On the Problems of Reference and Intentionality II

The following remarks were written when I was wrestling with these problems over an extended period. They might act as a stimulus to someone. Part I is here.

Part II

24. What do I mean when I say 'reference is no ordinary relation' in (22)? Perhaps the point would be better put in terms of reference-ascribing propositions.

25. I have long had the feeling that the desire for a classical analysis of reference ('x refers to y iff ...') comes from within a way of looking at things which involves something which might be called a 'transcendental illusion'.

26. For some reason I want to say: remember, in reference-talk, we are, as always, presupposing a connection between our concepts and their objects. But what this means, if anything, is very unclear.

27. There is the urge to say something like: 'Remember, you can't get outside your own mind'. But what the hell kind of reminder is that?! Who would ever think otherwise? (Satisfaction with such remarks belongs to a relatively low level of philosophizing.)

28. I can imagine someone proposing a two-component theory of reference which makes use of a "disquotational base" together with "equivalence criteria". So when I say 'my concept of John is of John' this comes from the base, and so is similar to 'I have a concept of John, who exists'. But I can also say, for example, this name 'X' seems to be refer to John, this being similar to: This word is used in the same way as my 'John'.

29. We might say: Reference is that relation which is presupposed when anything is spoken of.

30. This ignores some complications (e.g. fiction) but is more appropriate, I think, than:

(1) Existence is that property which is presupposed. and

(2) Identity is that relation which is presupposed

because that distorts the logical form of existence and identity propositions. Reference is more properly a relation (i.e. more properly a property-or-relation, i.e. thing which can be ascribed to things).

31. What do I mean by 'more properly a relation' here? Here is one way to explicate this. Imagine a form of representation where objects are represented with dots or boxes, which are then labelled with names (or otherwise made intrinsically unique). Properties can be indicated by labelled lines which have one point of contact with dots or boxes, two-place relations with labelled directed lines which have two points of contact, etc.

With such a technique, an existence property-representation and an identity relation-representation would serve no purpose. Every object would have exactly one point of contact with an existence line, and exactly two points of contact with a single reflexive identity line.

32. This is, of course, a consequence of the way I am imagining this mode of representation to work. There are of course more sophisticated possibilities, where one uses dots and boxes at a higher semantic level, so to speak - as representations of ideas of objects rather than objects themselves. In that case, one can meaningfully use something like existence and identity lines - but then aren't they more properly seen as ascriptions of the property of designating something and the relation of codesignating respectively?

33. Now, what about reference? Here we can start to see that we can sort reference propositions into classes, depending on how they work and get verified. One distinction we can make is between reference propositions which concern expressions belonging to the same language-system, and those which talk about expressions from another language-system. Let us call the former 'intrasystematic' and the latter 'extrasystematic'.

34. Now we may make a further distinction between disquotational intrasystematic reference propositions, and non-disquotational. (Some of the latter can be used to give all kinds of information, e.g. '”The winner of the race” refers to John' can inform someone that John won the race.)

35. Consider what happens when we use the graphical mode of representation described above, and a new (extralinguistic) object comes to our attention. We draw in a new box or dot (let us say "node" from now on).

But in so doing, we bring into existence a new object - one which stands in the reference relation to the original object. We can represent this fact now too. One way of going would be simply to draw in another node, which represents the last node drawn, and then to connect it to that node with a line indicating the reference relation. But of course this procedure can then be repeated for the new node. This procedure corresponds roughly to giving a name its own name in word language.

36. Another way of going, which corresponds roughly to quotation in word language, would be to introduce a kind of operator on points of contact. An unmarked point of contact is the ordinary case, a marked point of contact indicates that the contacted node is representing itself.

37. Consider the first technique, where we start at the bottom level, and then construct nodes to represent those nodes, etc., as required. This process thus "automatically generates" reference propositions. These automatically generated ones are the disquotational intrasystematic ones.

