A Fallacy in Hofweber's Arguments in Ontology

[This is a draft of a paper.]

Hofweber's ontological project crucially involves inferring negative existential statements from statements of non-reference, i.e. statements that say that some term or terms do not refer. Here, after explaining the context of this move, I want to show that it is fallacious, and that this vitiates Hofweber's ontological project.

Thomas Hofweber has for several years been developing a distinctive approach to ontological and metaontological questions.

One of his starting points is the way some ontological questions in philosophy can apparently be settled with trivial arguments - for example, since mathematics has established that there are infinitely many prime numbers, it follows that there are numbers, and so there is no room for a special philosophical discipline of ontology (if it is to respect mathematics) to deal with this as a substantial question, the way ontologists of mathematics seem to try to do. Call this the puzzle about ontology.

Hofweber attempts to solve the puzzle about ontology by independently motivating a distinction between two different readings of quantifiers, or two sorts of quantification: internal and external. Internal quantificational statements, unlike external ones, do not work by placing conditions on a domain of objects. (To see that we might need something like this, consider the quantifier in 'Santa Claus doesn't exist, therefore there is something that doesn't exist'.) He then argues that the trivial arguments go through, but only when the quantifiers are given an internal reading. Give the quantifiers an external reading, and it is not clear that their premises have been established - in the case of 'There are infinitely many prime numbers', for instance, it might be that mathematics has established this on its internal reading, but not on its external reading.

Hofweber doesn't just want to solve the puzzle about ontology with his internal/external distinction, however. He also wants to use it to establish answers to certain (external) ontological questions - negative answers. This is what I call 'Hofweber's ontological project'.

Taking the number case, the project goes roughly like this. Hofweber argues that, if we can establish internalism about number-talk, including arithmetic (i.e. if we can establish that the quantifiers involved in number-talk, including arithmetic, are internal ones which do not place conditions on a domain of objects), we can show that the external question of whether numbers exist is left open by this talk, and is thus free for the taking by ontology.

Next, Hofweber argues that numerals, number words like 'four', and terms like 'the number 2' are not referring terms. I.e., that they are not in the business of referring to things. They sometimes assume the superficial grammatical position of referring terms for sophisticated linguistic reasons involving the notion of a 'focus construction' (and other considerations, depending on the kind of occurrence).

Then, on the basis that number terms don't refer, Hofweber concludes (via a principle designed to enable one to infer non-existence of things from statements of non-reference) that numbers don't exist, i.e. that there are no numbers (the quantifier here being intended externally), thus answering one of ontology's fundamental questions. I will good-naturedly call this last step 'the Howler'.

A couple of years ago, I inconclusively argued that Hofweber's distinction between internal and external quantification is ill-motivated. Here, I want to grant that distinction, and even grant that it enables Hofweber to explain the validity of the trivial arguments.

I want to make it clear that the Howler is a fallacious move, and that this vitiates Hofweber's project for answering certain ontological questions (e.g. about numbers, properties and propositions) in the negative. I will not be concerned here with whether Hofweber succeeds in establishing internalism about number-talk - my point is only that his argument from internalism to negative answers in ontology contains a fallacy.

The Howler appears in Hofweber's contribution to the influential 2009 anthology, Metametaphysics: New Essays on the Foundations of Ontology. The contribution is called 'Ambitious, Yet Modest, Metaphysics'. (I include other relevant papers in the bibliography, to help readers piece together a more detailed view of Hofweber's overall project, but he gives a good sense of it in the paper just mentioned.)

I think my criticism will be most effective if I quote the Howler along with the argument in which it appears, rather than reconstructing it and insisting that that is what Hofweber was doing. Here is the argument:

Let’s briefly reflect on what seems to be a central thesis about reference or denotation: 

(REF) If Fred exists then ‘Fred’ refers to Fred. 

Of course, I am assuming that ‘Fred’ is unambiguous, or at least used in the same way throughout. (REF) is uncontroversial, I take it, and probably a conceptual truth. Note that it implies the following: 

(REF∗) If ‘Fred’ doesn’t refer to Fred then Fred doesn’t exist. 

There are two ways for an expression not to refer. One is to aim to refer, but not to succeed. A classic case of this are empty names. Although the details of any example one might try to give of this are controversial, let’s nonetheless take ‘Sherlock’ to be an empty name of this kind. That is, suppose Sherlock is a name and thus has the semantic function of picking out an object. But it fails in carrying out that function. It thus doesn’t succeed in referring, and thus doesn’t refer. Thus Sherlock does not exist. Nothing in the world is Sherlock, no matter what in general the world contains. There could be all kinds of people, with all kinds of professions, but no matter how general properties are instantiated in the world, nothing in it is Sherlock. And nothing could be. If ‘Sherlock’ does not refer then Sherlock does not exist. This is all fairly trivial, but I go over it to make it vivid for our next case. 

Names aim to refer, but they can fail to succeed in what they aim for. The second way in which an expression might not refer is when it does not even aim to refer. Non-referential expressions, like ‘very’, don’t refer since they don’t even aim to refer. If internalism is correct about talk about numbers, properties, and propositions, then the relevant singular terms are non-referential. They do not aim to refer, and thus they do not refer. According to the above version of internalism ‘two’ is just like ‘most’. But since it doesn’t refer we know that there is no such thing as the number two. Since ‘two’ and ‘the number two’ are non-referring expressions nothing out there is (or can be) the number two. There can be all kinds of objects, abstract or concrete, they can have all kinds of properties and relations to each other. Nonetheless, none of them is (or can be) the number two. Or any of the other numbers. Internalism thus answers the ontological question.

Note first that Hofweber says that '[a]ccording to the above version of internalism "two" is just like "most"'. But what do we get if we substitute 'most' for 'Fred' in Hofweber's (REF*) principle?:

(REF*-Most) If 'most' doesn't refer to most then most doesn't exist.

