Friday, 29 April 2011

Breckenridge and Magidor on Arbitrary Reference: An apparent counterexample

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.

Tuesday, 19 April 2011

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.

Sunday, 17 April 2011

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).

Thursday, 7 April 2011

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

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).