference Principle. I then use this result to show that Hale and Wright’s recent attempts to avoid this paradox by rejecting or otherwise defanging the Reference Principle are unsuccessful.
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In fact, this is just one half of what Wright (1998, p. 73) calls ‘the Reference Principle’; the other is the principle that co-referring expressions are intersubstitutable salva veritate in extensional contexts. For my purposes, however, it will suffice to focus solely on the half of the Reference Principle presented above.
The role of the Reference Principle in Frege’s thinking about the concept horse paradox is most visible in his (1892, p. 189).
See for example Frege (1923, p. 393).
Indeed, it seems plausible to suggest that ‘I’ and ‘me’ have exactly the same sense; the difference between these words would then be a product of the particular grammar of English. If that is right, then ‘I’ and ‘me’ are trivially intersubstitutable at the level of sense.
See also Frege (1892, p. 189).
From now on I will usually leave this qualification tacit.
At least, (4) is obviously true when our metalangauge is (an extension of) our object-language. In this paper I will ignore the case where the metalanguage is not (an extension of) the object-language; however, I believe that we could extend my argument to cover this case if we helped ourselves to an appropriate notion of translation.
I am here making the standard assumption that predicates are objects. I suspect that this assumption is false, but one thing at a time!
It is, however, grammatical to write ‘“\(\xi\) is a horse” refers to horses’, and it is well known that plurals in English at least sometimes play the role of predicates. Nonetheless, it is still not clear how to understand this sentence, nor is it clear that ‘horses’ is playing the role of a predicate in this particular sentence.
In fact, this claim is a little premature, and will not be fully justified until Sect. 6.
In order to explain fully how such a higher-order definite description operator works, we would have to say something about how to talk about identity in relation to properties. This is something that I will discuss in Sect. 6.
Compare Wright (1998, p. 79).
Hale (2013b, p. 142) recommends that we take ‘\(\square \forall x (\phi x \leftrightarrow \psi x)\)’ as a second-level analogue of identity; however, he also thinks that we can refer to properties with singular terms and so can also use ‘\(\xi =\zeta\)’ in relation to properties.
I am here assuming that if any property were an object, then every property would be an object. If you do not want to make this assumption, then you should weaken the requirement that \(\fallingdotseq\) be reflexive as follows: for any \(F\), if \(F\) bears \(\fallingdotseq\) to anything then it must bear \(\fallingdotseq\) to itself. For ease of expression I will leave this complication out of the main text.
They (2012, Sects. IV–VI) do ask whether there is something deeper that might motivate us to accept the Reference Principle; however, they then set about showing that there is not.
In fact, Wright (1998, Sect. VII) argues that the Reference Principle itself demands that we distinguish between reference and ascription. I will not discuss that argument here.
The idea that predicates ascribe rather than refer to properties is further developed by Liebesman in his (forthcoming).
Perhaps we could amicably agree to call Hale’s objects ‘bobjects’ and my ones ‘robjects’.
My predicament is exactly the same as the one in which Wittgenstein found himself at the end of the Tractatus (6.54). This should come as no surprise. As Geach (1976) convincingly argues, Wittgenstein’s saying/showing distinction has its roots in his reflections on the concept horse paradox.
See Button (2010, Sect. IV) for a similar line of thought.
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Thanks to Arif Ahmed, Daniel Brigham, Tim Button, Tim Crane, Owen Griffiths, Adrian Haddock, Bob Hale, Luca Incurvati, Colin Johnston, Fraser MacBride, Steven Methven, Michael Potter, Agustín Rayo, Lukas Skiba, Peter Sullivan, Nathan Wildman, Crispin Wright, Adam Stewart-Wallace and an anonymous referee. Thanks also to the Analysis Trust for their studentship, during which part of this paper was written.
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Trueman, R. The concept horse with no name. Philos Stud 172, 1889–1906 (2015). https://doi.org/10.1007/s11098-014-0377-x
- The concept horse paradox
- The reference principle