But this seems like ungrammatical nonsense. Furthermore, it doesn't seem that 'Does most exist?' or 'Is there such a thing as most?' are substantial, sensible questions. It may be argued that 'Does most exist? No.' is not complete gibberish, if it is construed as a kind of metalinguistic point - it's not true to say 'Most exists'. This question-and-answer does not appear to be about whether the domain of our external quantifiers meets certain genuine conditions (and not simply metalinguistic conditions such as 'being referred to by the word "most"').

So if 'two' really is just like 'most' in all relevant respects, Hofweber has a problem. There is, of course, an important difference. Consider:

(REF*-Two) If 'the number two' doesn't refer to the number two then the number two doesn't exist.

Unlike (REF*-Most), (REF*-Two) is superficially grammatical. It even appears not to be nonsense (if we consider it independently of Hofweber's views). Do either of these two differences help?

Superficial grammaticality doesn't help; consider the nonsensical but superficially grammatical question 'Is there a rock of eggs?'. This doesn't seem to turn on whether a domain of objects meets some genuine condition, and 'Is there a rock of eggs? No.', like the 'most' case, seems to be a metalinguistic point at best.

The appearance of sense doesn't help either, for Hofweber has no way of explaining it except in terms of internalism; he explains occurrences of number-expressions always in terms of their being non-referring terms that appear in the syntactic guise of referring terms (for sophisticated linguistic reasons). And it is not at all clear what these could possibly be doing in an external quantificational context.

In general, the point might be captured by the following principle: non-existence of something only follows by semantic descent from non-reference when the non-referring term plays the semantic role of referring. Otherwise you can't semantically descend to a well-formed, sensical proposition.

(This is a necessary condition for the non-existence of something following by semantic descent from non-reference, but it may not be sufficient. I say 'follows by semantic descent' rather than simply 'follows' because '"X" does not refer' may be argued to always imply 'The referent of "X" does not exist' - but there is no semantic descent there, as there is in Hofweber's arguments.)

This seems like the natural view, in lieu of some special story, and Hofweber hasn't given any such story.

In a forthcoming post, I will argue that Hofweber's ontological project is impossible, for a different (though related) reason: internalism at the strength he requires it is inconsistent with the thesis that there is a substantial ontological question about numbers left open by arithmetic and other non-metaphysical number-talk, since such a question would constitute a counterexample to internalism. This mistake is, I think, more profound than the one exposed here - making it involves a kind of sawing-off of the branch one is sitting on.

Bibliography

Main reference: 

Hofweber, T. 2009. 'Ambitious, yet modest, metaphysics', in David John Chalmers, David Manley & Ryan Wasserman (eds.), Metametaphysics: New Essays on the Foundations of Ontology. Oxford University Press.

Background reading for Hofweber's project:

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

A Problem for the Simple Theory of Counterfactuals

In a recent blog post called 'The Simple Theory of Counterfactuals', Terrance Tomkow argues extensively for a theory of counterfactual conditionals along broadly Lewisian lines, explicitly restricted to counterfactuals with nomologically possible antecedents. The theory, Tomkow says, was first proposed by Jonathan Bennett in 1984, but later abandoned. Lewis held a more complicated theory.

Tomkow argues successfully, in my opinion, against Bennett's reasons (given in his Philosophical Guide to Conditionals) for rejecting his own theory. (Tomkow tells me, in a private communication, that Bennett has agreed with these arguments of Tomkow's, also in a private communication.) There is much else of value in the post as well. However, I cannot agree with Tomkow that the theory as he states it, even with its restriction, is correct.

The Simple Theory, or the Bennett-Tomkow Theory, is this:

THE SIMPLE THEORY
A > C iff  C is true at the legal A-worlds that most resemble @ at TA.

('A > C' is a shematization of 'counterfactual statements of the form: If ANTECEDENT had been the case then CONSEQUENT would have been the case.'

'@' denotes the actual world. 'Tp' denotes the time that the proposition 'p' is about. 'Legal' worlds are nomologically possible worlds.

The restriction of the this theory is then given as follows: 'To keep things simple, we will only deal with cases where A is false at @ but nomologically possible.')

Now, before giving the objection which is the main point of the present post, I want to note a simpler but less powerful objection. Some counterfactuals with nomologically possible antecedents are categorical - that is, require that all A-worlds are C-worlds. For example 'If I had met a bachelor this morning, I would have met an unmarried man this morning', in the context of a language-lesson. I argue for this here. The Simple Theory seems to assign the wrong meaning here, since it says that such a counterfactual is true iff C is true at the legal A-worlds that most resemble @ at TA, and these won't be all A-worlds, as intuitively required by the counterfactual. This objection is less powerful than the one I am about to give, because it can be easily avoided by simply restricting the theory to non-categorical counterfactuals.

Now the more powerful objection. This is inspired by my cartoon understanding of the confirmation of relativity, but let's just treat it as a fiction. Einstein asserted a law in paper N which actually holds, and which, together with the facts of some experimental setup E, predicts that some light will bend.

Now, it seems to me we can evaluate counterfactuals where the relevant closest A-worlds are worlds where the law doesn't hold, for example ones with the antecedent '~L' (where L is the law in question). Tomkow seems to agree, saying in a comment that 'we do need an account of counterfactuals with contra-legal anteced[e]nts'. So far, no problem for the Simple Theory.

My idea is that there are counterfactuals whose antecedents are legal, but where the similarity relation is contextually understood in such a way that the closest relevant A-worlds are counter-legal. So, with the following counterfactual:

(H) If Einstein had been wrong in paper N, this light would not have bent.

both what Einstein wrote and the experimental setup may be held fixed during evaluation (i.e. match in these respects required for close similarity), while the actual laws of nature are not held fixed. The antecedent itself is legal, however, since there are legal worlds where Einstein is wrong in paper N, but where he writes something else.

I will now try to make this more precise, and spell the objection out.

For a given counterfactual and contextual understanding of it, call the 'focus set' the set of A-worlds at which C is required, by the counterfactual, to be true. (This of course assumes that a theory with broadly Lewisian/strict-implication outlines is basically right.)

The special property (H) was designed to have is thus: having a legal antecedent, yet being legitimately and naturally understandable such that its focus set contains counter-legal worlds.

