Mike Huemer offers the following summary of this first argument:

"1. In order to know something, I must have a good reason for believing it.

2. Any chain of reasons must have one of the following structures: Either

    (a) it is an infinite series,

    (b) it is circular, or

    (c) it begins with a belief for which there are no further reasons. But,

3.  I cannot have an infinitely long chain of reasons for any of my beliefs.

4. Circular reasoning cannot produce knowledge.

5.  Nor can I gain knowledge by structure 2c, for

    (a) I would not know my starting beliefs to be true (from 1) and

(b) I cannot gain knowledge by deriving it from assumptions that I do not know to be true.

6. Therefore I cannot know anything."        (9-10)

2.  The Problem of the Criterion

        Mike Huemer illustrates this second argument in terms of a community that uses the method of the “Magic Eight-Ball”, and that, when this method is challenged, uses that method to determine whether the method itself is reliable.

        Mike Huemer offers the following summary of this skeptical argument:

"1. All my beliefs are formed by some method.

2. I am justified in accepting a belief formed by method M only if I am first justified in believing method M is reliable.

3.  I do not have an infinite series of belief-forming methods.

4.  Thus, all of my beliefs must rest on beliefs formed by methods whose reliability has not first been established.  (from 1 and 3)

5. Therefore none of my beliefs are justified.  (from 2 and 4)"        (13)

3.  How Can You Get outside Your Head?

        This argument has the following overall structure:

(1) One cannot have any knowledge of a world external to one’s mind unless either direct realism is true or indirect realism is true.

(2) Direct realism is false.

(3) Indirect realism is false.

(4) Therefore, one cannot have any knowledge of a world external to one’s mind.

            How are (2) and (3) to be established?  In the case of (2), Mike Huemer mentions the sort of argument alluded to in the first chapter, which he summarizes as follows:

"1.  As your focus shifts to the background, the fingerlike thing you are seeing splits in two.

2.   No physical object splits in two at this time.

3.   Therefore, the thing you are seeing is not a physical object."    (14)

        The next question is how (3) is to be established.  Here Mike Huemer sets out Hume’s famous argument.  This he summarizes as follows:

"1.  In order to have knowledge of the physical world, we must be able to know that our sense data are caused by physical objects.

2.  In order to know that A causes B, one must have experience of A and B.

3.  We have no experience of physical objects.

4.  Therefore, we do not know that physical objects cause our sense data.  (from 2, 3)

5.  Therefore, we have no knowledge of the physical world.  (from 1, 4)"    (16)

4.  The Brain in a Vat

        Mike Huemer summarizes this argument as follows:

"1.  Your sensory experiences are the only evidence you have for propositions about the external world.

2.  The BIV scenario predicts that you would be having the same sort of sensory experiences as you are actually having.

3.  Therefore, the sensory experiences that you are actually having are not evidence that the BIV scenario isn’t true.  (from 2)

4.  Therefore, you have no evidence that the BIV scenario isn’t true.  (from 1, 3)

5.  Therefore, you don’t know that you are not a BIV.  (from 4)

6.  Therefore, you do not know anything about the external world.  (from 5)"  (17)

        Here is a different way of formulating the brain in the vat argument:

1.  Your sensory experiences are the only evidence you have for propositions about the external world.

2.  The BIV scenario predicts that you would be having the same sort of sensory experiences as you are actually having.

3.  When two theories predict precisely the same sensory experience for you, you cannot be justified in assigning a higher probability to one than to the other.

4.  Therefore, the sensory experiences that you are actually having cannot make it the case that the view that you are perceiving an external world is more likely to be true than that you are a BIV.  (from 3)

5.  If p and q are two incompatible propositions, and p is no more likely than q, then you are not justified in believing p.

6.  Therefore, you are not justified in believing that that you are not a BIV.  (from 4, 5)

7.  Therefore, you don’t know that you are not a BIV.  (from 6)

8.  Therefore, you do not know anything about the external world.  (from 7)