Breckenridge and Magidor on Arbitrary Reference: An apparent counterexample

[This is an early draft of a paper which, since being posted, has grown and changed title. Email me if you would like a copy. - TH 9/4/15]

In an interesting paper forthcoming in Phil. Studies, Breckenridge and Magidor argue for this thesis:

Arbitrary Reference (AR): It is possible to fix the reference of an expression arbitrarily. When we do so, the expression receives its ordinary kind of semantic-value, though we do not and cannot know which value in particular it receives.

Their primary argument in favour of AR is that it can be used to give an attractive account of 'instantial reasoning' such as this (their 'Argument 1'):

(1) There is someone x such that for every person y, x loves y [Premise]
(2) Let John be such a person
(3) For every person y, John loves y [Existential Instantiation on 1]
(4) Let Jane be an arbitrary person
(5) John loves Jane [Universal Instantiation on 3]
(6) There is some person x such that x loves Jane [Existential Generalisation on 5]
(7) But since Jane was an arbitrary person, for every person y there is some person x such that x loves y [Universal Generalisation on 6] 

I will not attempt to rehearse, or even summarize, their arguments, since they state them well and their paper is freely available on Magidor's website. My purpose here is to give an apparent counterexample to the claim that AR can be used to give an attractive account of instantial reasoning.

The following appears to be a logical truth: 

(Unref) If (all unreferred-to objects are white and there is an unreferred-to object), then there is a white object.

(By 'unreferred-to object', I mean an object which is never referred to by anyone or anything.) Here is a quasi-formal argument for (Unref):

(1) All unreferred-to objects are white and there is some unreferred-to object. [Assumption]
(2) All unreferred-to objects are white. [Conjunction Elimination on 1]
(3) There is some unreferred-to object. [Conjunction Elimination on 1]
(4) Let O be such an object.
(5) O is white. [Universal Instantiation on 2]
(6) There is some white object [Existential Generalization on 5]

(Unref) now follows from (1) - (6) by conditional proof.

This seems to be a valid argument. But the theory of instantial reasoning advanced by Breckenridge and Magidor seems to imply that the expression 'O' above refers to an unreferred-to object, which is absurd.

Tristan Haze
The University of Sydney

Reference
Breckenridge, Wylie & Magidor, Ofra (forthcoming). 'Arbitrary reference'. Philosophical Studies. 

There is a post about this paper on Ross Cameron's blog here.

On the Interpretation of the Propositional Calculus

I've just posted another (more recent) longer article on my homepage, On the Interpretation of the Propositional Calculus. The next post will be a short article, I promise.

Comments are welcome.  Here is the abstract:

The question considered is 'How can formulae of the propositional calculus be brought into a representational relation with the world?'. Four approaches are discussed: (1) the denotational approach, on which formulae are taken to denote objects, (2) the abbreviational approach, on which formulae and connectives are taken to abbreviate natural-language expressions, (3) the truth-conditional approach, on which truth-conditions are stipulated for formulae, and (4) the modelling approach, on which formulae, together with either valuation- or proof-theory, are regarded as an abstract structure capable of bearing (via stipulation) a representational relation to the world.

The modelling approach is developed here for the first time. The simple technical apparatus used for this is then applied to two issues in the philosophy of logic. (1) I demonstrate a corollary or converse to Carnap's result that certain 'non-normal' valuation-functions can be added to the set of admissible valuations of formulae without destroying the soundness and completeness of standard proof-theories. This sheds considerable light on a recent thread of the inferentialism debate which involves dialectical use of Carnap's result. (2) I show how the approach can be extended to quantification theory, by defining a model-theoretic notion of validity equivalent to the usual one, but making use of a proof-theoretic apparatus in place of the device of assigning values to formulae. This sheds light on the close relationship between proof- and valuation-theory.

On Identity Statements

I've just posted a longer article on my homepage, On Identity Statements: Against the ascriptional views.