If there are counterfactuals with that property, that's a problem for the Simple Theory as stated, since it says that 'A > C iff C is true at the legal [my emphasis] A-worlds that most resemble @ at TA'.

Their having legal antecedents puts them in the scope of the Simple Theory as stated, but the presence of counter-legal worlds in their focus sets (on the relevant understandings of them) conflicts with it.

The Or-to-If Argument

This is a sequel or appendix to 'The Truth-Functional Account of Indicative Conditionals'.

Before we look at this argument, it needs to be made clear exactly how, if valid, it would support the view that '⊃' can be taken as an abbreviation of 'If...then'. Canonically, the argument proceeds from a truth-functional disjunction of the form '~p ∨ q' to a conditional of the form 'If p then q'. And the truth-function associated with 'p ⊃ q' in the propositional calculus is equivalent (or identical, extensionally speaking) with the truth-function associated with '~p ∨ q'. Secondly, it is widely accepted that 'If p then q' implies '~p ∨ q'. (This has been questioned on subtle grammatical/syntactic grounds, but we will not discuss that here.) Thus it seems that if the Or-to-If Argument is valid,'⊃'-statements can be taken as logically equivalent to indicative conditionals.

Historical preliminary
 
The origin of the idea that one can infer a conditional from a disjunction appears to be unknown. There has been speculation that it originated with Stalnaker. Priest (2001, p. 17) says the Or-to-If Argument was 'given by' Faris (1968) - and it was, but not for the first time. While one of those authors might have made the first use of the inference form as an explicit argument for a truth-functional reading of 'if' after the issue had become controversial in our era, the form itself has a long and venerable history. We find it on p.64 of Cohen and Nagel (1934):
 

Equivalence of Compound Propositions
 
. . .
 

Consider next the alternative proposition Either a triangle is not isosceles or its base angles are equal. To assert it means to assert that at least one of the alternants is true. If, therefore, one of the alternants were false, the other would have to be true. Hence we may infer from the alternative above the following hypothetical If a triangle is isosceles, its base angles are equal.

This textbook, which was popular in its day, also contains quite extensive discussion of the relation between 'formal' and 'material' implication - including a resolution of the 'paradox' attending to the latter (there is no paradox, since the term 'implication' is just given a special technical use in the propositional caclulus). Curiously, this 'paradox' is not related to hypotheticals (conditionals). In fact, hypotheticals are not discussed in the chapter on 'the calculus of propositions' at all, but in two more old-fashioned chapters near the beginning called 'The Analysis of Propositions' and 'Relations between Propositions'. (No doubt this has partly to do with the dominance of the denotational approach to the propositional calculus at that time.) It is in the latter that the Or-to-If Argument and its conclusion appear as a bland lesson.

Even C.I. Lewis, who famously raised the 'paradoxes of material implication' in his 1918 Survey of Symbolic Logic (and articles written earlier), had no problem with '⊃' being read as 'If...then'. He appeared to regard the latter as ambiguous between an "extensional" and an "intensional" reading. A curious passage on p.225 reads [and bear in mind that Lewis was using the notation of the algebraic tradition]:

we can now prove that we have a right to interchange the joint assertion of p and q with p × q, "If p, then q", with p ⊂ q, etc. We can demonstrate that if p and q are members of the class K, then p ⊂ q is member of K, and that "If p, then q", is equivalent to p ⊂ q. And we can demonstrate that this is true not merely as a matter of interpretation but by the necessary laws of the system itself. We can thus prove that writing the logical relations involved in the theorems—"Either ... or ...," "Both ... and ...," "If ... , then ..."—in terms of +,×,⊂, etc., is a valid procedure.

In this case, the "proof" does not proceed from Or to If , but by the previously "established" theorem '(1 a) is equivalent to (a = 1)', together with the rather Tarskian postulate 'For any proposition p, p = (p = 1)', and a tacit use of something like Conditional Proof (which, we shall see, is crucial in the Or-to-If Argument). Today this reasoning would be regarded as metalinguistic, not 'by the necessary laws of the system itself'.

Earlier, we find the Or-to-If Argument given in support of the very first definition in Principia Mathematica, 'Definition of Implication':
 

*1 01. p ⊃ q . = . ~ p ∨ q Df.

. . .
 

According to the above definition, when 'p ⊃ q' holds, then either p is false or q is true; hence if p is true, q must be true. Thus the above definition preserves the essential characteristic of implication . . .

This was then taken to be authoritative in Hankin (1924), a widely-cited legal article on 'Alternative and Hypothetical Pleadings', with the groan-inducing remark:

"If A, then B" is equivalent to the statement "either A is false or B is true". To persons not engaged in the study of logic this may at first appear absurd; yet it can be proved.

In Boole (1847) p.54, the supposed equivalence - except with the negation in the conditional instead of the disjunction - is baldly stated:

To express the conditional Proposition, If X be true, Y is not true. The equation is obviously

                                                              xy=0, (37);

this is equivalent to (33), and in fact the disjunctive Proposition, Either X is not true, or Y is not true, and the conditional Proposition, If X is true, Y is not true, are equivalent.

Earlier still, according to Ashworth (1968), 'The Spanish scholastic, Petrus Fonseca ... [wrote] that the name 'hypothetical' most properly applies to conditional propositions, but can also be used of disjunctions, because they imply a conditional.' Ashworth tells us that Abelard discussed the point in his Dialectica.

It is known that Abelard learnt about the theory of hypothetical syllogisms from Boethius, whose De Hypotheticis Syllogismis, written during the years 516–22, contains what seems to be a related but distinct idea:

[1.3.2] Fiunt uero propositiones hypotheticae etiam per disiunctionem ita:

Aut hoc aut illud est.

Nec eadem uideri debet haec propositio quae superior, quae sic enuntiatur:

Si hoc est, illud non est.

haec enim non est per disiunctionem sed per negationem.

This may be translated as:

[1.3.2] But propositions become hypothetical also through disjunction, thus:

Either this is, or that is.

Neither should the proposition pronounced as follows:

If this is, then that is not.
 
seem the same as the one above. For this one is not through disjunction but through negation.

