One of Berkeley's principal arguments against material objects, the so-called Master Argument, begins with a challenge and a boast. ‘If you can conceive a material object’, he says, ‘I'll grant you that there are such.’ The challenge is to conceive a material object; the boast comes in the form of an extravagant reward if you can, a reward that he's sure you can never collect: admitting that he's wrong and that there are material objects. Taken together, the challenge and boast amount to his assurance that it's impossible that material objects exist. After all, common sense tells us that no one would ever admit that something exists merely on the grounds that it could be conceived, and so possibly exists. No one would admit that a three-headed dragon exists, for example, merely on the grounds that a three-headed dragon can be conceived. To be willing to admit that something exists merely on the grounds that it can be conceived means that you're sure it can't be conceived, certain that it isn't possible.

In Three Dialogues between Hylas and Philonous, the argument is presented dialectically:

Much the same argument is presented in quasi-dialectical form in the Principles of Human Knowledge.

1. 22. … I am willing to stake my whole position on this: if you can so much as conceive it possible for one extended movable substance – or in general for any one idea or anything like an idea – to exist otherwise than in a mind perceiving it, I shall cheerfully give up my opposition to matter; and as for all that great apparatus of external bodies that you argue for, I shall admit its existence, even though you cannot either give me any reason why you believe it exists, or assign any use to it when it is supposed to exist. I repeat: the bare possibility of your being right will count as an argument that you are right.

2. 23. ‘But’, you say, ‘surely there is nothing easier than to imagine trees in a park, for instance, or books on a shelf, with nobody there to perceive them.’ I reply that this is indeed easy to imagine; but let us look into what happens when you imagine it. You form in your mind certain ideas that you call ‘books’ and ‘trees’, and at the same time you omit to form the idea of anyone who might perceive them. But while you are doing this, you perceive or think of them! So your thought-experiment misses the point; it shows only that you have the power of imagining or forming ideas in your mind; but it doesn't show that you can conceive it possible for the objects of your thought to exist outside the mind. To show that, you would have to conceive them existing unconceived or unthought-of, which is an obvious contradiction. However hard we try to conceive the existence of external bodies, all we achieve is to contemplate our own ideas. The mind is misled into thinking that it can and does conceive bodies existing outside the mind or unthought-of because it pays no attention to itself, and so doesn't notice that it contains or thinks of the things that it conceives. Think about it a little and you will see that what I am saying is plainly true; there is really no need for any of the other disproofs of the existence of material substance. (Berkeley, Principles of Human Knowledge, Sections 22–23)

A material object, as commonly understood, and as Berkeley more or less understands it as well, is an object that (a) exists outside minds, (b) does not depend for its existence on minds, (c) is extended or spread out in a public space, and (d) is, in principle, accessible to many minds. Stars and rocks, for example, as usually thought of, are material objects. It's (a) and (b), and especially (b), that are the focus of Berkeley's argument for the impossibility of material objects. Both are essential to the notion of a material object, so if nothing could satisfy both, nothing is or could be a material object.

Informally, the argument is: try to conceive a material object as such, an object outside all minds, an object that no one ever has perceived or conceived or ever will perceive or conceive. That's easy, I might reply: I'm right now conceiving a stone that no one ever has perceived or conceived or ever will perceive or conceive. But Berkeley's reply is, No, you're not. You've conceived the stone, so the stone is not an object that no one has ever conceived or ever will conceive (dropping the idea of perceiving hereafter for simplicity's sake, as in effect Berkeley does as well). The stone doesn't exist outside all minds. The same is true of any supposed material object, be it a star or a street. A stone or a star or a street or any other supposed material object that exists unconceived, then, cannot be conceived, and so is inconceivable. But since what is inconceivable is impossible, material objects, because of their very nature, are impossible.

Like all philosophical arguments, this one seems strong at first blush, or, if it doesn't, at least where the mistake lies isn't immediately obvious. Its conclusion, however, is so counterintuitive that there is a strong suspicion that something must be wrong. That suspicion can be reinforced by a refutation by logical analogy, that is, an argument strictly parallel to the one to be refuted – in this case, Berkeley's – but with a conclusion that's obviously incorrect.

The analogous argument is this. Try to conceive a natural number as such, but a natural number outside all minds, a number that no one ever has conceived or ever will conceive, a number that doesn't depend for its existence on any mind at all. That's easy, I might reply: I'm right now conceiving a very large natural number but one that no one ever has conceived or ever will conceive. But Berkeley's reply is, No, you're not. You think you are but you're not. You've conceived the number, so the number is not one that no one has ever conceived or ever will conceive. The number doesn't exist outside all minds. The same is true of any number, be it a fraction or a transcendental number. A natural number or a fraction or a transcendental number or any other number that is unconceived, then, cannot be conceived, and so is inconceivable. But since what is inconceivable is impossible, there are not, nor could there be, any numbers that are not conceived.

