Identity Expressed with One-Place Predication

Introduction

Frege's famous paper On Sense and Reference begins with the question of whether identity is a relation. Frege then goes immediately on to ask whether it is a relation between objects, or their names. The latter question then sees most of the action.

This is a confusing issue. What is a relation, anyway? What does it mean to hold that identity statements ascribe a relation, as opposed to doing something else? Might there not be various ways of categorizing things, perhaps involing, or giving rise to, slightly different senses of 'relation'?

It is not my aim here to give an overall discussion of the main philosophical problems surrounding identity statements. My purpose is to show how easily we can modify first-order logic with identity (FOL=) so that identity statements are treated as one-place predications rather than two-place relational predications. Comparing the result with natural language identity statements such as 'Hesperus is Phosphorus' makes the occurence of 'is' look more like a copula ("the 'is' of predication") rather than a relation symbol (some special "'is' of identity"). Sentences like 'Hesperus is identical to Phosphorus' then look, by contrast, more comparable to the familiar '=' form in logic - that is, more like they contain a relation-symbol.

I had thought of this possibility before, at least in part, but it came forcefully to mind recently when I was reading Delia Graff Fara's draft paper, 'Names as Predicates'. The theory put forward there is sophisticated, but my basic thought was: if, as Fara argues, 'Hesperus is Phosphorus' is not an identity, but a statement attributing to Hesperus the property of being Phosphorus, then what do count as identity statements? Statements involving variables? But they can also be treated as one-place predications. Instead of saying these are not identity statements, why not let them be the paradigms of identity statements, and just say that identity statements can be construed as one-place predications? (For Fara, I think, an example of a genuine identity statement would be 'Hesperus is identical to Phosphorus' - cf. the paragraph above. That is, Fara makes it a requirement of identity-statementhood that the statement have a two-place relational syntax, whereas I don't wish to. This is a fairly unimportant terminological difference as far as I can see.) 

The Strategy

We make three modifications to the ordinary syntax and semantics of first-order logic with identity:

- Instead of having a special symbol '=' in our stock of two-place predicates, we add two pointy bracket symbols '<' and '>' to the vocabulary.

- Add the following clause to the recursive specification of the well-formed formulae: For all terms T, '<T>' is a one-place predicate. ('T' here is a syntactic variable, specifically a term placeholder.)

- Instead of mapping '=' to a set of repetitive ordered pairs - one for each object in the domain, containing that object twice, i.e. "the identity relation" construed extensionally - we add the following rule to the semantics: For any term T which has a referent, let the sole member of <T>'s extension be T's referent.

(Note on quantified formulae: this works most clearly with the style of semantics where one considers models which contain a new constant in place of the variables bound by the quantifier, but it also works with variable-assignment semantics, if we class assignments to variables as referents.)

Now, in place of, e.g., 'a = b', we write '<b>a'. In place of '∃x (x = x)', we write '∃x(<x>x)', etc.

Remarks

This way of doing things is interesting in that we can, in an important sense, say everything we said with '=', while using a language that doesn't suggest any talk about identity as a relation which holds between all objects and themselves. The illumination this affords is, I think, the sort of thing Wittgenstein was talking about when he wrote the following:

Each time I say that, instead of such and such a representation, you could also use this other one, we take a further step towards the goal of grasping the essence of what is represented. (Philosophical Remarks, sect. 1.)

I am also reminded, in an obscure way, of this unforgettable passage in Russell's Logical Atomism lectures:

There is a good deal of importance to philosophy in the theory of symbolism, a good deal more than at one time I thought. I think the importance is almost entirely negative, i.e. the importance lies in the fact that unless you are fairly self-conscious about symbols, unless you're fairly aware of the relation of the symbol to what it symbolizes, you will find yourself attributing to the thing properties which only belong to the symbol. That, of course, is especially likely in very abstract subjects such as philosophical logic, because the subject-matter that you are supposed to be thinking about is so exceedingly difficult and elusive that any person who has ever tried to think about it knows you do not think about it except perhaps once in six months for half a minute. (Logical Atomism Lectures, Logic and Knowledge, p. 185.)

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.)