(Thanks to P.V. Spade for this translation.) Boethius intends exclusive disjunction. To help corroborate the suggestion that this can be seen as a precursor to the Or-to-If Argument: Lagerlund (2010), discusing Boethius's work on hypothetical syllogisms, makes the following suggestion (without specific reference to the text):

Boethius also treats ‘P or Q’ as hypothetical, apparently because he thinks that disjunction can be translated in terms of a conditional sentence

Criticism of the argument
 
Here is the Or-to-If argument in schematic form:
 
1. ~A ∨ B. (Premise)
2. A. (Hyp)
3. B. (1, 2, Disj. Elim)
4. If A then B. (2 - 3, Cond. Proof)
 
Consider the following instance:

1. ~grass is green ∨ grass isn't green. (Premise) 
2. Grass is green. (Hyp)
3. Grass isn't green. (1, 2, Disj. Elim.)
4. If grass is green then grass isn't green. (2 - 3, Cond. Proof)
 
I think there is something wrong with this argument, and I suspect most unindoctrinated people who comprehend it would agree. If a demon somehow convinced me of the truth of '(~grass is green ∨ grass isn't green)', and if I were rational, I would conclude that grass isn't green. In that situation, it would not appear rational (valid, truth-preserving) to conclude further that if grass is green, then grass isn't green. Of course, a defender of '⊃' as 'if' will argue that I have been deceived by appearances on this point. I have tried to undermine the motivation for this in the post on the truth-functional account of indicative conditionals. However, the question remains: what should we say is wrong with the argument?
 
The fallacy occurs, I think, in the step of discharging the hypothesis and deriving a conditional. That is, in the application of the rule of Conditional Proof (roughly speaking, the natural language analogue of the Deduction Theorem for the propositional calculus - I say 'roughly' because DT is strictly a metatheorem, not a proof-rule). Notice that, together with (2) (whose scope it appears in), (3) is an absurdity; it can't be that grass is and isn't green. Accordingly, I propose that CP becomes unavailable once an absurdity has been derived within the scope of the supposition. (Here I count as an 'absurdity' anything which, when conjoined with the supposition, yields an absurdity in an ordinary sense.) That CP is unavailable in such circumstances should not be surprising; if it were not so, all sound reductio arguments could be used to establish bizarre conditionals.

(This constraint is arguably insufficient to make Conditional Proof valid. There will remain the problem of Strengthening the Antecedent, and perhaps others. For a more thorough treatment of this matter, see Thomason (1970) (thanks to Adrian Heathcote for the reference). According to King (2004), Abelard rejected something like Conditional Proof. Given his interest in the semantics of conditionals, it is conceivable that his reasons were closely related to ours.)

Essentially the same point can be seen from another side, if we change the premise to something we actually believe, such as: ~grass is blue ∨ grass isn't blue. Coming to step (2), in this case the hypothesis that grass is blue, if we really want to assume this hypothesis for the sake of argument, then we can hardly use the above disjunction in the ensuing reasoning, unless we are trying for a simple reductio of the proposition that grass is blue. And that would be epistemically queer, since it is hard to see how we could rationally be more sure of the disjunction than the "conclusion" that grass is not blue.
 
What I think all this shows is that the Or-to-If Argument form is not generlly valid, as it would have to be if '⊃' could be read as 'If...then'. Therefore '⊃' cannot be read as 'If...then'. There is, of course, much more to say, in particular concerning the wide range of cases in which one seemingly can argue from Or to If; it seems that while '⊃'-sentences aren't conditionals, assurance of the truth of a '⊃ '-sentence can in many cases serve as a basis for a conditional. The common talk about ordinary conditionals differing from '⊃'-sentences in asserting some kind of natural "connection" between antecedent and consequent is, for this reason, highly suspect.

For a differently orientated discussion of the Or-to-If Argument which culminates in the same verdict - that it is not valid - see Bennett (2003).

References

Ashworth, E.J. 1968. 'Propositional logic in the sixteenth and early seventeenth centuries' in Notre Dame Journal of Formal Logic, Vol. 9, No. 2, 179-192.

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

Boethius, Anicius Manlius Severinus. 516–22. De Hypotheticis Syllogismis.
Original Latin available at Peter King's website:
<http://individual.utoronto.ca/pking/resources.html>
Published in Italian:
(Istituto di Filosoofia dell'Università di Parma, Logicalia 1).
ed. Obertello, L. Brescia, 1968.
 
Boole, George. 1847. The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning, Cambridge: MacMillan, Barclay and MacMillan. London: George Bell.

Cohen, Morris R. and Nagel, Ernest. 1934. An Introduction to Logic and Scientific Method, London: Routledge & Kegan Paul Ltd.

Faris, J.A. 1969. 'Interderivability of "⊃" and "if"' in Logic and Philosophy: Selected Readings, ch. 7, Iseminger, G., ed., Appleton-Century-Crofts, New York.

Hankin, Gregory. 1924. 'Alternative and Hypothetical Pleadings' in The Yale Law Journal, Vol. 33, No. 4 (Feb., 1924), pp. 365-382.

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

Lagerlund, Henrik, "Medieval Theories of the Syllogism", The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2012/entries/medieval-syllogism/>.

Lewis, Clarence Irving. 1918. A Survey of Symbolic Logic, University of California Press, Berkeley.

Priest, Graham. 2001. An Introduction to Non-Classical Logic, Cambridge University Press.

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

Whitehead, Alfred North and Russell, Bertrand. 1910. Principia Mathematica, Vol. 1.
Cambridge: Cambridge University Press. Second edition 1925.