38. Also, among extrasystematic reference propositions, two kinds of verification criteria can be distinguished. Direct comparison with our systems (looking at arithmetic talk for instance), vs. coordination which "involves the object" more. (The field-linguist would be doing both of these things.)

39. There is a close connection here to the Twin-Earthable/Non-Twin-Earthable distinction. (Perhaps it is that the direct comparison verifications yield propositions about the referents of non-Twin-Earthable expressions, and the “object-involving” ones yield propositions about the referents of Twin-Earthable expressions, but that may be an oversimplification.)

40. Consider Idealism here. On Idealism, perhaps all extrasystematic reference criteria can be reduced to system-coordination. A kind of collapse of Twin-Earthability.

41. For example: I see a man point at a rock and say 'N', and I form the hypothesis that 'N' is a name (which is not part of my system) representing the rock I see in front of me. This is a paradigm case, it might seem, of acting on criteria for extrasystematic reference which cannot be reduced to system coordination. I coordinate a part of his system directly with its object, not with a part of my system.

42. An Idealist may insist that this is not so. Rather (they may say), I am correlating my perceptual representation of the rock with something. But it would seem that if we are to be consistent, we can't really say that the perceptual representation of the rock is correlated by us with the name 'N', since 'N' is not part of our system. Mustn't we now say that we correlate our perceptual representation with our representation of the extrasystematic name 'N'?

43. But still, this does not give the Idealist the distinction I want to have between external-object-involving verifications of extrasystematic reference-propositions and verifications of extrasystematic reference propositions which involve "merely internal" comparison of systems. For in the internal case they also have to talk about our representations of some extrasystematic expression.

44. Why is this interesting? Not 'in case Idealism is true'!

45. When '”A” refers to B' is not disquotational, it seems that for practical purposes it means the same as the statement that 'A' and 'B' codesignate, and can therefore be understood as being verified by correlation of (aspects of) the role of two signs.

46. x refers to y iff x codesignates with 'y' iff x's referent = y.

47. But not all reference propositions which look like disquotational ones are such. For example: '”N” refers to N in German too'. Or when setting up a new language, as in formal Peano arithmetic: '"0" refers to 0'.

48. We should take note of the singularity of a truly disquotational reference-thought.

49. In philosophy, we think: 'N' refers to N. Then we think: how?

50. Now, that first thought is a thought to the effect that a certain symbol stands in the reference relation to a certain object. That much is clearly true to say.

51. Compare the thought, as had by an English speaker, that 'Deutschland' refers to Germany. This too can be truly said to be a thought to the effect that a certain symbol stands in the reference relation to a certain object. Likewise the thought that 'John' refers to the man I met yesterday.

52. But clearly the first, disquotational thought is a very different beast from the latter ones. What worries me is the effect of their assimilation under the rubric: reporting a reference relation between symbol and object. Reference propositions are propositions which report such relations (connections), reference thoughts are thoughts that such relations hold.

53. We might want to say that the disquotational reference propositions and thoughts are a subset of all reference propositions and thoughts, characterized by the fact that the symbol which the thought is about is also used to represent the related object. And now we may ignore this subset and concentrate instead on the non-disquotational subset.

54. There is a problem with this formulation, however. For example, suppose we are told that someone used a certain expression which itself is named E, but one does not know which expression E is. One may learn something about E, namely that it refers to Venus. Suppose E is actually 'Venus', the same word used in the same sort of language system. In that case, on my formulation above, the thought that E refers to Venus would be classed as disquotational, even when the thinker doesn't know that E is the name 'Venus'.

53. So it seems what we really wanted is: the disquotational subset is characterized by the fact that the same symbol is used twice over – to specify the referer, and the referent. But even this doesn't quite work, for '”The bearer of 'N'” refers to N' fulfills that condition.

54. The difference between disquotational and other reference statements is reminsicent of the difference between trivial (repetitive) and informative identities. However, the problem is in a sense inverted: the naive relational view of identity statements makes all instances look trivial, including the nontrivial ones, whereas the naive relational view of reference statements makes all instances look nontrivial, including the trivial ones. (This use of 'trivial' may not be fully warranted, but suffices for making the point.)

