Objection to Berkeley's Master Argument

Question

The Master Argument (roughly) states that it is not possible for sensible objects to exist without a mind. Now part of Berkeley's Argument goes as follows:
Suppose something exists without being conceived of by anyone, in thinking of such an such an object you are actually conceiving of it and that is a Contradiction with it not being conceived of.
Now my problem with this is that there is no way of formalizing this. Since any such formalization must include a premise Q(x) which states that x is inconceived of. Now for Berkeley's Statement to hold that "in thinking of such an such an object you are actually conceiving of it", there must be a proposition such as Q(x) => ~Q(x) as a premise,so he can then arrive at the contradiction, given the semantics of Q that is simply not true.
Now my question is this:
Whether the Argument I provided is correct and whether it's the same as Russell's objection, concerning mixing the Representation with the Representative.

Answer 1

You seem to be wanting to argue that Berkeley's argument isn't even valid. I don't think that's right. Berkeley's point could be made formal as so:

1. There is an object o such that nobody conceives of it. (Premise)
2. If (1) is true, then o is such that somebody conceives it. (Premise)
3. o is such that somebody conceives it. (1, 2 modus ponens).
4. Therefore, o is such that nobody conceives it and somebody conceives it. (By conjunction introduction on 1, 4)
5. Contradiction.

The argument given is valid--if the premises were true, the conclusion would be true. But since no contradictions are ever true, then we have to give up either (1) or (2). Giving up (1) is what Berkeley wants us to do. To force us on to this desperate path though Berkeley needs to give us reason to think (2) is true.

That's what I interpret him to be doing in the snatch of quoted dialogue. Presumably he'd have to say something like, "You have to recognize (2) is true, because just by reading and understanding (1) you have begun to think about the object o!" But that doesn't sound right, and Russell gives the reason why: It confuses the properties of our mental representation of an object for properties of the object so represented. Russell's theory of knowledge by description (as opposed to acquaintance) in Problems of Philosophy pp. 42-45 seems to explain how this works.

It is like Berkeley is saying that just by reading premise (1) I have direct acquaintance with o, and in that sense if I do have acquaintance with it, I am able to "conceive" of it, perhaps, as (2) seems to say. However, if we take (1) to give me merely knowledge by description instead, then (2) looks clearly false since I can know of the existence of o by description without being myself acquainted with it, which is what (2) required.

Answer 2

Berkeley's argument might be formalized as follows:

• Ax = x is an assumption
• Cx = x is conceived of

{1}      1.   ∀x[Ax → Cx]                  Prem.
{2}      2.   ~Ca                          Assum.
{3}      3.   Aa                           Assum.
{1}      4.   Aa → Ca                      1 UE
{1,3}    5.   Ca                           3,4 MP
{1,2,3}  6.   Ca & ~Ca                     2,5 &I
{1,3}    7.   Ca                           2,6 RAA
{1,3}    8.   ∀x[Cx]                       7 UI - INVALID
{1,3}    9.   ~~∀x[Cx]                     8 DNI - INVALID
{1,3}    10.  ~Ǝx[~Cx]                     9 QI - INVALID

First of all, line 3 involves a meta argument which I believe might be formalized in a better way than how I did it. Even so, I don't find it very controversial because it only asserts that the assumption of a on line 2 implies that a is an assumption. With that, I invoke the premise which I added that everything which is assumed is conceived of. Even though there might be a better way to formalize it, the main point is that it can be formalized. Even so, I believe his argument has weakness for other reasons.

What I find most questionable is the application of a rule of inference in a way for which it wasn't intended. It's common in logic to assume a typical instance (as on line 2) in order to infer a general principle (as on line 8). However in this case the general principle that it being inferred involves the act itself of assuming. I believe it's clearly fallacious to conclude that the ability to make an assumption implies that all such assumptions are actually made; it only implies that they can be made. Unless it can be shown that all things are actually conceived of, the argument doesn't hold.

It might be argued that this weakness only has to do with the particular way that I represented it, but Berkeley's argument is really asserting that very thing. He is not allowing for the possibility of something to exist without an actual instance of it being conceived. The potential to conceive is not the same as actually conceiving, so no valid inference can be drawn. For that reason, Berkeley's argument doesn't hold.

Edit:

The problem with the proof as I originally presented it is that line 3 should have been entered as an assumption and not as a consequence of line 2. (I've edited it to show the correction.) Because of that, a is a free variable, so the application of Universal Introduction is not valid:

"If we are tempted to generalise universally on a formula containing a particular name we can always look back at previous lines of proof, checking carefully that there is no line which contains a formula telling us something special about that named individual, e.g. that it has a particular property or properties. In other words, we must always take care to ensure that the particular name we use in such a context refers to an element of the domain which is genuinely typical in the context of that use." (Paul Tomassi, Logic, p. 277)

Answer 3

I don't follow your exact objection. It seems like taken seriously, your objection would rule out proofs by contradiction altogether. From Q(x) -> ~Q(x), we do not deduce Q(x) is not a meaningful statement, only that the statement is not true. And that is exactly what Berkeley is after, for your Q not to be true of any x.

But back off and look at examples. By Berkeley's argument, as you have presented it, the bicycle had been conceived of before fire was discovered. Once any caveman reached the concept that there were things of which he had not yet conceived, every one of those things was immediately conceived of, at least in the sense Berkeley seems to be using here. So this first human to doubt his own omniscience, at that very time, conceived of bicycles and microwave ovens? That is clearly nonsense.

The problem is that conceiving of something is not transitive (or more formally 'idempotent'): conceiving of conceiving of something is not the same thing as conceiving of it. I can conceive of conceiving of the most important technological innovation for the next thousand years. But I cannot conceive of the innovation itself, or I would be able to start development on it right now. I would at least be able to tell you the basic idea. And I can't.

Any attempt to formalize the argument is going to show the telescoping of two nested "failures to conceive" into one, and point up the flaw in the argument.

---

An attempt at a formalism:

C = 'concepts of'

1. Choose y in {x: C(x) = {}}
2. The reference to y in 'Choose y' in 1 is in C(y)

   (** This is erroneous)

3. So C(y) and not C(y)
4. So there is no y
5. Therefore for no x is C(x) = {}

Unfortunately the reference to y in 'Choose y' is not a concept of a given thing, it is a concept of a thing lacking a concept.

This involves the concept of the thing having a concept, so we are close. But we can close the gap only if C(C(x)) = C(x). By some slippery English, you can make it seem like the concept of the concept of something implies there being a concept of it. But it is not so.

It falls to the bicycles-and-cavemen example. The concept of that which I have not yet imagined does not provide the concept of a bicycle just because bicycles are something I have not yet imagined. The technical reason is that intensionally defined sets need not have extension (evaluation of an iterator can be lazy, so the constructors implicit in the enumeration may remain uncalled, for the functional programmers out there).