Blackburn's Interpretation of Wittgenstein as a Proto Quasi Realist

In his 1993 'Review of Paul Johnston's Wittgenstein: Rethinking the Inner' (published in Ethics vol. 103 that year, pp. 588 - 590), Simon Blackburn tries to demonstrate that Wittgenstein has shown us the way to a fruitful theoretical perspective, while not going all the way himself. The view in question is Blackburn's quasi realism. According to the Oxford Dictionary of Philosophy (which Blackburn wrote!), quasi realism is the view that projectivism or expressivism about ethics can make legitimate sense of the 'realist-sounding' aspects of ethical discourse. This contrasts with error theory, on which these aspects are held to reflect a false 'realist metaphysics'. (End of quotation.) Though quasi realism is by default regarded as applying to ethical discourse, analogous claims can be made about other kinds of discourse with 'realist-sounding' aspects. Some 'realist-sounding' discourse - for example discourse describing the present location of observable objects - is fully realistic, and can be taken at "face value" - that is, as describing reality. Terms like 'truth', 'fact', 'correspondence' and 'description' primarily apply to this kind of discourse, but also find legitimate uses elsewhere, e.g. ethical discourse, which are less transparent but can be explained by the quasi realist. That is the view; problems may have suggested themselves to you already, but this is not the place to criticize quasi realism directly.

Blackburn begins by noting that Wittgenstein 'is constantly suggesting that underneath the superficial similarity of linguistic form there is deep difference of function'. He gives several examples: philosophical statements are treated as 'rules of grammar', mathematical statements 'do not have the use of statements but of rules', apparent self-descriptions are 'forms of self-expression', ethical, aesthetic and theological assertions are 'not what they appear', and more. Blackburn is very approving of all this, but then comes the question:

So can we continue to talk of truth, fact, knowledge and the rest in these nondescriptive areas without blushing? It seems a good question, and I do not think Wittgenstein ever confronted it squarely. His answer is going to be that we can, but it is not at all plain how he gets to it, for the difference of activity he harps on is introduced precisely by contrast with describing and representing how things are, and those are the activities that most obviously must conform to norms of truth and fact. Wittgenstein seems to leave unfinished business ... taken up by the character I call the quasi realist, with whom he should therefore be allied.

Against this, I think it can be demonstrated that while Wittgenstein may be an inspiration for quasi realism, he cannot truly be regarded as its ally. To begin with, is Blackburn right in saying that the question above was never squarely confronted by Wittgenstein? There doesn't appear to be any extended philosophical treatment of this question in his corpus, but it would be too quick to conclude from this that Wittgenstein has left unfinished business here, something he might have got around to. Let us try to get clearer about the question of whether we can 'continue to talk of truth, fact, knowledge and the rest in these nondescriptive areas without blushing', in order to see where Wittgenstein might stand in relation to it.

Blackburn is expressing himself figuratively; actual blushing is not in question. I didn't even take note of that when I read it, which is noteworthy in itself; one feels here that Blackburn is making things rhetorically easier for himself by expressing his question this way. Since he does after all have a definite agenda, namely to ally Wittgenstein with quasi realism, I think we are justified in asking: what did Blackburn really mean by this figure?

A simple answer would be that Blackburn means to ask whether can we truly speak of truth, fact and knowledge in what he calls 'nondescriptive areas'. This will not do, however, since it is possible to express truths in a way which is misleading and confused, and perhaps this could be cause enough for 'blushing' in Blackburn's sense. Hence it cannot be merely truth and bare meaningfulness which is in question, but also the appropriateness of certain forms of words in certain uses. And what kind(s) of uses are relevant here? There are very diverse cases of talk about truth, fact and knowledge in what Blackburn calls 'nondescriptive' areas. As a rough heuristic, we may divide this talk into two categories: everyday and philosophical (or, to put a different slant on the matter, practical and idle). Intermediate and other cases are no doubt common too, but let us consider some clear examples of practical uses and philosophical uses.

Practical:

- "There are four primes between 10 and 20." 'That's not true! ..oh, wait, yes it is.'

- "I have very little knowledge in topology."

- "I know it's a bad screenplay, in fact it's terrible, but the performances were somehow wonderful nonetheless."

- "At that point, he knew he had done the right thing."

- "...and that was when I first felt the pain. As a matter of fact, it's come back; I'm going to lie down."

- "She is basically a decent person." "I just can't believe that. If that were true, she wouldn't have..."

Philosophical:

- "The knowledge we possess about the realm of natural numbers is eternally valid, and more certain than any empirical knowledge."

- "There are ethical facts."

- "I know with certainty that I am conscious, but I can only hypothesize that others are."

- "There are many facts about the properties of my sense-data which I cannot express, for lack of a proper phenomenal language."

There is reason to think that Blackburn's question largely concerned with the practical uses. Firstly, quasi realism is inconsistent with full-blown Realist philosophical claims about the 'nondescriptive' areas, which can after all be made in terms of fact, truth and knowledge. Blackburn would not want to say we're entitled to make them. Secondly, Wittgenstein himself would probably take such utterances as symptomatic of philosophical confusion. This may not mean that all philosophers who say such things should in any sense blush, or be ashamed of themselves, but such assertions do seem like the sort of thing Wittgenstein would want to investigate critically. Therefore we shall focus on Blackburn's question as it applies to the practical uses.

On this understanding, it does seem Blackburn is right in saying that Wittgenstein's answer will be that we can speak in this way without blushing. But what of the next, quasi-realism-motivating claim, that it is 'not at all plain' how Wittgenstein 'gets to' this answer?

The trick is to see that Wittgenstein doesn't get to it at all. The propriety of such talk is not something Wittgenstein establishes with philosophical considerations; it is his starting point. Wittgenstein's acceptance of our ordinary employment of language does not derive from philosophy but from life. In his philosophy, it is the given. As evidence for this, I offer the following remarks from the Philosophical Investigations:

§124:

Philosophy may in no way interfere with the actual use of language; it can in the end only describe it.

For it cannot give it any foundation either.

(...)

§126:

Philosophy simply puts everything before us, and neither explains nor deduces anything. (...)

§132:

(...) we shall constantly be giving prominence to distinctions which our ordinary forms of language make us easily overlook. This may make it look as if we saw it as our task to reform language.

Such a reform for particular purposes, an improvement in our terminology designed to precent misunderstandings in practise, is perfectly possible. But these are not the cases we have to do with. The confusions which occupy us arise when language is like an engine idling, not when it is doing work.