55. It is instructive to compare ' "N" refers to N ' with 'The word "N" has reference, and it is used in the way it is used.' Or even just: 'The word "N" has reference, and it refers to the object which does in fact refer to'.

56. What looks like (and in some sense is) a tautologous appendage changes the modal profile radically.

57. It certainly seems that having purely disquotational reference propositions in the mix can blind us to how the rest work, owing to the special direct way the former are verified. But would it be right to discount these as degenerate?

58. 'London' refers to London. This fact could have an interesting historical explanation. (Contrast 'London is London'.)

Part III coming soon.

Philosophers' Carnival #144 + Statement of New Direction

Welcome to the 144th edition of Philosophers' Carnival, the first edition since I took over organizational duties from Richard Yetter Chappell of Philosophy et cetera. The next edition will be hosted at Philosophical Pontifications on the 10th of November.

Those who have been following the Carnival lately will have noticed its decline. Brian Leiter recently decided to stop linking to it, due to declining quality. Last edition was a complete scandal. Depressed, I wrote to Richard Chappell offering to host the next edition, and to try to make it a better one. As an afterthought, I offered to take over the organization of the Carnival, and Richard accepted. 

I'm very excited about this opportunity to promote the free online dissemination of serious philosophy (something I believe in strongly), and I expect to give Brian Leiter reason to reverse his decision.

The two main things I think were wrong with the Carnival before are:

(1) It wasn't ambitious enough. It presented, and was regarded, too much as just a bit of fun.

(2) The work of highly visible and established bloggers, instead of being the backbone of the Carnival, was neglected.

Regarding (1): I see no reason why a significant amount of the serious, groundbreaking philosophy being produced now and in the future can't appear in the form of blog posts. Insofar as it can, the mandate of the Carnival should be to locate that stuff and help bring it to a wider readership.

Regarding (2): I suspect this is partly due to misguided attempts at fairness - people thinking 'Why should I link to Eric Schwitzgebel (or Berit Brogaard, or Brian Weatherson, or Richard Chappell etc.)? Those guys get tonnes of traffic already, and the Carnival is all about giving less visible writers a leg up'. Firstly, the Carnival isn't all about that - that's part of it, but it comes second to the main aim, which is to showcase the best philosophical writing of the blogosphere. Secondly, neglecting the best bloggers doesn't help anyone; there are surprisingly few people regularly posting serious philosophy on blogs - at least, blogs which aren't hidden away in dark parts of the web. Excluding the pillars of the online philosophy community, the central pool of talent, only reduces the Carnival's quality and interest. It doesn't make it any more open to new writers.

Look forward to a more serious Philosophers' Carnival, with higher aims and higher standards!

The Carnival Proper

Is Lewis's ontology qualitatively or ideologically parsimonious? - by Ross Cameron of Metaphysical Values. Highly recommended!

Impredicative Truth - by Abram Demski of In Search of Logic.

Whewell on Newton's Laws III: The First Law - by Brandon of Siris.

Bader against contingent and occasional identity - by Wolfgang Schwarz of wo's weblog.

Thoughts on evolutionary debunking arguments against religious belief - by Helen De Cruz of Prosblogion.

The Value Component of Plantinga's Free Will Defense - by Kenny Pearce of Prosblogion.

643 Words About Inferentialism, Disquotationalism, the Tonk Problem and the Liar Paradox - by Ben Burgis of Blog&~Blog.

Soames on the Abstract View - by Jeffrey Ketland of M-Phi.

When is true belief knowledge? - thoughts on Richard Foley's book of the same name, by Clayton Littlejohn of Think Tonk.

The Nature of Phenomenal Consciousness - by Richard Brown of Philosophy Sucks!

An arithmetico-temporal paradox - by Sam Alexander of xamuel.com.

Simmons' Paradox and the Barber - by Martin Cooke of enigMania.

What Experimental Philosophy Might Be - by Eric Schwitzgebel of The Splintered Mind.

