Structured Modal Operators

The task

Propositions have modal characters and truth-values. For now, we will distinguish two modal characters and two truth-values: necessary character, contingent character, truth and falsity.

Necessary character is what necessarily true and necessarily false propositions have in common. Contingent character is what contingent truths and mere possibilities have in common.

In effect, the modal operator 'Necessarily' (box), ascribes necessary character and truth to a proposition. 'Contingently' ascribes contingent character and truth. 'Necessarily, it is not the case that' and 'It is impossible that' ascribe necessary character and falsity. 'It is merely possible that' ascribes contingent character and falsity.

But not all modal operators ascribe a particular character/truth-value pair. Some merely rule out certain combinations. For example, 'Possibly' merely rules out the combination of necessary character and falsity.

(NB that I am here talking about what are often called alethic modal operators, rather than modal operators in a more general formal setting in which these claims only hold for certain choices of accessibility relation.)

It is common to see the following list of four modal operators presented, sometimes as though it were exhaustive: possibility, necessity, contingency and impossibility.

But reflect again that, of these four modalities, possibility is an odd one out, since it is non-commital on truth-value. Also, note that systems have been developed where other operators, e.g. one for non-contingency, are taken as primitive.

This can give rise to an uneasy, lost feeling. Are the usual four modal operators just a hodge-podge? What modal operators are there (could there be)? Is there a systematic way of producing them all? And is there then a systematic way of determining logical relations between them?

In this post, I try to begin answering these questions.


The notation

The notation I want to introduce here can be said to stand to the box, the diamond and such symbols roughly as truth-tables stand to truth-functional connectives. (Or instead of truth-tables, Wittgenstein's ab-notation, Venn diagrams, or the shuttle diagrams pioneered by Martin Gardner and extended by Gregory Landini.)

We have said that 'Necessarily', 'Contingently', 'It is impossible that' and 'It is merely possible that' all ascribe a particular character/truth-value pair, or: they all rule out all but one character/truth-value pair.

We can represent operators as matrices containing four cells, one for each character/truth-value combination. We can then mark the fields representing pairs which are ruled out by the operator in question.  A blank canvas, not representing any modal operator,  looks like this:

(The box and diamond here represent modal characters.)

The four aforementioned operators then look like this:

We can also consider the class of operators which  rule out two character/truth-value pairs:

And finally the class of operators which rule out just one character/truth-value pair:

A syntactical test for implication

For any two modal operators A and B, Ap implies Bp iff all boxes crossed in B are crossed in A. (This could license a simple rule of cross-elimination.)

Then, rules could be given allowing detachment of the truth-operator, conversion of the falsity operator to negation, attachment of the truth-operator, possibility operator, etc.

Flipping and inversion

An operator can be negated by inverting its markings. Its operand can be negated in effect by flipping the operator's marking vertically. The dual of an operator can be obtained by inversion and flipping.

Relation to model theory 

For now, atoms are treated as, in effect, simply being assigned a truth-value and a modal character in the semantics, but this can be brought into connection with the standard Kripke semantics for modal logic. Given an S5 frame, for example, an atom's having necessary character (at a world, if you like) amounts to its truth-value being invariant across all worlds. Contingent character amounts to its not being so invariant.

To do

Among other things: study iteration of operators. Iteration will raise philosophical issues about the application of the formalism. These will turn, at least in part, on how propositions are individuated. Similarly, a case could be made for distinguishing a third character, impossible character, when propositions are individuated in a fine-grained way such that our proposition 'Hesperus isn't Phosphorus' is not the same as the Babylonian. (Our version has impossible character. Theirs, necessary.)

Against Quine's Argument in Sect. 31 of Word and Object

ADDED: Here is a followup post from 26 December 2015.
UPDATE (Nov 2019): I have recently published a paper on this topic, 'Quine's Poor Tom', in the European Journal of Analytic Philosophy.