Blackburn pre-empts this exactly, by speaking of 'cluster of interpretations' of Wittgenstein on which there is no unfinished business. As Blackburn says, on this view:

The true Wittgensteinian reaction is just to find more motley. Talk of truth, knowledge, certainty is itself a patchwork. For these notions do not just arise in connection with descriptions and representations but with rules, ejaculations, and so on as well. Provided we have a correct "übersicht" of what we are doing, nothing needs explanation, nothing is hidden.

I think this is right, and that it is the truly Wittgensteinian reaction. (For the sake of simplicity, I will anticipate my conclusion by referring to it as such.) Blackburn's objection to this view gets us to the heart of the problem. I quote it in full, so that we may see in detail how it fails:

It is true to much in Wittgenstein, yet its problem is obvious: it denies Wittgenstein any words to say what he wanted about the differences that the position starts by celebrating. Maybe "description" and "representation" are a patchwork, or what might be called mottled themselves ("this is what we call describing ethical facts"). Drunk on clusters, we evade the problems that torment the quasi realist by reaching once more for the mottle. Of course ethical (mathematical logical, philosophical, psychological) statements are true, describe the facts, can be known, say how it is. No contrasts there! But they do not do so in the way that empirical statements do. Don't they? Find an interesting way in which they allegedly contrast and watch me mottle it!

The first claim, that the truly Wittgensteinian reaction denies Wittgenstein 'any words to say what he wanted' about the relevant differences, is simply false: all it says is that the propriety of 'realist-sounding' talk in (what Blackburn calls) 'nondescriptive' areas is part of the given, and not to be interfered with, and that such talk is variously used. This might deny Wittgenstein certain words to say what he wanted about the relevant differences, but he doesn't need them anyway. In fact, they can get in the way. There are many ways of talking about, and showing, relevant differences. Some basic examples:

- In Wittgenstein's simple language-games involving 'slab', 'there', etc., it is unnatural to regard the counting-words 'a', 'b', 'c', etc. as 'names of objects', and even if one does call them this, their completely different function is nonetheless manifest.

- It makes sense to say 'I doubt whether he is in pain', but not 'I doubt whether I am in pain'.

- Our method of verifying the proposition '25 x 25 = 625' is different in kind from our method of verifying the proposition 'It is raining'.

- It makes sense to talk about the destruction or disappearance of chairs, but not numbers.

This shows that Blackburn's first claim is wrong: there is plenty Wittgenstein can say about differences of function in language without employing the 'realist-sounding' words to do the distinguishing. The next claim which needs to be put right is: 'Drunk on clusters, we evade the problems that torment the quasi realist by reaching once more for the mottle.'

It is hard to know how to object to such a sentence. For a start, I think the quasi realist ought to be tormented by certain problems so long as they remain a quasi realist, since their view is not tenable. That the truly Wittgensteinian reaction avoids ('evades') these problems is a strength, not a weakness. However, it neither avoids nor evades the problem, shared with the quasi realist, of clarifying important differences in the working of language. Wittgenstein took that task very seriously. We 'reach once more for the mottle' only to show that Blackburn's particular approach, quasi realism, won't work. This is not to say: 'Find an interesting way in which ['nondescriptive' and 'empirical' statements] allegedly contrast and watch me mottle it!'. Here, Blackburn gives the impression that the quasi realist has proposed contrast after contrast, only to have each one mottled by the true Wittgensteinian. But not at all! He has only proposed one - that between quasi realistic and really realistic uses of language.

The examples given earlier, on the other hand, show some genuine contrasts which no Wittgensteinian would want to mottle. Unlike Blackburn's proposal, they are not distinctions which require extensive theoretical elaboration in order to have a chance of getting off the ground. Perhaps Blackburn finds them insufficiently 'interesting' or general, but this is a prejudice which Wittgenstein did not share, and which we need not share.

The Non-Indexical Core of Presentism is Monism

In recent discussions about temporal ontology, there has been interest in the question of the prospects for a non-indexical, or externalist, formulation of presentism. Some say there are no prospects. Thus Hinchliff (2000):

[A substantive distinction between presentism and eternalism] cannot be formulated in nonindexical terms. That is why I have formulated [presentism] with the aid of the indexical 'presently'.

Others have tried with elaborate means to formulate something which could reasonably be called a non-indexical version of presentism. Matthew Farr, for example, is currently exploring a strategy involving two temporal dimensions. (This note was inspired by a talk he gave at the University of Sydney on 25/3/13 called 'Supertemporal Ontology and the "Triviality" Problem'.)

The issue is obviously interesting, assuming that the question of the meaning of presentism is interesting. One special connection in which it is interesting is this: if there is no non-indexical core of presentism, then perhaps the metaphysical dispute between presentism and non-presentism can be dissolved, on the grounds that presentism is not a distinctive thesis about the structure of reality, but something else (a "view from inside", or something).

I am certainly sympathetic to the idea that the debate in temporal ontology (between presentism, eternalism, the growing block view etc.) is something which should be dissolved or transcended. I seriously doubt that this is a debate about some real subject matter, let alone that it is a debate about some real subject matter where one of the positions is right and the others wrong. I think the confusion here is deep, philosophically important, and deserves to be investigated carefully, not just dismissed. However, I do not think that the idea that the debate cannot be formulated non-indexically is a way to make progress on this, because I think that idea is wrong.

My suggestion here is that we can analyse presentism as the conjunction of two claims, one of which is non-indexical and incompatible with eternalism, growing blockism and shrinking blockism, the other of which is indexical but agreed to by all these parties.

(P) Only the present moment exists.

may be analysed as

(P-conj) Only one moment exists, and this moment exists (where 'this' indicates the present moment).

Call this the conjunction analysis. The non-indexical core of presentism, on this suggestion, is simply the first conjunct: monism about moments (or times, instants, timeslices or whatever).

This non-indexical core is, of course, compatible with strange propositions like:

(S) Only one moment exists, and it is the moment of Napoleon's birth.

But so what? No one believes that, and everyone (presentist, eternalist, growing blockist, shrinking blockist) believes that the present moment exists. Forget the label 'presentism' - look at the conjunction analysis, and it becomes clear that what is really distinctive about this view - i.e. what distinguishes it from other actual contenders - is its monism.