Next edition at Philosophical Pontifications on November 10.

On the Problems of Reference and Intentionality

The following remarks were written when I was wrestling with these problems over an extended period. They might act as a stimulus to someone.

Part I

1. What determines reference?

2. Not, in general, descriptive or conceptual content associated with the name. Although the usual counterexamples to descriptivism do not apply to a range of names - for example, names of mathematical objects.

3. Not, in general, a causal connection between the object and the name. Both because the kind of connection in question is unclear, and because there is clearly no such thing in many cases - e.g. mathematical objects. But there is no clear counterexample in a range of cases: names of spatiotemporal individuals which someone in the linguistic community at some time has been perceptually acquainted with, for example.

4. What determines reference? - What gives this question its power?

5. The question seems to be: given that someone is referring to a particular object, what determines which particular object that is?

6. This is to be distinguished from: What is it which characterizes all cases of representation of, or reference to, an object? I.e. what do they all have in common which distinguishes them from other phenomena, apart from 'involving representation of an object'? But it is close to: Given some particular object O, what characterizes all cases of representation of, or reference to, O?

7. The thing we just can't accept is that we have nothing definitive to say in answer to the question: what determines reference?

8. It is extremely remarkable and important that people have tried hard to say something definitive here. It not a matter of course that anyone is moved in this way by that question. (Many people innocent of philosophy would find the whole thing quite unnatural and meaningless.)

9. I would like to talk about the feeling that, when the investigation goes empirical and scientific, we have left the realm of essence, so to speak. Is that right, anyway? Could one not maintain that by increasing our knowledge of the contingent facts - how reference actually gets going - we may be better able to penetrate to the essence? I.e. to what it takes in principle for reference to get going?

10. Well, historical development itself cannot be part of the essence. It is clear enough that there is no logical impossibility in a whole lot of organisms being spontaneously generated out of nowhere and beginning to speak of things.

11. When we think of the representation of external, physical objects, some kind of correspondence or covariance theory seems attractive. One thinks of primitive representation (or proto-representation) which is of this kind.

12. However, the merest thought of arithmetic, and Wittgenstein's builders, at once makes all this look inessential.

13. There is no reason why one cannot erect a concept which applies, for example, only to the representation of physical phenomena. But if one is inclined to say that we refer also to abstracta, and to wonder how reference in general is possible, then an investigation of the first concept alone will never satisfy.

14. It is clear enough that the idea of referring, which is primitively connected with the ideas of looking-at and pointing - the picture of representation and represented - is widely useful, and widely used.

15. Now, it looks very much as though one might, having noticed this, ask for an explanation of why it is. I.e., what makes the picture of representation and represented apply in this wide range of cases. Or better: what is it about all these cases which makes the picture of representation and represented apply to them?

16. This traces the problem down quite deeply. At this point we see that none of the common 'theories of reference' is going to help us here.

17. No answer comes to this question. This seems to have something to do with the fact that, once you are talking about a potential object of reference, you have already prejudged, so to speak, that the picture of representation and represented applies. Wherever we think of an object, there already we have the material for a higher-order thought about our representation of that object.

18. 'The problem of the essence of the representation of things leads, in a certain sense, to the problem of the essence of things.' The problem of the essence of things is obviously no ordinary one! 

19. To explain this tracing again: whenever we find ourselves in conditions where we can refer to an object, we thereby find ourselves in conditions where we can apply the picture of reference.

20. But, this is not the same as saying: wherever the conditions for objecthood are fulfilled, thereby are the conditions for a representation. For this is refuted by imagining a world in which there is just one object and nothing else. (This sort of distinction is important all through logical philosophy. It appears to be neglected, for instance, when people try to analyze existence and identity statements as designation and codesignation statements respectively.)

21. Two senses of truth conditions, two conceptions of what I will call fulfilment worlds for representations:

(1) All conditions/worlds in which what the representation actually represents to be the case is the case.

