I attempted, in the “Theories of Knowledge … in Parallel Outline 1 Revisited” of January 2, 2014, to simplify that which I had presented in my December 30, 2013, post, “Theories of Knowledge and Theories of Linguistic Representation in Parallel Outline 1.” I did this in order to make the things said in the latter post a bit more approachable for those who, as I said in the former, have not wasted spent much of their adult lives in the study of academic philosophy. In the present post I wish to effect a simplification proceeding in a rather different direction. That is, I wish to present the defining theses of some theories of knowledge and some theories of linguistic representation in such a way that their logical “skeleton” or “form” is rendered absolutely obvious, at least for those who have delved a bit into elementary “mathematical” or “symbolic” logic.

The presentation will suffer from the minor practical difficulty that I don’t know how to get the WordPress apparatus upon which this blog depends to express the upside-down “A” which logicians use for the universal quantifier and the reversed “E” which logicians use for the “existential”quantifier. What I will do, then, is resort to expressing “for any existent x” by means of “(Ax)” and “there is an existent x” by means of “(Ex).” In the present context that will lead to no confusion.

1. So, then, I will express by means of:

(Ex)(Ey)(Kxy)

This abbreviates the thesis that:

There is an existent x and there is an existent y, such that x is a knowledge of y.

or, more idiomatically:

Some existent x is a knowledge of some existent y.

or, even:

Some existent is a knowledge of some existent.

This you will recognize as similar to the thesis identified in “Theories of Knowledge and Theories of Linguistic Representation in Parallel Outline 1” as a version of epistemism, i.e.,

There actually is a knowledge of some existent.

Now, seven and seven only additional theses can be expressed by means of the negation of one, two, or all three of the three components of (Ex)(Ey)(Kxy).

2. Thus:

(Ex)(Ey)~(Kxy)

This abbreviates the thesis that:

There is an existent x and there is an existent y, such that x not is a knowledge of y.

or, more idiomatically:

Some existent x is not a knowledge of some existent y.

or, getting rid of the variables:

Some existent is not a knowledge of some existent.

3. And:

(Ex)~(Ey)(Kxy)

This is logically equivalent to:

(Ex)(Ay)~(Kxy)

This abbreviates the thesis that:

There is an existent x such, for any existent y, x not is a knowledge of y.

or, more idiomatically:

Some existent x is not a knowledge of any existent y.

or, getting rid of the variables:

Some existent is not a knowledge of any existent.

4. And:

(Ex)~(Ey)~(Kxy)

This is logically equivalent to:

(Ex)(Ay)~~(Kxy)

in turn equivalent to :

(Ex)(Ay)(Kxy)

This abbreviates the thesis that:

There is an existent x such, for any existent y, x is a knowledge of y.

or, more idiomatically:

Some existent x is a knowledge of any existent y.

or, getting rid of the variables:

Some existent is a knowledge of any existent.

5. And:

~(Ex)(Ey)(Kxy)

This is logically equivalent to:

(Ax)~(Ey)(Kxy)

and to:

(Ax)(Ay)~(Kxy)

This abbreviates the thesis that:

For any existent x and for any existent y, x is not a knowledge of y.

or, more idiomatically:

Any existent x not is a knowledge of any existent y.

or, getting rid of the variables:

Any existent not is a knowledge of any existent.

A still more idiomatic equivalence would be:

No existent is a knowledge of any existent.

This you will recognize as similar to the thesis identified in “Theories of Knowledge and Theories of Linguistic Representation in Parallel Outline 1” as a version of skepticism, i.e.,

There is no knowledge of any existent.

6. And:

~(Ex)(Ey)~(Kxy)

This is logically equivalent to:

(Ax)~(Ey)~(Kxy)

and to:

(Ax)(Ay)~~(Kxy)

and to:

(Ax)(Ay)(Kxy)

This abbreviates the thesis that:

For any existent x and for any existent y, x is a knowledge of y.

or, more idiomatically:

Any existent x is a knowledge of any existent y.

or, getting rid of the variables:

Any existent is a knowledge of any existent.

7. And:

~(Ex)~(Ey)(Kxy)

This is logically equivalent to:

(Ax)~~(Ey)(Kxy)

and to:

(Ax)(Ey)(Kxy)

This abbreviates the thesis that:

For any existent x there is an existent y such that x is a knowledge of y.

or, more idiomatically:

Any existent x is a knowledge of some existent y.

or, getting rid of the variables:

Any existent is a knowledge of some existent.

8. And, finally, the last of the pertinent possibilities:

~(Ex)~(Ey)~(Kxy)

This is logically equivalent to:

(Ax)~~(Ey)~(Kxy)

and to:

(Ax)(Ey)~(Kxy)

This abbreviates the thesis that:

For any existent x there is an existent y such that x is not a knowledge of y.

or, more idiomatically:

Any existent x is not a knowledge of some existent y.

or, getting rid of the variables:

Any existent is not a knowledge of some existent.

It remains to take note that we can quite easily present statements of the corresponding eight defining theses in the theory of linguistic representation by substituting “LR” and “linguistic representation” for “K” and “knowledge” in the eight formulae and theses presented above. And, while we are at it, we might as well also take note that we can just as easily present statements of the corresponding eight defining theses in the theory of representation tout court by substituting but “R” and “representation” for “K” and “knowledge” in the eight formulae and theses presented above.

Again, still while we are at it, we might as well also take note that we can just as easily present statements of the corresponding eight defining theses in the theory of love by substituting but “L” and “love” for “K” and “knowledge” in the eight formulae and theses presented above. Etc.

The far more important steps are those of understanding those theses and of discovering and weighing the arguments for and against them. I am hopeful that I will take at least some of those steps and report on where they have taken me in, if not due course, at least in some course.