Apart from minor revisions, it is about 18 months old now. I would not write it in the same way now, but I still hold the views expressed there. Comments welcome, here or by email (my email address is on my homepage and on the 'About/contribute' page here).

UPDATE 21/06/2016: I have removed the link, as a descendant of this paper called 'On Identity Statements: In Defense of a Sui Generis View' has finally been accepted for publication.

Comment on Brogaard and Salerno's 'Counterfactuals and Context'

This is a draft of a paper.

It is quite commonly believed by contemporary logicians that contraposition, strengthening the antecedent and hypothetical syllogism fail for counterfactuals. In their (2008), Brogaard and Salerno argue that the putative counterexamples to these principles are actually no threat, on the grounds that they involve a certain kind of illicit contextual shift.

Here I suggest that this particular kind of contextual shift, if it is properly so called, is not generally illicit, and therefore the counterexamples cannot be blocked with the kind of blanket restriction Brogaard and Salerno appear to advocate. This sort of restriction, I suggest, ought to be made at the level of particular inference rules.

Brogaard and Salerno conduct their discussion within the framework of the standard Lewisian account of counterfactuals, which says that

a subjunctive of the form ‘if A had been the case, B would have been

the case’ is true at a world w iff B is true at all the A-worlds closest (or

most relevantly similar) to w.¹

They introduce the term 'background facts', by which they mean to designate 'the respects in which A-worlds are relevantly similar to w'. Thus every counterfactual, once understood on the standard theory, is attached to a set of background facts. Now, the central claim of their article is that 'the set of contextually determined background facts must remain fixed when evaluating an argument involving subjunctives for validity'. One set of background facts per argument. Let us call this the Brogaard-Salerno Stricture. Brogaard and Salerno say that to break this stricture is to commit an illicit contextual shift, and since the putative counterexamples to contraposition etc. break the stricture, they should not be accepted.

For an argument to comply with Brogaard-Salerno Stricture, all counterfactuals occurring within it have to be alike in background facts. What I wish to point out is that this condition is plainly unsatisfied by a great many arguments, including the following:

If Mary hadn't had breakfast, she would have lunched sooner.

If John had worn black shoes, he would have worn black socks.

Therefore, if Mary hadn't had breakfast, she would have lunched sooner, and if John had worn black shoes, he would have worn black socks.

For the first premise, one of the background facts might be that Mary has a normal appetite. Another might be that she does not like to go hungry. These are plainly irrelevant to the second premise, i.e. these are plainly not background facts for the second premise. Conversely, John's sense of style has nothing to do with the first. We cannot stipulate that these premises are attached to the same set of background facts without doing obvious violence to their meaning. These two premises, if they are to be understood the way they are meant to be understood, cannot figure in the same argument without breaking the Brogaard-Salerno Stricture. But the above argument is obviously valid. Therefore the stricture is not generally appropriate. I suggest that a better course would be to restrict particular rules - starting with contraposition, strengthening the antecedent and hypothetical syllogism - in respect of background facts pertaining to counterfactual evaluation, rather than deductive argumentation in general. Other rules may be fair game too. In this connection, consider this passage:

But suppose we are wrong about this. Suppose shifting context mid-inference is no fallacy at all. Then a rather surprising consequence follows. Modus ponens - which many possible world accountants love and cherish - fails too. (2008, p. 44).

On my suggestion, the evidence for the claim of the last sentence might motivate the view that modus ponens needs to be restricted too - but still, not all deductive argumentation. Conjunction introduction, for example, is prima facie OK without such a strong restriction.

Tristan Haze

The University of Sydney

References

Brogaard, B. and Salerno, J. 2008. Counterfactuals and context. Analysis 68.1: 39–46.

Lewis, D. 1973. Counterfactuals. Oxford: Blackwell.   

1 This is the formulation used by Brogaard and Salerno. It is adapted from Lewis (1973). 

A Note on Hofweber's Distinction between Internal and External Quantification

ABSTRACT: Thomas Hofweber's distinction between internal and external quantification is crucial to the solution he offers to his now well-known puzzle about ontology. Here I argue that this distinction is not well motivated by the considerations he employs.