(2) The narrower set of all conditions/worlds in which the representation exists, represents what it actually represents, and where what it represents is the case.

22. What I was saying above now comes to: all the fulfilment worlds in the second sense for 'x exists' are automatically fulfilment worlds in the first sense for '"x" refers to x'. Or: once 'x exists' is fulfilled in the second sense, '"x" refers to x' is fulfilled in the first. (This holds not just for 'x exists' but for any proposition which contains 'x' in an extensional context.)

22. By dwelling on this we can start to see that reference is no ordinary relation.

23. As long as one is prepared to ignore the distinction between (1) and (2), or to openly switch from one to the other mid-analysis, then one can analyze existence and identity both in terms of reference. (Early Frege, Geach.)

Part II.

On Conceptually Progressive Propositions

The following remarks were written around 2010 - early 2011, when I was training myself in Wittgenstein's later methods.

1. In some cases where we wonder at the (metaphysical, conceptual) possibility of something, imaginability, conceivability in full detail seems to be close to what is at issue - i.e., whether the thing can be 'worked out'. In other cases however, it is quite different. In other cases, the question is closer to: ought we conceive of this? (This is connected with what I want to call railed vs. unrailed sponteneous concept formation in Wittgenstein's philosophy of math, knowing how to go on vs. discovering how.)

2. An example: suppose someone says that when a cat weaves around someone's legs, they are asserting ownership of this person. (I was actually told this as a child.) We might now consider whether this is so, get puzzled, and then consider whether it is even possible for part of a normal cat's normal behaviour to turn out to amount to an assertion of ownership. Here, it is quite unnatural to think that any kind of conceivability or imaginability (however detailed) is at issue. We have seen the behaviour a hundred times. We can imagine it perfectly well. What, now, does it mean to imagine it being an assertion of ownership? (Furthermore, actually being such an assertion - by which I mean that it does not count to merely imagine that one is dealing with some extraordinary cat who can think and communicate with conventional signals, or indeed anything out of the ordinary.)

3. The assertion that the cat is asserting ownership, and countless others like it, have a special character - it is granted in advance, so to speak, that the cat is not asserting ownership by the usual criteria. Conceptually progressive propositions, but also linguistically progressive propositions. (The former are perhaps well seen as a proper subset of the latter.)

4. One feels like saying: here, a decision is needed (although this is misleading in that no conscious decision is needed, no freedom to decide either way need be felt, etc.).

5. Within such cases, one might distinguish those which as it were call for full-blown (literal?) acceptance, while others are not concerned with that, being, rather, quasi-metaphorical, quasi-analogies. But - and hopefully this is seems trite - there is no clear line between literal truth and analogy, figure, simile, metaphor. However, with the latter, one of the characteristic things is that a decision isn't really needed in the full sense. However, one may still reject or embrace ('work with') the idea, criticize it, modify it, etc.

6. (Simile as a minimal literalization of metaphor - in effect, the literal truth one gets to when one accepts a metaphor as a metaphor. Which isn't to say that all similes are such literalizations, but rather: given an acceptable metaphor, one can get a literally true simile.)

7. An interesting feature of the cat case is that, in a sense, the verification and falsification conditions are already in place. It is already clear that, if one accepts this kind of proposition at all, one will count 'the cat asserted ownership' true when the cat weaved around the person's legs, false otherwise. Obviously it will be a more subtle than that, for many reasons - vagueness about whether weaving took place, the possibility of the cat being in some pathological state which just happens to involve weaving, etc. - but still, the point remains that the working criteria are not the focus of the problem. Reflection on this leads to the comparison of the cat proposition (in its progressive use) with a proposal of the adoption of a norm of expression.

8. Whenever something like that is said (i.e. 'a norm of expression'), it looks as though something important is being skirted over too easily. This is connected with the fact that this is no merely conventional, arbitrary norm of expression. It is highly charged with significance, with meaning. Also: to say it is a norm of expression might suggest that it is not a norm of thought. That thought goes on underneath, and this norm relates to how the thought gets 'put into words'. But the cat case is obviously not like that.

