Quick Thoughts on the Identity of Indiscernibles

The law of the Identity of Indiscernibles is said to be the converse of Leibniz’s law. This is the Indentity of Indiscernibles:

If a and b have all their properties in common, then they are one and the same thing.

In symbols:

(x) (y) (F) ((F) x ≡ F (y) ⊃. x = y)

The basic question is:

Imagine two steel balls which have all their properties in common. Could they still be two?

Intuitively, most people (I think) would say 'yes'. It doesn't seem inconceivable prior to modal philosophising.

For a start, wouldn’t the balls still be spatially or temporally separate? If that is the case, then surely they wouldn't have all their properties in common. (That's if you accept spatial and temporal properties, which many philosophers do.)

Now we go deeper into this thought experiment.

One could say (I suppose) that if the balls were suddenly frozen in space, then their positions would be different. Hence they'd have different spatial properties. However, what if the balls will never be frozen in space and never have been frozen in space? (Is that a hypothetical scenario about a hypothetical scenario?)

As they are now, and in five minutes, etc., the balls are constantly on the move relative to one another. Thus they have all their spatial (as well as temporal) properties in common. And because they're both in an empty world, there can be no relational properties (care-of other objects, conditions, events, etc.). Any relational properties a has relative to b, b has relative to a. Thus they have all their properties in common.

This, then, appears to break Leibniz’s law in that the balls are indiscernible; though not identical!

Leibniz’s Law & Intensional Contexts

What is Leibniz’s law? This:

If a is the same as b, then everything true of a is also true of b.

Or in symbols: (x) (y) (F) (x = y ⊃. F (x) ≡ F (y))

There's at least one way in which it can be taken to be false. Take Roger Scruton’s example:

“Suppose John is thinking tenderly of Mary, and Mary is the person who, unknown to John, ate his beloved cat. Is John thinking tenderly of the person who ate his cat?” 

This is primarily a question of John’s beliefs about Mary (de dicto): not about Mary herself (de re). It's indeed the case the cat-eater is the same person as the person thought tenderly of by John. However, John doesn't know that Mary ate his cat. This is a fact about John; not a fact about Mary. (Can there be psychological facts?)... Unless what is thought by other people about Mary also constitute facts about Mary. In that case, there would be a multitude of facts about Mary that she and others (as individuals) couldn't know about Mary (which isn't in itself implausible).

Does this story break Leibniz’s law? No.

What we would require to save the day is a theory of contexts: 'intensional' contexts. Do we need such intensional (or belief contexts) at all? Quine said no – at least not in science, logic and perhaps in philosophy too. That is, what John is thinking of doesn't belong to this particular extension (or reference) – that is, to Mary. Thus it plays no part in science or logic. (Though it would, perhaps, play a part in the psychological descriptions of John.) Then again, it could also be said that Mary (as a single person or human being) couldn't really play a part in science or logic either.

James Ladyman & Don Ross on Metaphysicians' Intuitions

 

You will note how important the criticism of the (analytic) metaphysicians' reliance on 'intuitions' is in James Ladyman and Don Ross's paper/book.

Since I've read many analytic philosophers (if not analytic metaphysicians) criticising not only the notion of “intuitions” - but also the philosophical reliance on them, I find it hard to make sense of Ladyman and Ross's stress on such a thing.

Sure, some philosophers have noted and even relied on intuitions; though many haven't. Though, as I argue later, it's almost impossible not to begin one's philosophical pursuits without utilising one's intuitions. And, it may follow from that, that if one's intuitions are acknowledged as a starting point, that starting point is bound to have an affect on much of what follows.

Having said that, it's indeed ironic, at least prima facie, that metaphysicians rely at all on intuitions. Isn't it far more likely that an epistemologist or a philosopher of mind (for reasons I hope are obvious) would stress or even rely on intuitions?

                          A Case for Intuitions

There are many arguments in favour of intuitions... and not all of them use intuitions.

For example, you must start from somewhere. And the best - or even the only - place to start from in philosophy (as in most things) is from one's own intuitions. Indeed it's hard to even make sense of the idea of starting from anywhere else. And if you start from your own intuitions (I stress the word 'start'), then it may be equally - or more - wise to take on board collective (as it were) intuitions as well.

Bearing all that in mind, it's hardly a cardinal sin when metaphysicians begin by using phrases such as "it is intuitive that..." or "it is counter-intuitive that..." (the examples given by Ladyman and Ross) when, presumably, such people won't end their philosophical pursuits with such phrases.

So when Ladyman and Ross say that intuitions aren't scientific data, the metaphysician may simply say: “Yes, I know. And?”

