## Monday 16 January 2012

### Structured Modal Operators

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.)

1. Why *arent't* you considering modal operators in the more general formal setting involving accessibility relations?

2. In a word, because that setting gives one a greater multiplicity or generality than is required for alethic modal notions such as metaphysical/subjunctive necessity, possibility etc. and a priori necessity (a priority), possibility, etc.

In a possible worlds semantics for those notions, one would (for instance) always want the accessibility relation to be reflexive. (I need to think more about transitivity and symmetry in this connection - it connects to questions about the individuation of propositions.) So, while I do intend to say more in future about how the notation here can be given a full-blown possible worlds semantics, the whole point in a sense was to start with something more specific and purpose-built, to better enable a clear overview of the space of metaphysical/subjuctive modal operators (or, equally, modal operators based on the notion of a priority) without raising separate issues simultaneously. (These modalities could be brought together under the heading 'alethic', although I think the original motivation for that label might be somewhat confused.)

3. What kind of necessity is alethic necessity, then? I suppose my thinking is that answering this question would imply a specific possible-worlds structure, automatically answering all of the questions mentioned in the first section of your post. More generally, I'm fairly well convinced that accessibility-relation type analysis is all that can be said of the English notion of "necessary" (that is, specific English usage will imply whatever accessibility relation and world structure is convenient for the conversation).

On the other hand, I'm curious about causality and the kind of possible world semantics needed for causal reasoning via counterfactuals. In particular, must we believe in those as "true possibilities" (realism concerning possible worlds)? I guess my tentative answer is "no", but I'm generally worried that this has some troublesome implications. (Rejecting the merely possible, in general, feels like it might remove a bunch of needed structure.)

4. Your approach looks quite interesting. Would it perhaps be possible to combine with some of Moretti's ideas about the geometry of modalities?
http://alessiomoretti.perso.sfr.fr/GeometryForModalities.pdf

5. Thanks very much for the interest, Rael, and for the link. This paper you've pointed me to looks rich and exotic, not to mention difficult! I'll have to take a good look some time before I have an answer for you. Meanwhile, any hints from you regarding possible points of contact are welcome.

6. Exotic indeed. What intrigues me is that your approach is very clear and logical, but that i don't see how to make sense of it in terms of the modal square of oppositions. We can take the extended square on p. 3 in Morretti (the S5 one to the right), and interpret the point to the left as T and the one to the right as F. This is I think quite accepted, not wild and exotic as many other things in Moretti's paper. (Now, to compare with your diagrams, we must remeber that the diamond in the square (hexagon) is what you call contingent character, whereas your diamond is contingency as opposed to necessity.)

What bothers me is that I don't see the easy way of translating your scheme into the hexagon, or the hexagon into your scheme, that surely must exist. Perhaps you can find the right way to do this.

7. - Sorry, this is confused: "Now, to compare with your diagrams, we must remeber that the diamond in the square (hexagon) is what you call contingent character, whereas your diamond is contingency as opposed to necessity."

8. I'll stop spamming you now (for today at least). Just a final thing: what I would like to accommodate your scheme to is the modal octagon as presented on p. 13 in this piece by Beziau:
http://www.jyb-logic.org/papers/risingsquare.pdf

(Take p (extreme left) in this diagram to correspond to T, and ~p (extreme right) to correspond to F. Note also that Beziau doesn't draw lines for implications (e.g. subalternation), as is normally done in these figures.)

Your "contingent character" is the lowest point of the octagon, "necessary character" is the highest point". So the highest point, the lowest, the left and the right are the four parameters that you use for constructing your matrices. However, there is still something I don't grasp...

9. This isn't spam! Thanks kindly and don't hesitate to comment again. I will have a look at the Beziau piece soon.

10. All right! I've been looking around to see if there is something similar to your approach. It seems that in the early days of modal logic, people had some hope for a truth-functional account based on matrices of this sort. But they ran into problems; see pp. 3-4 here:
http://ocw.mit.edu/courses/linguistics-and-philosophy/24-244-modal-logic-fall-2009/lecture-notes/MIT24_244F09_lec03.pdf

But the idea lives on in some systems:

Kearns, Modal Sematics without Possible Worlds, Journal of Symbolic Logic, No 1 1981.

Beziau (again), 2011, esp. p. 4:
http://www.jyb-logic.org/papers/la4.pdf

Fariñas del Cerro & Herzig 2011: http://commonsensereasoning.org/2011/papers/DelCerro.pdf

These systems start from necessary truth, contingent truth, contingent falsity and necessary falsity. They tend to use matrices too. Whether the further moves they make are compatible with your project I don't know. I look forward to see how you will develop the idea, which is nice and intuitive.

11. Thanks for the references. I'd seen the Kearns piece before. The Stalnaker notes seem particularly illuminating.

'I look forward to see how you will develop the idea' - in terms of formal developments, I have no idea what if anything I'll do next, but stay tuned. (If I knew who you were I'd say 'I'll keep you posted'!)

In any case, the basic strategy of clearly separating the issue of modal character from that of truth-value will be a key ingredient in the book I am working on, the object of which is to investigate and clarify the topic of metaphysical or subjunctive modality.

12. I am struck by something in the Stalnaker notes which I would not go along with (at least not without qualification). He writes that 'Necessitation of a necessary truth should indeed yield a necessary truth: BOX(1) = 1.'

I think that on one natural and important way of individuating propositions, which I will in this comment call 'the internal mode of individuation', this does not hold. Consider the necessary a posteriori truth, 'Hesperus = Phosphorus'.

On the internal mode of individuation (where sameness of extension/truth-value is not required for sameness of content), instances or tokens of this same proposition(-type) can come out true in some environments (such as ours) and false in others (such as a Twin Earth on which the morning star there is not the evening star there). This hangs together with this proposition's a posteriori status: its truth-value (on the internal mode of individuation) is not an internal property of it.

So, individuating proposition-types internally in this way, the claim that (our tokens of) 'Hesperus is Phosphorus' is necessary amounts to something like this conjunction:

(Our tokens of) 'Hesperus is Phosphorus' are of necessary character AND true.

And this claim itself will not be necessary, but contingent.

(The way I have expressed all this is slightly confusing, due to type/token issues, but I think and hope that a clearer and more explicit terminology will fix this.)

13. Are your boxes Boolean. If they were then the field that is rules out is a 0 and we can represent your squares as Boolean strings i.e.

1100 Falsity
0011 Truth
0001 Nec
0010 Cont
0100 Imposs
1000 Mere Poss
1010 Chont.Char.
0101 Nec.char
1001 Nec. Or Mere pos
1001 Cont. OR Imposs
1101 Non-cont
1110 Non-nec

Are your boxes Boolean. If they were then the field that is ruled out is a 0 and we can represent your squares as a 16 valued logic with 4-tuple Boolean strings i.e.

1100 Falsity
0011 Truth
0001 Nec
0010 Cont
0100 Imposs
1000 Mere Poss
1010 Chont.Char.
0101 Nec.char
1001 Nec. Or Mere pos
1001 Cont. OR Imposs
1101 Non-cont
1110 Non-nec

Is this compatible with you box notation?

Here is something very similar to your work.

http://www.semantic-cube.com/modalities16v.png