Reference

Hinchliff, M. 2000. 'A defense of presentism in a relativistic setting', Philosophy of Science 67, pp. S575-S586.

Philosophers' Carnival #150

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On the Truth-Functional Account of Indicative Conditionals

The "if/⊃"-question has an interesting history. It had evidently been considered (in essentials) by the Stoics, and by some mediaevals (Abelard especially). By the 19th century, many logicians endorsed the view that '⊃' (or whatever symbol was used) could be read as 'if...then'. This continued through the early years of the 20th century, but conscientious objectors came into view. This is socially and historically interesting, in that (as we shall see) the essential matter of the controversy had lain dormant in logic books for years beforehand, without being much discussed. It is as though logic had started to come to life again: gradually, more people were moved to think critically (but without complete dismissal) about what they read in logic books.

In the English-speaking world, MacColl was one of the earlier dissenters, though his criticisms were partly obscured by his own unpopular doctrines and procedures.

In 1908, in a short polemic against Russell, MacColl wrote: 'For nearly thirty years I have been vainly trying to convince [logicians] that this assumed invariable equivalence between a conditional (or an implication) and a disjunctive is an error'. (This is a reference to the Or-to-If Argument, which we will consider in a future post.) Russell's reply was made easy by the fact that MacColl had, in his objection, overlooked the former's distinction between propositions and propositional functions. After correcting this, Russell addressed the main issue swiftly, writing 'I say that p implies q if either p is false or q is true. This is not to be regarded as a proposition, but as a definition', and admitting happily that this definition does not give 'implies' its usual meaning. But this does not square well with the justification of the 'Definition of Implication' given in Principia.

More successful criticisms came later from Strawson. By the time of Quine's (1953) review of Strawson's Introduction to Logical Theory, the former was able to treat the semantic divergence between '⊃' and 'if...then' as rather old news:

The well-known failure of the ordinary statement operators 'or', 'if-then', 'and', and 'not' to confirm in all cases to the precepts of truth- functional logic is well expounded by Mr. Strawson. Because 'and' and 'not' deviate less radically than the others, I have found it pedagogically helpful (in Elementary Logic) to treat the translation of ordinary language into logical form, at the truth-functional level, as funnelled through 'and' and 'not'; and Mr. Strawson follows suit.

And later:

Mr. Strawson is good on '⊃' and 'if-then'. He rightly observes the divergences between the two, and stresses that 'p⊃q' is more accurately read as 'not (p and not q)' than 'if p then q'.

This state of affairs did not last. A series of post-1960 events has changed things irrevocably, so that Quine's comments above seem to come from a bygone era when things were much simpler. In my own view, the Quine-Strawson view was basically right, but one cannot make a respectable case for that today without discussing the post-1960 events. Therefore I shall now give a summary of the events, followed by a series of critical comments.

The resurgence of '⊃': a potted history

Phase 1: In his William James Lectures at Harvard in 1967, Grice makes public his theory of implicature and conversational maxims. People are impressed by this idea: 'John is poor but honest' has the same truth-conditions as 'John is poor and honest', but the former (in some contexts) strikes people as objectionable and unassertable, even when the latter may be both true and assertable, the difference being that the former can carry an implicature that poor people aren't honest. Secondly, the maxim of 'Assert the Stronger' is developed; if someone asks where John is, and I know he's at the library, it's not proper to respond that he is either in the library or at the pub. Similarly, Grice argues, sentences like 'if snow is green then I am king' are true (just because snow isn't green), but unassertable, since we should assert the stronger: that snow isn't green. (The work is published in Grice (1975).)

Phase 2: Meanwhile, other philosophers had been continuing to develop more sophisticated accounts of the truth-conditions of conditionals. Among these is the possible worlds account of Stalnaker (1968), who, following Adams (1965) (who himself wasn't interested in the question of truth-conditions), conjectured that the probability (in some sense) of a conditional 'If A then C' is the probability of 'C' given 'A'. That is: P(If A then C) = P(C/A) = P(C & A)/P(A) (where P(A) is positive).

Phase 3: David Lewis proves his triviality results in Lewis (1976), to the effect that 'there is no way to interpret a conditional connective so that, with sufficient generality, the probabilities [of truth] of conditionals will equal the appropriate conditional probabilities'. He considers the possibility of accommodating this with a theory on which conditionals do not have truth-values (i.e. are not truth-apt): 'Why not? We are surely free to institute a new sentence form, without truth conditions, to be used for making it known that certain of one's conditional subjective probabilities are close to 1. But then it should be no surprise if we turn out to have such a device already.' He writes: 'I have no conclusive objection to the hypothesis ... . I have an inconclusive objection, however: the hypothesis requires too much of a fresh start. ... [W]hat about compound sentences that have ... conditionals as constituents? We think we know how the truth conditions of compound sentences of various kinds are determined by the truth conditions of constituent sentences, but this knowledge would be useless if any of those subsentences lacked truth conditions.' This boosts Grice's proposal, which Lewis has come to endorse: 'It turns out that a quantitative hypothesis based on Grice's ideas gives us just what we want: the rule that assertability goes by conditional subjective probability.' And so the truth-conditions of indicative conditionals are identified with those of '⊃'-statements. And for sophisticated reasons.

(To complete the story, though this is less important for what follows: in a postscript to his (1973) in his Philosophical Papers, Volume II, Lewis admitted that in 'special cases', assertability and conditional probability diverge. Secondly, he abandoned the 'Assert the Stronger' explanation of apparent counterexamples to the '⊃'-analysis, due to apparent counterexamples to the 'Assert the Stronger' maxim itself, in favour of an ingenious alternative theory devised by Frank Jackson: one may assert 'if A then C', even when one is in a position to assert the stronger 'C', if one wants to give information which is robust with respect to 'A' (which could have low probability): information which, even if 'A' turned out true, would still hold. For more details on how this theory works, see Lewis's postscript and Jackson (1979).)

