By Paul Hsieh

Date: Tue, 6 Sep 1994 18:09:53 EST

From: Paul Hsieh hsiehpcrl.com

To: Multiple recipients of list AYN-RAND ayn-rand@iubvm.ucs.indiana.edu

Subject: Objectivist Epistemology and the A/S Dichotomy

Kudos to Jimbo for his eloquent essay on epistemology! In his essay, he briefly touches upon the topic of "analytic" vs "synthetic" truths (some people use the terms "necessary" vs "contingent" truths). He states his position (which is in basic agreement to Peikoff's position in ITOE) as follows:

Every true proposition can be recast in a form analogous to "X (which

means X, the existent, including all its characteristics) is: one or

more of the things which it is." That is, "X is what it is." Every false

proposition, similarly, can be recast analogously to "X (which means X,

the existent, including all its characteristics) is: one or more of the

things which it is not." Or, "X is not what it is." (See ITOE, p. 100.)

It is critical here to go back to our discussion of truth for a moment,

and reconsider the fact that it is possible to form a valid proposition

without knowing the truth status of that proposition. It is therefore

possible to examine a contradictory proposition without realizing that

the proposition is contradictory. Contradiction is a fact about

valid false propositions which depends on the proposition's relationship

to reality, not upon our full awareness of that relationship. Failure

to realize this is one of the greatest sources of confusion with

respect to the traditional dichotomies of "analytic versus synthetic"

propositions or "logically true versus empirically true" or "logically

possible versus empiricaly possible."

It is true that some _sentences_ violate the rules of construction

appropriate to man's means of cognition, e.g. "Quarks are not quarks,"

such that we can know without further investigation that the sentence

does not designate a true proposition. In all other cases, some degree

of additional evidence is necessary to determine truth value -- as we will

see in the next section.

I'm not sure, but I think Jimbo is saying that there is no dichotomy between two true statement like: (1) "This chunk of ice in my glass is shaped like a cube ." and (2) "This chunk of ice in my glass is made of H2O."

(Trust me, the chunk of ice really is shaped like a cube!) Those who advocate the analytic-synthetic dichotomy would claim that (1) is a synthetic truth, since that chunk of ice could have been shaped like, say, a sphere. Those same people would say that (1) is an analytic truth since "ice" is defined as the solid phase of water (i.e. H2O), which under normal conditions becomes solid when the temperature falls below 32 degrees Fahrenheit. Hence, they would say, (2) is *necessarily* true in a way that (1) isn't.

On the other hand, Objectivists would take a different approach. They would state that both sentences use the phrase "this chunk of ice" to refer to a specific existent. Hence, anyone who makes a statement about that existent, must (at least implicitly) grant as truths all attributes about that existent. Hence, statement (1) is just as necessarily true as statement (2), since one of the attributes of the existent (the chunk of ice) is that it is shaped like a cube.

Jimbo acknowledges that to confirm the truth of (1) might take more investigation and effort than to confirm the truth of (2). (After all, (2) can be confirmed simply by recalling the definition of ice, whereas (1) can be confirmed only by taking the time to look at the ice cube and examine its shape.) But that is an epistemological issue. In a metaphysical sense (according to Objectivists), both are "necessarily" true statements -- there is no dichotomy.

I'd like to explore this a little further. For the sake of discussion, I want to examine three propositions. To simplify things, I will make them mathematical propositions, just so we don't have to worry about potentially confounding issues of evidence and certainty. We'll take as our context, Euclidean plane geometry. We can define a triangle as: a polygon in a Euclidean plane with three vertices and three sides, where the "vertices" are three points on a plane which are non-collinear (i.e., they don't all lie on the same line), and where a "side" is a straight line segment connecting adjacent vertices.

Let's assume that I have drawn a specific triangle on this Euclidean plane. We'll call this triangle T. My three statements are as follows: (1) Triangle T has three sides.(2) Triangle T has internal angles that add up to 180 degrees.(3) Triangle T is equilateral -- all three sides are of equal length.All three of these statements are either true or false. For the sake of argument, let's say that I picked my triangle T so that it is an equilateral triangle. Hence all three of these statements are true. But are they all true in the same way? Let's see what it takes to verify the truth of these propositions.

Number (1) is easy. We can tell right away that it is true, since the definition of a triangle explicitly includes the condition that it has three sides.

Number (2) is a bit harder. The definition actually does include all the information necessary for us to conclude that it is true. But the logical steps we have to take to reach this conclusion are less obvious (at least for those of us who aren't math geniuses). Everyone who has taken high-school geometry has worked through the steps of the chain of reasoning behind this particular statement. The truth of (2) follows as directly and surely from the definition of a triangle, although it follows *implicitly*, not explicitly as in statement (1).