Section 31 of Quine's Word and Object contains an arresting fallacious argument. In 1966, R.C. Sleigh Jr. published an objection to it. In 1977, David Widerker published an objection to Sleigh's objection. More recently, in 2007, Charles Sayward has published a paper where Sleigh's objection is further criticized. (References below.)

I will not engage directly with these three papers, but rather aim to give a clearer objection to Quine's argument.

Here is Sayward's apt description of the argument's point:

To a first approximation, the argument purports to show that if Tom has a certain minimal level of logical acuity—a level many of us possess—then if ‘belief’ has a sense in which it is a transparent operator, then Tom, if he in that sense of the word believes anything, he in that sense of the word believes everything. (Sayward 2007, p. 54.)

Quine assumes that Tom believes at least one true sentence and one false one. In fact, he assumes something much stronger: that Tom believes the true sentence 'Cicero denounced Catiline' and the false sentence 'Tully did not denounce Catiline'. (Cicero is Tully.) That these sentences are are (in a sense) contradictories, and that they are about the same object, is not essential for Quine's argument. These features of Tom were needed for earlier, separate arguments in chapter IV of Word and Object. 
Here is the argument:

Where ‘p’ represents a sentence, let us write ‘#p’ (following Kronecker) as short for the description: 

     the number x such that ((x = 1) and p) or ((x = 0) and not p). 

[In place of '#', Kronecker and Quine used a different symbol, which I can't easily reproduce here. - TH.] 

We may suppose that poor Tom, whatever his limitations regarding Latin literature and local philanthropies, is enough of a logician to believe a sentence of the form ‘#p = 1’ when and only when he believes the sentence represented by ‘p’. But then we can argue from the transparency of belief that he believes everything. For, by the hypothesis already before us, 

     (3) Tom believes that # (Cicero denounced Catiline) = 1.

But, whenever ‘p’ represents a true sentence, 

     # p = #(Cicero denounced Catiline). 

But then, by (3) and the transparency of belief,

     Tom believes that #p  = 1,

from which it follows, by the hypothesis about Tom’s logical acumen, that 

     (4) Tom believes that p. 

But ‘p’ represented any true sentence. Repeating the argument using the falsehood ‘Tully did not denounce Catiline’ instead of the truth ‘Cicero denounced Catiline’, we establish (4) also where ‘p’ represents any falsehood. Tom ends up believing everything. (Quine 1960, pp. 148–149).

First, to rehearse Quine's definition of referential transparency. (Familiar readers can skip this paragraph.) Quine defines transparency in terms of 'modes of containment ... of singular terms or sentences in singular terms or sentences'. Definite descriptions count here as singular terms. For Quine, a mode of containment M is referentially transparent iff, 'whenever an occurrence of a singular term t is purely referential in a term or sentence C(t), it is purely referential also in the containing term or sentence M(C(t)). ' (p. 144, schematic letters changed). For a singular term t to be purely referential in a term or sentence is for it to occupy a purely referential position there. Quine's 'criterion' for a position's being purely referential is that the position 'must be subject to the substitutivity of identity' (p. 142). That is, to the substitutivity of co-extensive singular terms salva veritate.

Let us begin by simply granting (3) for the sake of argument, ignoring its justification - Quine's 'by the hypothesis already before us'. (After we have identified a later fatal flaw in the argument, we will return to (3)'s justification briefly, since it seems to suffer from essentially the same flaw.)

Now, note that Quine's 'hypothesis about Tom's logical acumen' (hereafter 'the acumen hypothesis') and the steps of his argument are at different semantic levels. The hypothesis is framed in terms of belief in sentences, while in the argument, sentences appear unquoted as the contents of 'that'-clauses. Thus, the acumen hypothesis does not apply directly to 'Tom believes that #p = 1', since that sentence says nothing about Tom's belief in any sentence. Quine is, apparently, suppressing a quotational and a disquotational step here. An expanded version of this part of the argument, in which the acumen hypothesis could be applied directly, would have to run something like:

(i) Tom believes that #p = 1.