In a series of interesting papers (2000, 2005b, 2007), Thomas Hofweber has identified a puzzle about ontology and developed a novel solution. Briefly, the puzzle is that questions such as 'Do numbers exist?' seem trivial from one point of view, but highly contentious from another. On the one hand, it is obvious that, e.g., there are even numbers smaller than 6. It follows trivially from this obvious statement that there are numbers. On the other hand, it is hotly disputed among philosophers whether or not there are numbers. Hofweber himself denies their existence. Nonetheless, he agrees that, e.g., there are even numbers smaller than 6.

Hofweber's solution to this puzzle crucially involves a distinction between two kinds of quantification which he calls 'internal' and 'external'. External quantification is familiar; externally quantified statements impose conditions on a domain of objects. Hofweber plausibly argues (2000, 2005b) that we must also recognize a kind of quantification which does not impose such conditions. His strategy is to highlight a certain 'inferential role' which quantifiers play in natural language, which enables them to function as place-holders for incomplete information; suppose we knew that Fred admires Thomas Edison, but then forgot this, remembering only that there is someone Fred admires. Hofweber argues that 'this situation is completely general', and that 'the only instances of the quantifier might be things that don't exist'.¹

With this distinction between internal and external quantification on board, Hofweber's solution to the puzzle about ontology is that the "trivial arguments" to the existence of contentious entities are indeed trivially valid, on the proviso that the quantifiers in their conclusions are given an internal reading. Questions about what there is, where 'what there is' is construed externally, thus remain as a non-trivial subject matter for ontology.

My object here is to show that the distinction Hofweber intends to make is not what it may appear to be at first glance, and furthermore that it cannot in fact be motivated solely by means of the considerations (indicated above) which he employs.

Let us begin with the question: why can't external quantification play the role of facilitating the expression of incomplete information? From the considerations offered, it seems that the only reason is that, as Hofweber says, the only instances of the quantifier might be things that don't exist. Thus we might think of internal quantifiers as characterised by the fact of ranging over both merely intentional objects and not-merely-intentional objects, in contrast to external quantifiers, which range over not-merely-intentional objects only. (I will call this 'the simple intentional-permissive understanding' of internal quantification.) This, however, is not how Hofweber conceives the distinction.

This becomes clear once we look at his views about arithmetical discourse with the distinction between the merely intentional and the not-merely-intentional in mind.² You can look for a prime between 24 and 28, and thus be looking for something. However, you will not find one: in this sense, there is no such thing. Hofweber fully recognizes this distinction, while nonetheless believing all quantification in arithmetic to be internal. Thus Hofweber's distinction between internal and external quantification cannot be understood in terms of the pre-existing distinction between the intentional and the not-merely-intentional. And yet this pre-existing distinction seems a natural and sufficient basis for a notion of quantification fit to play the inferential role Hofweber identifies. Therefore his consideration of this role is not by itself a good motivation for his internal-external distinction.

Note carefully that this argument does not require that the simple intentional-permissive understanding of internal quantification be a suitable basis for a solution to the puzzle about ontology. Furthermore, it does not rule out Hofweber's using the puzzle itself as a motivation for a special reading of quantification. The point is that he has not succeeded in establishing an independent motivation for such a reading.

It might be objected that I have not made an adequate case for the possibility of the simple intentional-permissive understanding of internal quantification. But I am not seeking to establish this conclusively; only, given that Hofweber has identified an inferential role which calls for a non-external reading of the quantifiers, the simple intentional-permissive conception is prima facie a better candidate than one based on Hofweber's internal-external distinction (considered apart from any puzzle about ontology). It may seem as though I'm not being quite fair, since I haven't really made his distinction clear in its own right. But I have no idea how to do this. Hofweber wants a reading of quantification such that the following comes out true:

There is an x such that x is not a merely intentional object, and x does not exist (in the external sense).

It has not been made sufficiently clear that such a reading is available.