Number (3) is a little different. To ascertain the truth value of , we actually have to examine specific details of triangle T. With statements (1) and (2) we didn't have do this.

This suggests that there is some sort of difference between (1) and (2) on one hand, and (3) on the other hand.

The difference is not in the amount of mental energy we have to expend. After all, to ascertain the (3) is true is probably easier than to verify that (2) is true. (Maybe not for all people. But in that case, we can substitute a different version of (2) which requires a more difficult geometric proof, yet still follows from the definition of "triangle".) No, I think that the main difference is in the fact that one can take statements (1) and (2) and modify them to be true statements about *any* triangle. If we had several other triangles, T1, T2, T3, ... Tn, and we just replaced the term "T" with "T1" (or "T2" or whatever), then the new versions of (1) and (2) would still *always* be true. On the other hand, the new versions of (3) might be true or false, depending on specifics of the individual triangles.

What's the best way to describe this difference between the two types of truths? Since this is an Objectivist mailing list, I will use Objectivist terminology (even if I personally find it a little bit awkward and unfamiliar.) The various triangles (our original triangle T, as well as the new triangles T1, T2, etc.) are all specific existents.

However, they can be subsumed into the concept "triangle" by what Objectivists call "omitting the measurements" (i.e. leaving out the specifics about the length of the sides and focussing on the fact that they all have exactly 3 sides and 3 angles).

We've seen that statements (1) and (2) can be modified to refer to any triangle as long as it is a triangle and they will still remain true statements -- the exact measurements of the triangle are irrelevant. On the other hand, the truth of statement (3) is critically dependent on the specific measurements of the triangle being considered. And the truth value of versions of (3) will change as the specific measurements change.

So perhaps we should say that true statements like (1) and (2) are "measurement-independent" whereas true statements like (3) are "measurement-dependent". Although all three statements are true with respect to triangle T, this distinction is important. Much of mathematics (and philosophy in general) is concerned with identifying fundamental principles and truths about broad categories of existents. At least in mathematics, many (?most) of the important truths are of the "measurement-independent" kind.

Examples would be the following: (In mathematics) "This triangle (and any other triangle) has internal angles that add up to 180 degrees." or "This plane map (and any other plane maps) only requires 4 colors to guarantee that all adjacent regions have different colors from one another." or "This matrix with a determinant of zero (and any other matrix with a determinant of zero) is non-invertible".

(In philosophy) "This man (and anything which is a conscious rational living being) has a right to his own life and his own private property."

The fact that these truths are measurement-independent makes them far more powerful (and interesting) than truths which are measurement-dependent. Because they are measurement-independent, they can be applied to every member of the category in question ("triangle" or "plane map" or "matrix with zero determinant" or "man").

When I read Peikoff's original essay in ITOE on the analytic-synthetic dichotomy, I got the sense that he was attacking it because some philosphers then take that notion and use it to conclude (erroneously) that so-called "analytic" truths are merely definitional and devoid of meaningful content, whereas so-called "synthetic" truths are always just a little bit uncertain.

I agree with Peikoff that those conclusions are unwarranted. But I don't want to throw out the baby with the bath water. I think it is important to recognize the difference between "measurement-dependent" and "measurement-independent" truths. If we are aware of this distinction, we can direct our thinking in the direction we want. A mathematician or philospher might wish to spend more time working on what he considers to be the more challenging measurement-independent truths, and less time on the other type.

Brief aside: I'm not saying that measurement-dependent truths are worthless. There are plenty of mathematicians that devote their lives to evaluating the properties of a specific entity, such as the real number line, and the truths they discover are not extendible to other entities like general topological spaces (of which the real number line is merely one example). There are several (actually an infinite number of) general topological spaces. But they all have a few key characteristics in common, and the concept of "general topological space" can be derived by considering these key characteristics, and "omitting the measurements" that distinguish these specific mathematical spaces/structures from one other.

And even in this case, a good mathematician who knows the ins-and-outs of various true statements about his particular entity (such as the real number line), is also keenly aware of which truths are specific to that particular structure, and which are a consequence of the fact that the real number line is a general topological space. In other words, he knows which truths are measurement-dependent and which are measurement-independent, in this setting. Again, this knowledge helps guide his thinking.

So let's return to our original ice chunk example. Statement (1) about the piece of ice is an example of a measurement-dependent truth, since it depends on some of the specific measurements that distinguish this particular chunk of ice from other chunks of ice. Statement (2) is an example of a measurement-independent truth, since it applies to any chunk of ice.

Paul S. Hsieh

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