On the one hand, it may be understandable to argue against intuitions regarding, say, quantum mechanics or the nature of DNA. However, many mathematicians and scientists (ranging from Kurt Godel and Alan Turing to Roger Penrose) have happily stressed the importance of intuitions in both mathematics and physics. (Though, admittedly, perhaps not in quite the same way the guilty metaphysicians do.)

You can also defend the existence and utilisation of intuitions without using the phrase (which I noted in Ladyman and Ross's paper and elsewhere) “the faculty of intuition”. That sounds like the kind of reification that Gilbert Ryle warned against some seventy years ago. Indeed if people do believe in such a faculty, it will take on a role similar to that of Kant's a priori 'categories' or even the amygdala. In that case, just as philosophers could have asked Kant why he thought that the mind's concepts or categories were a-historical and universal; so a contemporary critic can ask why (some) metaphysicians think that our faculty of intuition is reliable and/or static from (say) an evolutionary or biological point of view.

Though, again, our intuitions need not be seen as a priori, a-historical or even as constituting a faculty as such.

It would be wise to say, then, that when contemporary metaphysicians appeal to intuitions, they don't (or, at least, they ought not to) refer to some magical ability which only they possess. Rather, they're simply using semi-rhetorical language; of which there are many other examples in analytic philosophy (such as "surely...", "it is obvious that...").

The Basics of Ladyman and Ross's Case Against Analytic Metaphysics

It can be seen that the basic idea is simple.

Before the rise of modern science it was philosophers who investigated “the fundamental structure and nature of physical reality” (as it's often put). However, after the rise of modern science, philosophers shouldn't be still doing this without the help of science.... at least not in 2015!

As a consequence of that, The Every Thing Must Go position is against any “a priori metaphysics” or the search for “a priori truths”.

Prima facie, it's hard to make sense of this because I can't really believe that there's a 21st-century (or 20th century) metaphysician who would claim to be engaged in an entirely a priori pursuit. (Though perhaps I'm wrong.) In fact I'm not even sure what the words “a priori metaphysics” mean or if it would be achievable (in principle).

Anyway, if such a thing does exist, then James Ladyman and Don Ross class it as “neo-Scholasticism”.

Sometimes Ladyman and Ross's main criticisms of analytic metaphysics seem rhetorical – at least as they stand. For example:

i) That metaphysics "contributes nothing to human knowledge”.
ii) That metaphysicians are "wasting their talents”.
iii) That metaphysics “fails to qualify as part of the enlightened pursuit of objective truth, and should be discontinued”.

Sure, these positions can be argued for. However, it must now be said that some commentators say that they aren't argued for by Ladyman and Ross: they're simply stated.

What may happen here, then, is that those who follow the every-thing-must-go position will simply end up talking a different language to those who practice analytic metaphysics (or just plain metaphysics). And then it will come as no surprise that there's no mutual ground between them (or even a conversation). It will become like the situation between much Continental philosophy and analytic philosophy (at least until, say, the 1980/90s).

What Ladyman & Ross Do

The central metaphysical position of the everything-must-go school (if there is such a thing) is one of “ontic structuralism realism”. The fundamental aspect of this is the importance it gives to the mathematical relations or structures which capture the nature of physical reality (i.e., in physics). Clearly, then, if one wants to explore this position, then that's where to begin.

ETMG philosophers also class themselves as “neo-positivists”. They acknowledge that there were big problems with the original logical positivist school (if it ever was a school). That's not a problem because the logical positivists themselves realised that there was a problem with (much of) logical positivism. In fact it was mainly - or only - former logical positivists who destroyed logical positivism. (I doubt that other philosophers would have had the knowledge or skill to carry out that feat.)

It also seems that Ladyman and Ross are saying that metaphysicians should be scientifically-literate holists who should try to show us “how everything fits together in a broad sense” (as Nelson Goodman or Wilfred Sellars once put it).

In other words, the “ontological structure” of the universe is the domain of physics and science generally. Metaphysics, on the other hand, should attempt to find a unified and “cross-disciplinary” philosophical synthesis and analysis of how the sciences tell us the universe is structured. (Put that way, it's similar to Quine's position; though he didn't really emphasise cross-disciplinary unification as such.)

The Debate?

When a metaphysician says that analytic metaphysics is concerned with problems which aren't (strictly speaking) scientific (as well as when he says that it uses analytical and logical methods that aren't those of of science), then, I suppose, Ladyman and Ross may give the obvious reply:

The problems and tools of metaphysics shouldn't be distinct from science – even if they aren't identical.

Though if you were to take this position too far, metaphysics will simply become physics/science; or, at the least, a part of science/physics.