Finally, one might wonder how Hofweber's internalism about arithmetical discourse avoids trivializing arithmetic. For on this conception, so-called "existence statements" about merely intentional objects (e.g. the largest prime) can easily come out true. Hofweber handles this with a supplementary doctrine to the effect that quantification in arithmetic is generally restricted to statements which have instances containing number words or numerals ('one', '46', etc.). However, and as Hofweber himself acknowledges, this sort of account cannot be extended to the reals, since we do not have number words for all of them.³ This gives rise to the worry that whatever the truth is about our quantification over the reals, it may also account for our quantification over natural numbers in arithmetic, rendering internalism about arithmetic theoretically superfluous.

Tristan Haze

The University of Sydney

References

Hofweber, T. 2000. 'Quantification and Non-Existent Objects', in Empty Names, Fiction and the Puzzles of Non-Existence, eds. Everett, A. and Hofweber, T. CSLI Publications.

Hofweber, T. 2005a. 'Number Determiners, Numbers, and Arithmetic', The Philosophical Review 114:2.

Hofweber, T. 2005b. 'A Puzzle about Ontology', Nôus 39:2.

Hofweber, T. 2007. 'Innocent Statements and their Metaphysically Loaded Counterparts', Philosophers' Imprint 7:1, <www.philosophersimprint.org/007001/>.

These papers are available on Hofweber's homepage: http://web.mac.com/hofweber/Thomas_Hofwebers_homepage/Papers.html

1 (2000), p 16.

2 These views are indicated in his (2005a).

3 Thanks to Thomas Hofweber for helpful correspondence on this and related points.

A Liar Paradox of Material Implication

[UPDATE 19/8/17: This post is the subject of a journal article by Matheus Silva, who thinks the argument fails.]

Here I present a new objection to the material or "hook" analysis of indicative conditionals - the thesis that an indicative conditional 'If A then C' has the truth-conditions of the so-called material conditional - based on Liar-like reasoning. This objection seems invulnerable to any Grice-Lewis-Jackson-inspired pragmatic rejoinder.

(1) If (1) is true, (1) is false.

Let us call (1)'s antecedent 'A', and its consequent 'C'. I propose that the following sentence is intuitively true, or true based on intuitive and unproblematic reasoning:

(S) On the assumption that (1) is neither true nor false, A and C are false.

The reasoning is: Assume that (1) is neither true nor false. Then A is false, since it says that (1) is true, and C is false since it says that (1) is false.

If we accept the hook analysis, however, this reasoning is not secure. For if A and C are false, and the hook analysis is right, then (1) must be true. But if (1) is true, then A must be true (since it says that (1) is true), and C must be false (since it says that (1) is false). But then, by the hook analysis, (1) must be false, since it has true antecedent and false consequent. But if (1) is false, then A must be false since it says that (1) is true, but then by the hook analysis (1) must be true. But then...

The point is, the hook analysis treats (1) as a truth-functional compound, and this places it squarely in the Liar family, making our straightforward argument to (S) veer into paradox. Yet (S), and our argument for it, seem clearly correct. Therefore we should reject the hook analysis.

There are already plenty of intuitions around which seem to cast doubt on the hook analysis. This one has more bite, I submit, since it cannot be explained away with the customary sorts of pragmatic story. Take a case like 'If I die tonight, I will be alive tomorrow'. The typical proponent of the hook analysis will maintain that, given that I will not die tonight, this sentence is true but not assertable, since I should assert something stronger,¹ or something robust with respect to the antecedent,² etc. But (S) is an example of an intuitively true sentence which comes out as paradoxical (i.e. leads to paradox) if we apply the hook analysis to (1). It is hard to see how any Grice-Lewis-Jackson-inspired pragmatic story could account for our asserting, or treating as true, sentences which are "really" paradoxical.