9. To think of a badly treated mechanical device as suffering, feeling sympathy for it - this is a perfectly possible form of thought which, as a matter of fact, ordinary adults do not engage in. (Here my saying 'as a matter of fact' may give a wrong impression, namely that I think it would be 'just as good' to do that, or that we can't say anything against such a viewpoint, etc. But not at all. I'm not getting into that.) 

10. The cat case, and the case of the suffering vacuum cleaner, are interesting partly because they make trouble for a certain oversimplification of the working of language which is, in certain circumstances, natural for us.

11. The oversimplification is something like this: in cases where the working criteria are not an issue, but where judgement one way or the other sets a kind of precedent (for the remainder of a conversation or train of thought (internal monologue), at least), it may look like the issue is 'merely verbal', 'merely pragmatic', or something of the sort. But this is liable to be extremely misleading; it focuses on the comparison of such a case to that of, say, the adoption of a perfectly arbitrary label, to the exclusion of other comparisons. The consequences are very different here from those which follow an arbitrary labelling decision. (Associations.)

12. It is instructive to reflect that sometimes we accept provisionally a conceptual precedent for the purposes of a conversation (or even a private train of thought), but have misgivings about it. We feels, as it were, that quite possibly we can get on OK with it for now, that to start critiquing here might be impractical at this point (e.g. in an involved conversation where we will have to go to bed soon, or get off the phone or something), but that, were the stakes to be raised, so to speak, we would have to try to sort it out.

13. Consider a conversation of this sort. The limitations one characteristically feels in such discussions. (Consider especially the case where your conversation partner doesn't see any difficulty.) Thoughts have potential problems other than falsity! Truths, expressed in a certain way, can be almost false. One almost wants to talk here of a 'higher' kind of truth and falsity, of verification and falsification. (I think Hegel actually does something of this sort at the beginning of his Logic.)

14. 'They believe a machine suffers!' - this could be thought to be an inadequate, misleading statement in a similar way to 'Cantor showed that there are numbers larger than the number of natural numbers' - except in the first case, the thing is rejected as absurd, in the second case, accepted as mind-bogglingly wonderful. (There are, of course, reasons for these different treatments.) In both cases, an extended conception is being regarded from the point of view of a narrower one.

15. ((There is a false note here, interestingly. I feel like saying: no, those who believe that the machine suffers do not have an extended concept, it is we who have a restricted one. We've got more structure, so to speak. We could have a concept similar to but distinct from our concept of suffering, which we apply to machines. Thus we have a sort of distinction where they have none. Or, we could apply something like that more minimal concept across the board, and then add something for the case of "real suffering".))

16. The proposition that a machine suffers could actually be embedded, quite deeply, into a highly sophisticated conceptual framework. The point being: it is not a proposition which can only survive in very primitive or childish systems. One could talk of panprotopsychism (Chalmers), a continuum of organizational complexity, and with it a continuum of suffering. So complex human computers in disorder, on such a view, perhaps suffer about as much as an insect. How far can it be taken? To vacuum cleaners? Hot water bottles? Not the latter. (The human teleology (purpose) of the object, if such there be, would naturally not play any part in its placement on the continuum.)

17. How do we know that rocks aren't incredibly sad? If this is metaphysically impossible, how do we know that? This is another case where imagination seems quite irrelevent. There's nothing to picture except the rock - or one might imagine being sad. Both procedures are clearly idle!

18. I believe that any reason given to think that rocks aren't sad is going to be bunk. What is disturbing here is the strong inclination toward finding such reasons nonetheless. It suggests deep infelicities in our thinking.

Changing the Past: An Examination of the Debate

Draft paper from the first half of 2011 here.

Coming soon: a post outlining recent developments in my account of subjunctive or metaphysical necessity. (Preview: A proposition is necessary iff it is, or is implied by, a proposition which is both counterfactually invariant and true. What remains is to explain what I mean by 'proposition' and 'counterfactually invariant'.)

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.)