(ii) Hence Tom believes the sentence '#p = 1'. (Quotation step.)

(iii) Hence Tom believes the sentence 'p'. (Acumen hypothesis together with (ii).)

(4) Tom believes that p. (Disquotation step.)

Secondly, note that 'believes' in 'Tom believes that #p = 1' is to be taken in a transparent sense, as the piece of reasoning preceding it makes clear. (In case of any residual doubt about this: in the very next sentence after the argument as quoted, Quine summarizes it by saying 'Thus in declaring belief invariably transparent ... we would let in too much.')

Putting these things together, we can see the invalidity of Quine's argument: when (i) is taken in a transparent sense, it does not imply (ii).

To see this, consider that Delia Graff Fara believes (in the transparent sense) that Quine wrote Word and Object. We hereby introduce a new name for Quine, 'G6'. Now, since G6 is Quine - since 'G6' and 'Quine' are co-extensive - we may infer that Delia Graff Fara believes (in the transparent sense) that G6 wrote Word and Object. Plainly, we cannot infer from this that Professor Fara, who knows nothing of my convention (at the time of writing), believes the sentence 'G6 wrote Word and Object'.

The problem with Quine's argument as it stands, then, is in the first instance a use-mention confusion. (None of the papers cited makes anything of this point.) We have now seen that the problem cannot be fixed by expanding the argument to contain a quotational and a disquotational step; the quotational step is invalid. Can it be fixed by rephrasing the acumen hypothesis as a schema containing placeholders for unquoted sentences?

It cannot. Such a schema would run: 'John believes that #p = 1 when and only when he believes that p'. The dilemma here is that, if 'believes' is taken transparently, the schema is not a defensible principle of rationality (even for logicians), and if it is taken opaquely, the principle doesn't apply to Quine's argument.

Finally, to return to the first step of the argument, namely (3)'s justification. The 'hypothesis' Quine cites here is, as far as I can tell, the acumen hypothesis. And so this step is just as invalid as Quine's inference to (4). For the case where 'p' is true, however, (3) will be true anyway, so long as Tom believes that 1 = 1.

Tristan Haze
The University of Sydney

References

- Quine, W. V. (1960). Word and Object. The MIT Press.

- Charles Sayward (2007). Quine and his Critics on Truth-Functionality and Extensionality. Logic and Logical Philosophy 16:45-63.

- R. C. Sleigh (1966). A note on an argument of Quine's. Philosophical Studies 17 (6):91 - 93.

- David Widerker (1977). Epistemic opacity again. Philosophical Studies 32 (4):355 - 358. 

In with the True - An interview with Peter Boghossian

Prof. Boghossian writes,

Love your blog. I think your readers may enjoy this highly controversial piece: http://www.philosophynews.com/post/2011/12/05/Interview-with-Peter-Boghossian.aspx

Disclaimer: Sprachlogik is not styling itself here as an atheist blog. Theists and agnostics are most welcome. (My father's a non-churchy theist. I'm all over the place, but basically agnostic.)

Philosophers' Carnival - December 12, 2011

Welcome to the December 12, 2011 edition of Philosophers' Carnival.

There weren't many good submissions this time around, so I went hunting. If you write substantial philosophical blog posts, please consider making a habit of submitting to the Carnival.

Submitted

Should grad student stipends be tax exempt? - by Matt Hoberg of The Consternation of Philosophy.

Satisficing by Effort - Richard Yetter Chappell of Philosophy et cetera discusses effort-based satisficing consquentialism, a view about ethics.

Is the right to self-defense "unlimited"? - by NChen of Gavagai!.

Found

Reviews of this Blog - a blog, by Brian Rabern, consisting solely of reviews of itself. Inspired by Douglas Hofstadter. (Sorry.)

Ethics and Affectivity - by Levi Bryant of Larval Subjects.

Some Consequences of Perdurantism - by Andrew Brenner of the Florida Student Philosophy Blog.

Comparative Desert and the Bounds of Well-Being - by Antti Kauppinen of PEA Soup.