The problem is that no only may Ladyman and Ross throw out metaphysics and even all philosophy (if you follow their logic), it may even be the case that much science will also be thrown out too. (This point was famously made against certain positions advanced by the logical positivists.)

For example, what about empirically-untestable string theory? Is that “neo-Scholasticism”? What about some of the well-known mathematical and logical problems? That is, the ones which can be seen as “intellectual puzzles” and nothing more?

Everything must go?

Michael Loux & Philip Goff Discuss Nominalism About Universals

 

[Although the quotations in this piece have page numbers, I can't remember the books or papers from which they were taken. Some readers may know.]

Nominalist Primitivism

The realist as regards universals believes that many particulars exemplify a particular universal and that this is a ‘primitive’ or ‘basic’ fact about the world. The nominalist, on the other hand, thinks more or less the same about the fact that “different objects agree in attribute by all being triangular” (1).

The question now is: What do they mean by ‘primitive’ or ‘basic’?

Is anything in the world that is truly primitive? What is the ‘exemplification’ of universals in things? What do the nominalists mean by ‘agree’ in ‘different objects agree in attribute’? Perhaps Michael Loux will explain:

“… we are to take agreement in attribute to be a fundamental or unanalysable feature of the world… There are no prior facts that serve to explain these facts; they constitute the primitive materials out of which we construct our story of the world.”

Of course the notion of fundamentality or unanalysability has been used many times before in the history of philosophy. For G.E. Moore, for example, the property goodness was deemed to be unanalysable. For Wittgenstein (in the Tractatus), simple objects were seen in the same way. In addition, truth is seen this way by many philosophers.

There must be a reason (or reasons) why we “take agreement in attribute” between different particulars. Surely we can't get away with simply saying that this agreement is ‘fundamental’ or ‘unanalysable’ – why should anyone else accept these predicates at all? Aren’t they cop-outs or neat sidesteps of the issue? Just to argue that “every ontological account must take some facts as primitive or basic” (or that physics does the same) is simply not enough of a reason or answer. We need to ask why and how they are primitive or unanalysable. There may, for example, be an argument that such things halt some kind of infinite regress. However, we still need to know why we have chosen the agreement in attributes between objects - or the exemplification of universals as properties - as such primitives.

Just as the nominalist takes the Platonic position "one step earlier" by taking “the original fact that certain things are triangular as basic” rather than relying on the basic fact of the exemplification of universals in particulars, why can’t we also take the nominalist position one step earlier by giving a more substantive and descriptive account of agreement, resemblance or whatever?

Nominalist Reformulations

The problem is linguistic, according to the nominalist. That is, we believe in the existence of universals because sentences like

“Sphericity is a shape.”

have a property (or attribute) as the subject-term (or noun-phrase) of a subject-predicate sentence. And because the name ‘Tony Blair’ in

“Tony Blair is a liar.”

refers to the person Tony Blair, we assume that the ‘sphericity’ in “Sphericity is a shape” must refer to a universal – or at least to something.

Some philosophers (including Michael Loux) believe that nominalists “deny the existence of properties” (1). If that's indeed the case, then the nominalist

“is obliged to offer analyses of all sentences which contain words apparently referring to properties in terms of sentences which contain no such words” (1).

For a start, do nominalists really deny the existence of properties or do they simply think that they're not the instantiations of universals and are thus only found in objects?

In any case, how will a simple reformulation of a sentence alone get rid of universals or properties? If I choose never to refer to God (or use the word ‘God’), that won't take God out of existence. If no one used the word ‘God’, and if everyone reformulated all the sentences which refer to God in a bona fide atheistic manner, that alone wouldn’t get rid of God (as it were).

Loux goes on to explain the nominalist position thus:

“… the sentences appear to express claims about universals, but are really just disguised ways of making claims about familiar concrete particulars… For every sentence incorporating an abstract singular term, it is possible to identify a sentence in which the term does not appear but the corresponding general term does, such that the latter sentence gives the meaning of the former.” (1, Loux, 2002: 65-66)

Of course to the layperson the sentence “Sphericity is a shape” doesn't “appear to express [a] claim about [a] universal”. It simply makes a claim about spherical objects (or, less likely, about the property being spherical). However, if ‘sphericity’ is a noun (or an abstract singular term), then this may not be logically or philosophically acceptable.

In any case, perhaps the layperson is tacitly or implicitly referring to the universal. In that case, both the realist and the nominalist are offering us (or discovering) the true logical form of the aforementioned sentence. The nominalist is arguing that the sentence is a disguised way of making a claim about a universal. The realist is saying that the logical form is apparent; though not (as it were) taken literally when it's not taken as a claim about a universal.