Even on mathematical constructivism, however, this isn't correct. The number of natural numbers is infinite but we're finite creatures. Some natural numbers, then – in fact, an infinity of them – will never be conceived by us. Thus the conclusion of the above argument is false. The refutation by logical analogy is possible because Berkeley's argument is topic neutral, and very nearly purely formal: it speaks of material objects but material objects aren't essential to it. What is essential is simply the idea of my conceiving something, that and nothing more. It's for that reason that it applies just as well to anything that supposedly can exist unconceived, anything that satisfies (a) and (b). If Berkeley is correct, all such objects would be proved to be inconceivable, and so impossible, and everything that exists would have to be, of necessity, mind dependent.

The last point can be bolstered by a second, related refutation by logical analogy, one that focuses on a different aspect of Berkeley's argument. The original argument challenges me to conceive an object – a purported material object – that exists unconceived, and when I can't, it's concluded that the object is impossible. But since Berkeley's argument is topic neutral, a parallel argument, an argument parallel to the original argument or the one about numbers, shows that I should also conclude that it's impossible that minds other than my own exist unconceived – exist unconceived by me, that is. Other minds are on a par with material objects and numbers as far as the argument is concerned. The result is, in effect, solipsism. I would apologize to my readers for this conclusion, but since there are no readers other than myself to whom to apologize, apologies are apparently unnecessary.

Worse still, on the argument I myself can't exist unconceived. The argument applies to me, the conceiver, just as surely as it does to any mind. In other words, the topic neutrality of the argument catches me and everything else in its net. It's impossible that I exist unconceived by me; my existence is dependent on my conceiving me. Thus it's not that I can and do conceive things because I exist; it's not that my existence makes my conceiving possible. Rather, it's that I can and do exist because I conceive myself; it's my conceiving me that makes my existence possible. Descartes, then, is wrong in the cogito, for he there assumes that his existence makes his conceiving possible. Descartes, a wag might say, put da cart before the horse.

But the axe falls with a thud when the last point is extended. If we assume, as Berkeley does, that his argument shows not only that stones, trees and the rest depend for their existence on being conceived by a mind, but that they exist in and by being conceived by a mind, the position becomes utterly incoherent. My conceiving me would be a content of my mind, and my mind would be the content of my conceiving me. That seems unintelligible. Taken to the bitter end, Berkeley's argument self-destructs.

These increasingly strong form-alone-essential refutations by logical analogy show that something is wrong with Berkeley's original argument. They don't, however, pinpoint where and what the problem is. In some sense, the situation is like that of a reductio ad absurdum in mathematics. A reductio will show that a given mathematical proposition must be true, but only because deep trouble, an inconsistency, is the result if it's not. With a reductio, a positive argument, pointing directly at the truth of a proposition, is lacking. Similarly, the refutations by logical analogy above show that there must be something wrong with Berkeley's argument, but only because deep trouble, ultimately an incoherency, is the result if there isn't. A positive argument, pointing directly at the problem, is lacking.

It can be provided, though. Berkeley's argument is basically:

1. (1) I conceive x, x being an object that has never been or will be conceived. [Assumption]

2. (2) x is conceived by me and x has never been or will be conceived. [From (1)]

3. (3) x has metaphysically incompatible properties. [From (2)]

4. (4) It's impossible that x, an object with metaphysically incompatible properties, exists. [From (3) and the principle that nothing can have metaphysically incompatible properties]

5. (5) It's impossible that an object that has never been or will be conceived by me exists. [From (1) and (4)]

The problem in the argument concerns (1), (2) and (5). Interpreted correctly, (1) means only that I form a concept whose content is an object that has never been or will be conceived. Premise (2), however, interprets premise (1) as stating that I form the concept of a particular thing (e.g. a particular stone, number or mind) that has never been or will be conceived, that is, that no one ever forms a concept of. But a particular thing that I conceive and that is never conceived is, of course, an impossibility, just as a horse that I ride and that is never ridden is. Generalizing from this, it's concluded that no object that has never been or will be conceived can exist. That would be a legitimate conclusion if the content of the concept of an object that has never been or will be conceived is internally inconsistent. But that's not what has been proved. What's been proved is that it's impossible to give an example of something of which no one has ever or will ever give an example. The principal fallacy of the argument, then, is equivocation: (1) must be understood one way in order to capture the concept of a mind-independent object, but then, in (2), must be interpreted differently, in terms of a particular object that is never conceived, in order to provide the materials for the deduction of an impossibility. The impossibility is then reinterpreted, in (5), in terms of the content of the concept of a mind-independent object being internally inconsistent. It's the conclusion of the deduction that invites the common, but inaccurate, characterization of the fallacy of the argument as the introduction of the act of conceiving into the content of what is conceived. Rather, an equivocation makes both possible and explicable that inaccurate characterization. The real problem is that one relatively trivial thing is proved, while another far from trivial thing is thought to be proved, and the mistake in so thinking is facilitated by an equivocation. We must speak by the card, as Hamlet says, or equivocation will undo us – or at least undo us of material objects.