Monday, 16 January 2012

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.)