More technically, Loux makes a distinction between the nominalistically-unacceptable abstract singular term (‘sphericity’) and the nominalist general term which will be its substitution. What would that general-term substitution of the abstract singular term ‘sphericity’ be?

What is the substitution of

1) “Sphericity is a shape.”

that's given by the nominalist? This:

2) “All spherical things are shaped things.”

Prima facie, it's very hard to accept the latter as an analysis (or anything else) of the former! There are many differences between the two.

For a start, 2) is about spherical things. 1) isn't about things at all. In fact 1) is about a shape, not about a thing or things. 1) attributes or predicates something to the property sphericity; whereas 2) predicates something to things. We can even say that 2) is analytic (or partly analytic) in that the subjects (‘spherical things’) are by definition ‘shaped things’. 1), on the other hand, seems to offer us information about sphericity itself – that it ‘is a shape’. However, we can say that this is partly analytic too; though not in the direct manner of 2).

The bizarre thing is that 2) still contains what can be taken as references to universals anyway. Spherical things and shaped things are themselves universals, aren’t they?

Philip Goff writes that

“analysing all sentences which make reference to properties into sentences which do not make reference to properties is a tricky business”.

Isn’t it the case that in

“All spherical things are shaped things.”

the predicates ‘spherical’ and ‘shaped’ do indeed “make reference to properties”? It's just the case that ‘spherical’ and ‘shaped’ are adjectival terms; whereas ‘sphericity’ is an abstract singular term or a noun. Does that difference alone make the above fully nominalist or even nominalist at all? Surely adjectives like ‘spherical’ and ‘shaped’ are just as parasitical on universals as the abstract singular term ‘sphericity’ is!

I would say that

“All spherical things are shaped things.”

proves Goff’s point that there

“are true sentences containing reference to properties which are not analysable into sentences which do not contain reference to properties” (2).

Yes, the above ‘analysis’ is proof of this. Goff calls this the principle of semantic irreducibility (PSI). The “reference to properties” is unavoidable in the nominalist analysis. Thus it's semantically irreducible.

Despite the problem for such nominalist reductions, Loux argues that such reductions or analyses are “essential for avoiding a commitment to properties” (2). That is, not properties per se; but property-terms which are abstract singular terms which themselves can’t help but refer to universals. Thus we can ask if Loux is against both ‘sphericity’ and ‘spherical’ (or ‘shaped’) or only against the nominalist use of ‘sphericity’. It must be both because in

“All spherical things are shaped things.”

the nominalist does get rid of the predicate ‘sphericity’. The question is, is it an acceptable analysis (or reduction) and is it the case that the predicates ‘spherical’ and ‘shape’ don’t refer to properties and therefore to universals? According to Goff, then, Loux is committed to the principle of the necessity of reduction (PNR). Or, rather, Loux thinks that the nominalist must, or should, be committed to this principle and thus be able to formulate an acceptable reduction of all the questionable sentences which include references to universals. If the nominalist can't carry out such a reduction, according to Loux, “then properties exist” (2).

Isn’t all this very linguistic in nature? That is, even if the nominalist can’t successfully offer us reductions or analyses of the suspect sentences which include references to universals, does that automatically mean that universals must therefore exist? Similarly, if the nominalist can (or could) carry out such reductions, would that automatically mean that universals don't exist? Surely what we can and can't do with our sentences (or with our grammar) doesn't affect the existence or non-existence of universals. People use the predicates ‘God’, ‘Pegasus’ and even ‘the round square’, Quine uses the word ‘meaning’ and Paul Churchland uses the word ‘belief’. These people don't believe that these things exist. So why should the sentence “Sphericity is a shape” bring the universal sphericity into existence (as it were)? Likewise why should the sentence

“All spherical things are shaped things.”

stop universals from existing or even prove their non-existence (even if the above is an acceptable analysis or reduction)? Perhaps, then, such concerns are normative and grammatical in nature. That is, if you think that universals don't exist, then say

“All spherical things are shaped things.”

and that will be fine. On the other hand, if you think that universals do exist, then say

“Sphericity is a shape.”

and that too will be fine. Your particular formulation will show other people your ontological commitments. It will neither prove nor disprove the existence of universals, just as Bertrand Russell's

“There is at least one thing, such that this none thing is king of France.”

didn't prove (or even show) that the king of France did not in fact exist. It showed us, instead, that the formulator himself doesn't believe that the king of France exists. (Perhaps the formulation alone doesn't show us that the king of France doesn't exist.) And neither does

“All spherical things are shaped things.”

show us that the universal sphericity doesn't exist, just as

“Sphericity is a shape.”

doesn't show us that the universal sphericity does exist.