Thus the Grice-inspired Lewis-Jackson version of the '⊃' analysis is today regarded as a serious proposal, even if it is not widely accepted. Some other major accounts on the market deny truth-aptness, either completely (cf. Edgington 1991, 1995) or in certain cases, such as when the antecedent is false (cf. McDermott 1996). All these accounts have in common that they are error-theoretic with respect to many or most competent speakers: the '⊃' analysis implies that competent speakers often get a conditional's truth-value wrong, while accounts which partially or totally deny truth-aptness have it that competent speakers often mistakenly ascribe truth-values to sentences which have none.

Comments on the resurgence

Comment on Phase 1: Note a fundamental difference between the cases of 'but' and 'or' on the one hand, and the case of 'if' on the other: people do not generally judge it false to say that a poor and honest person is poor but honest, but rather wrong in some other sense. This is even more pronounced in the case of 'or'. In that case, we can see perfectly well that the misleading statement about John is true. By contrast, competent speakers will confidently classify a sentence like 'If grass is blue, it isn't blue' as not true. Thus it seems any view which says that for every '⊃' sentence, there is a corresponding 'if' sentence with the same truth-conditions, will inescapably be an error theory with respect to competent speakers.

Comment on Phase 2: The notion that assertability or probability of conditionals goes by conditional probability may seem initially appealing, but apparent counterexamples abound: sentences such as 'If 6 is greater than 5, then 7 is greater than 6' and 'If Gödel's proof really was valid, the sun will thankfully rise again' do not seem at all assertable or probable. They seem like bits of nonsense. Furthermore, the idea that assertability can be quantified, and that it equals any sort of probability, seems odd; if I attach a probability of only .5 to some proposition P, why would I assert it? Such a proposition seems not assertable at all in a normative sense - and therefore not 'half assertable' either, whatever that means. A common proposal in response to this is that assertability remains low until probability gets high, at which point it shoots up. This has been criticized by Dudman (1992), using lottery cases: someone who has a ticket in a lottery will usually not be prepared to assert that they won't win, even though they may realize that not winning is very highly probable indeed.

Comment on Phase 3: Lewis, wanting to maintain that assertability of conditionals goes by conditional probability ('A = CP' for short), uses his triviality results to argue in effect that, since we can't give any truth-conditional analysis of conditionals such that probability of truth will equal conditional probability, any truth-conditional account will (by A = CP) have to explain divergences between assertability and probability of truth, so why not at least start with something simple like the '⊃' analysis? The quite different course of denying truth-aptness remains open, but - says Lewis - that requires too much of a fresh start.

The first thing to note about this line of argument is that, for reasons given in the previous comment, A = CP is really not independently attractive, once you consider certain examples. So perhaps no 'divergences' need explaining at all, and philosophers can go on looking for a non-gappy truth-conditional account of conditionals which is more plausible than the '⊃' analysis.

The second thing to note is that the logical space between giving a truth-conditional analysis of conditionals and denying truth-aptness remains largely unexplored. Consider the case of subject-predicate statements about explanatorily basic things possessing explanatorily basic properties: this is a class of truth-apt statements for which no non-circular truth-conditional analysis can be given - what we might call an 'analytically basic' class of statements. A view on which conditionals are analytically basic - an antitheory of conditionals - can happily avoid the error-theoretic consequences of prevailing views, although it could be retorted that such a view is error-theoretic with respect to analytic philosophers. Surely the response to that is: when faced with a choice between a set of accounts which are error theoretic with respect to (almost) all competent speakers, and an error theory with respect to some philosophers, one of whom also believed in other universes inhabited by donkeys which speak, the latter should at least be examined properly. (This, of course, would go beyond the scope of the present inquiry.)

There is a different family of accounts, known as "support" theories, which are not strikingly error-theoretic. Such accounts are for the most part out of favour today, but a highly sophisticated one has been developed by my teacher Adrian Heathcote, in unpublished work. In my view, all such accounts - if they purport to be reductive - will face circularity problems. (A defence of this view is beyond our scope here.) However, even if they don't succeed as reductive analyses, the key ideas behind "support" theories seem important for understanding the logic and context-sensitivity of conditionals.

In a post coming soon, I will discuss the Or-to-If Argument. This is a simple, initially-compelling deductive argument-form which, if valid, would suggest that '⊃' can be read as 'if...then'.

Adams, Ernest W. 1965. 'The Logic of Conditionals', Inquiry 8, pp. 167-197. Adams, Ernest W. 1975. The Logic of Conditionals, Dordrecht, Reidel.

Dudman, V.H. 1992. ‘Probability and Assertion’, Analysis, 52:204-11.

Edgington, Dorothy. 1991. 'Do Conditionals Have Truth-Conditions?' in Jackson. ed. (1991, pp. 176-201).

Edgington, Dorothy. 1995. 'On Conditionals', Mind 104.414., (Apr. 1995), pp. 235-329.

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).

Grice, Herbert Paul. 1989. Studies in the Way of Words, Cambridge MA, Harvard University Press.

Jackson, Frank. 1979. 'On assertion and indicative conditionals.' in The Philosophical Review 88, 565-589. Reprinted in Jackson, ed. (1991, pp. 111-135). 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.

Lewis, David. 1986. Philosophical Papers, Volume II. Oxford University Press, Oxford.

McDermott, Michael. 1996. 'On the truth conditions of certain “If”-sentences' in The Philosophical Review, Vol. 105, No. 1 (Jan., 1996), pp. 1-37.

Quine, W.V.O. 1953. 'Mr. Strawson on Logical Theory' in Mind, New Series, Vol. 62, No.248 (Oct., 1953), pp. 433-451.

Russell, Bertrand. 1908. '"If" and "Imply", A Reply to Mr. MacColl' in Mind, New Series, Vol. 17, No. 66 (Apr., 1908).

Stalnaker, Robert: 'A Theory of Conditionals', Studies in Logical Theory: American philosophical quarterly monograph, Oxford, Blackwell 1968, pp. 98-112.

Whitehead, Alfred North and Russell, Bertrand. 1910. Principia Mathematica, Vol. 1. Cambridge: Cambridge University Press. Second edition 1925.