A Note on Craig's Standard Reply to Mackie on the Kalam Cosmological Argument - by ex-apologist. 

Ruling Out the Ruling Out Principle? - by Dan Lopez de Sa of the blogos, the blog of the LOGOS research group.

The insoluble problem of future tense statements - by Edward Ockham of Beyond Necessity.

Philosophy by Humans, 2: Living Metaphorically - by Luke Meuhlhauser A.K.A. lukeprog of Less Wrong. lukeprog is one of the more brilliant and prominent nerds in the Less Wrong community led by Eliezer Yudkowsky. Like it or not, something is happening there.

The truth about infinitesimals - by Denis Des Chene of Philosophical Fortnights.

Extensionalism & Presentism - by Philippe Chuard of BrainPains. The author is discussing extensionalism and presentism about temporal experiences.

Byrne & Hilbert on Color Physicalism - by Justin Fisher of BrainPains. See comments for a response from Hilbert.

Finally, a Wittgensteinian experiment of mine, Twenty Remarks on Necessitarianism and its Negation.

Thanks for visiting. The next Philosophers' Carnival will be at Cognitive Philosophy.

Assessing Epistemic Modals

Draft paper, here.

After critically examining contextualism and relativism (MacFarlane's view) about epistemic modals, I propose an approach I call 'the dimensions of assessment approach'. Comments welcome.

Twenty Remarks on Necessitarianism and its Negation

Disclaimer: the following remarks are highly exploratory and are not to be read as expressing anything final. 

1. What is the status of the claim with which Lewis begins On the Plurality of Worlds? (That there are other ways the world could have been other than the way things are.) We have a tendency to hear it in the wrong key. Imagine someone who had lost a loved one saying 'Things really could have been different'. (There seems something self-deluding in this, although it is no ordinary self-delusion. You don't fix it with negation.)

2. Ayer, Carnap and others wanted to say: the necessities fall out of our linguistic/conceptual scheme. Codify 'the rules of correct English' (cf. Language, Truth and Logic) and you've codified the necessities (or a base from which they may be logically derived). But the Kripkean necessary a posteriori, semantic externalism etc. make it clear that one needs a lot more fine structure than 'rules of correct English', and that some structures are in some sense not correct, in a way which has nothing to do with internal incoherence, but rather to do with inadequacy to an external referent. 

* * *

3. 'I went to X, but could have gone to Y instead.'

How might such a proposition be used? A retrospective epistemic use. Or to point to various facts - e.g. that Y wasn't too far away, or that I was under no compulsion to go to X, or that I was seriously considering going to Y, etc. But what if I say that none of that is quite to the point? 'Sure, those things might be true, but over and above that, I'm saying that I really could have gone to Y. The world could have gone differently here, and my going to X was just how it happened to turn out.' What could this mean? (We can imagine someone being bothered, preoccupied, by the thought that things really could have gone differently.)

4. Consider these propositions:

(1) There are things which could have gone otherwise.

(2) There are contingencies.

When a philosopher asserts (1) or words to that effect, typically the reaction they want is 'of course!'. Now, this would happen sometimes, but more common among non-philosophers might be perplexity - 'what do you mean?' - and skepticism - 'but how do you know?'. (I have some experience of this.)

5. Both in and out of serious philosophy, there is a tendency to take (1) and (2) in what I would call the wrong way. Roughly: as though they described something extremely fundamental and general about reality. And yet that is not an entirely wrong-headed thought, for it is immensely important that we talk and think about non-actual scenarios.

6. It is as though, when someone pushes a statement of counterfactual possibility in a certain way, we take them to be making a very strong claim, quite beyond our ken.

7. When one hears (1) and (2) in an "inflated" way - whatever that comes to - necessitarianism can seem like the antidote: something wrong has been said, and so it should be denied. But then the denial turns out to be just as troublesome, if not more! (Now it is trivially false, instead of trivially true - like Idealism in relation to Realism.)

