If All Knowledge Comes from Experience, is All Knowledge Subjective?

This is the kind of e-mail I like, brief and pointed:

Recently I've encountered an argument that runs like this:

1. All knowledge comes from experience
2. All experiences are subjective
3. Ergo, all knowledge is subjective.

I think I can argue somewhat against this argument, but I need a nice snappy response to it.

The snappiest response to this invalid argument is that it falls victim to a fallacy of equivocation: 'experience' is being used in two different senses.  Hence the syllogism lacks a middle term and commits the four-term fallacy (quaternio terminorum).

To experience is to experience something.  So we need to distinguish between the act of experiencing and the object experienced.  The act is subjective: it is a mental occurrence.  The object is typically not subjective.  For example, how do I know that there is a cat on my lap now?  I experience the cat via my outer senses:  I see the cat, feel its weight, hear it purr.  The experiencing is subjective; the cat is not.  I have objective knowledge of the existence and properties of the cat despite the fact that my experiencing is a subjective process.

Now I don't grant that all knowledge comes from experience; I grant only that all knowledge arises on the occasion of experience.   But suppose I grant premise (1) arguendo.   What (1) says is that all knowledge is knowledge of the objects of the senses.  (There is no a priori knowledge.) So we can rewrite the argument as follows:

1*. All knowledge is knowledge of sensory objects (either directly or via instruments such as microsopes).

2*. All acts of experiencing are subjective

Ergo

3*.  All knowledge is subjective.

This syllogism is clearly a non sequitur since there is no middle term.

The subjectivity of experiencing is logically consistent with the objectivity of knowledge via the senses.  There is no knowledge apart from minds.  And yet minds have the power of transcending their internal states and grasping what is real and true independently of minds.  How this is possible is a further question, and perhaps the central question of epistemology.

One way to embarrass an empiricist is to ask him how he knows propostions like (1*).  Does he know it by experience?  No.  Then, by his own principles,  he doesn't know it.  Why then does he think it is true?