That's the objection. While not exceedingly complicated, it is quite easy to misunderstand, so I shall conclude with a few clarifications. Firstly, the argument is not: when we apply the hook analysis to (1) we get Liar-like paradox, and since Liar-like paradox is undesirable, we should reject the hook analysis for (1). It is irrelevant to my objection whether Liar-like paradox is good, bad or indifferent. It is also irrelevant whether there is (or could be) a solution to these paradoxes. The point is simply that, intuitively, we do not get into Liar-like paradox with (1) and (S), and so the hook analysis seems to deliver the wrong answer on this point. For a truth-functional analysis of English conjunctions, on the other hand, generation of Liar-like paradox would be the intuitively right answer for certain sentences (e.g. 'This sentence is false and this sentence is false').

Secondly, I put (S) in the form 'On the assumption that X, Y', because if I had used the conditional form, the objection would have become messy through having to avoid begging the question against the hook analysis.

Finally, I am not maintaining that (1) is in no way paradoxical. It is paradoxical. To illustrate:

Suppose (1) is true. Then by (1) and modus ponens, it is false. Therefore, by conditional proof, if (1) is true then it is false. But that is just what (1) says, so it is true, but then by modus ponens (this time not within the scope of any assumption), it is false. Paradox.

What should be said about this and similar arguments, I regard as an open problem. Some thoughts: rejecting unrestricted conditional proof seems like a promising avenue, since several authors have done this for independent reasons.³ However, it is hard to shake the feeling that if (1) is true, it is false. Perhaps the object of this feeling could be accounted for as a 'Mackie conditional' or 'telescoped argument',⁴ and thus kept semantically distinguished from (1) read as an ordinary conditional. But if these telescoped arguments turn out to be truth-apt in some sense, and to sustain modus ponens, we would seem to be back where we started.

In sum, (1) does not appear to be a (full) member of the Liar family. An important difference can be expressed thus: while there are arguments involving the assumption that (1) is true which lead to paradox (and not just within the scope of the assumption), the bare assumptions that (1) false, or that it is neither true nor false, do not intuitively yield any such arguments (as they do with Liar-like sentences). Hence, we can and should accept (S) as straightforwardly true. And this means rejecting the hook analysis.

Tristan Haze

The University of Sydney

References

Beall and Murzi. draft. 'Two flavours of curry paradox'.

[draft available on the authors' websites, where it is listed as under review]

Bennett, Jonathan. 2003. A Philosophical Guide to Conditionals. Clarendon Press, Oxford.

Grice, Herbert Paul. 1975. ‘Logic and Conversation’, in The Logic of Grammar, D. Davidson and G. Harman (eds.), Encino, California, Dickenson, pp. 64-75. Reprinted in Grice (1989).

Jackson, Frank. 1979. 'On assertion and indicative conditionals.' in The Philosophical Review 88, 565-589.

King, Peter. 2004. "Peter Abelard" in The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2008/entries/abelard/>.

Lewis, David. 1976. 'Probabilities of conditionals and conditional probabilities.' in Philosophical Review, 85(3):297–315. Reprinted with Postscript in Philosophical

Papers, Volume II, pp. 133-152.

Mackie, J.L. 1962. ‘Counterfactuals and causal laws’, in R.J. Butler, (ed.), Analytical Philosophy, 1st series, Blackwell.

Thomason, Richmond H. 1970. 'A Fitch-style formulation of conditional logic' in Logique et Analyse, 52:397–412.

1 cf. Grice (1975), Lewis (1976).

2 cf. Jackson (1979).

3 According to King (2004), Abelard rejected something like conditional proof. More recently, cf. Thomason (1970), Bennett (2003), and Beall and Murzi (draft).

4 cf. Mackie (1962).

Welcome!

Here, until further notice, I will occasionally post short and often critical articles on various topics in philosophy, mainly the philosophies of logic and language. For now (at least) these will be written by me, but anyone who wishes to contribute can email an article to tristanhaze at <the domain-name of Google's email service> dot com, and I will consider it. (Needless to say, this is not a peer-reviewed journal.)

My name is Tristan Grøtvedt Haze, and I am a PhD student at The University of Sydney.