8. The disease is not fully containable in any one proposition. 

* * *

9. It can be very important to realize that some propositions are necessary - 'One cannot feel another's pain', 'The opium worked (if it worked at all) by means of a dormitative power'. Likewise, contingent - 'There are two sexes', 'We have one body each', 'We use the decimal system to count'. It can be very instructive to get clear about the "modal status" of these things. And of course this does not mean leaving the view that they are contingent (or necessary) and coming over to the other side, but rather: noticing something, which one might have been quite blind to. Think of the conceptual development of children - one cannot truly say that they go through a phase of believing that everyone knows the same as them, but we can see what this is getting at. In the place where we have a fundamental distinction or concept of great importance in adult life, they do not have anything to speak of.

10. So: it can be important to realize the modal status of certain propositions. But - and this is crucial - it does not at all follow from this that it is important, desirable or possible to draw a line which clearly divides all propositions according to modal status.

11. What is the modal status of 'Our conceptual scheme has moving parts'? I.e. how could anything else be a conceptual scheme? (Mind of God.) And yet the proposition easily acquires the flavour of a general fact of nature. Why? Well, lots of important facts lie immediately underneath.

* * *

12. The notion of the mind of God. It can look as though everything is an approach to this ideal, and yet we have absolutely no grip on the ideal. We (i.e. myself and most active philosophers) do not have any positive belief in the mind of God in the old sense, and all the minds we do believe in are completely un-Godly.

13. Our conceptual scheme has moving parts. But if God knows all, he doesn't need any moving parts. (Spinozistic necessitarianism, Leibniz's difficulties.)

14. So now it looks like, in a sense, the ideal conceptual scheme has no moving parts - no need to dance further with experience or reason. And yet we will always need a conceptual scheme with moving parts, and can really make no sense of anything else. Thus how could the mind of God be our ideal?!

15. From this perspective, one could imagine pantheism as a kind of scholastic solution to our present problem. (For it would block the assimilation of our mind with God's.) ('A full, non-redundant model of an object must be a duplicate of the object, or the object itself.')

16. We say things could have been otherwise - I could have walked somewhere else today. But then someone says 'but could that really have happened?' and it's like the stakes are raised somehow. But isn't this a shift to a more specialized modality?

17. "What makes you think things could have really been different, and that this isn't just a function of our ignorance?" - The first thing to say is there's no simple answer here: one can't just point to evidence. The question is symptomatic of something.

18. One pernicious kind of confusion when talking about metaphysical necessity - the modality identified, and separated from other concepts, by Kripke - is to think of it in a manner which would be appropriate for some kind of deep and remote form of retrospective epistemic possibility. This is illustrated by the notion of the Creation as an event. God did something, and it may have been reasonable to expect this to go various ways. It is important for our concerns here that such an idea is natural to some degree.

19. It is possible to feel that any kind of conceptual view of necessity, even if we grant everything to the concepts including external adequacy, somehow makes modality flimsy, shadowy - somehow still overlooks reality in some way. "The way things really could have been, concepts aside." (The notion of an inconceivable object. If such a thing is impossible, how is that not just a piece of luck? Thus it can seem that the conceptualist must rely on some kind of pre-established harmony between possible concepts and possible realities, which would of course be highly suspicious.)

20. There is a strong philosophical tendency to think of knowledge of how things must be as penetrating deeper into the world than mere contingent knowledge. As though we saw how a machine behaved, but then looked into its workings and saw that it had to behave that way. Or, we form a hypothesis about the workings based on, and in order to make sense of, the behaviour. (This gets us into difficulties, but it would be stupid to call it incorrect - or correct for that matter.)

For more on modality see here, here and here.

Toward an Understanding of De Dicto Subjunctive Necessity (draft paper)

Here. UPDATE March 2013: This document has been completely superseded by An Account of Subjunctive Necessity.

UPDATE March 2014: This latter document will in turn be superseded soon by a forthcoming blog post which will fit together with other posts.