[10.4.2.4.] Principles vs. Laws.

 

Consider once again Peirce’s claim that

 

anything is general in so far as the principle of excluded middle does not apply to it and is vague in so far as the principle of contradiction does not apply to it. (EP 2:351; CP 5.448; 1905)

 

If we take Peirce to have meant LEM and LNC rather than his own PEM and PC, then it appears that he meant to claim that…

 

universally quantified propositions are neither true nor false; i.e., the principle of bivalence (according to which any proposition is true or else false) does not apply to universally quantified propositions.   e.g. “All men are mortal” is neither true nor false.     Why think that this proposition, which seems  straightforwardly true, is neither true nor false?

existentially quantified propositions are both true and false.         e.g., “Some man is conceited” is both true and false.   Why think that one and the same proposition, such as the one above, is both true and false?

 

Once we see that by “principles of excluded middle and contradiction” he did NOT mean LEM and LNC, we see that this is not what he was claiming.

 

 

[10.4.2.5.] Excluded Middle and Generality.

 

Peirce’s PEM is a principle about individual subjects and subject-terms:

·         it states a necessary condition of individuality: (in the material mode) if S is an individual, then, for any property P, either S is P or S is not-P; or (in the formal mode) if “S” is an individual subject-term, then, for any predicate “P”, either “S is P” is true or “S is not-P” is true.[1]

·         So PEM (formal mode) is equivalent to the claim that for any individual (non-general) subject-term “S” and for any predicate “P,” the proposition “S is P or S is not-P” is true.

 

Peirce said that PEM does not apply to the general because it is not the case, with regard to every predicate “P” and every general subject-term “S,” that “S is P or S is not-P” is true; sometimes such propositions are false. For example:

 

“All Georgians are residents of Carroll County, or all Georgians are not residents of Carroll County.”

 

“All Georgians” is a general subject-term.

 

So some instances of “All Georgians are P or All Georgians are not-P” are false.

 

So Peirce’s claim that PEM does not apply to the general does NOT imply that general propositions are neither true nor false.

 

It therefore does not imply that the principle of bivalence is false.

 

However... Peirce apparently does reject the principle of bivalence! Only he does so for different reasons...

 

Affirmation and denial are in themselves unaffected by these concepts [viz. determination, generality and vagueness], but it is to be remarked that there are cases in which we can have an apparently definite idea of a border line between affirmation and negation. Thus, a point of a surface may be in a region of that surface, or out of it, or on its boundary. This gives us an indirect and vague conception of an intermediary between affirmation and denial in general, and consequently of an intermediate, or nascent state, between determination and indetermination. There must be a similar intermediacy between generality and vagueness. Indeed, in an article in the seventh volume of The Monist [“The Logic of Relatives,” the 1897 article in which Peirce rejects the IR account of substantial possibility] there lies just beneath the surface of what is explicitly said, the idea of an endless series of such intermediacies. We shall find below some application for these reflections. (“Issues of Pragmaticism,” EP 2:353, CP 5.450, emphasis added)

 

We will return to this point later and consider in detail Peirce’s philosophical motivations for thinking that not all propositions are either true or else false.

 

 

[10.4.2.6.] Contradiction and Vagueness.

 

Peirce’s PC is a principle about definite subjects and definite subject-terms.

·         it states a necessary condition of definiteness: (in the material mode) if S is definite, then S is not both P and not-P, or (in the formal mode) if “S” is a definite subject-term, then “S is P” and “S is not-P” are not both true.

·         So PC (formal mode) is equivalent to the claim that for any definite (not vague) subject-term “S” and for any predicate “P,” the proposition “S is P and S is not-P” is false.

 

Peirce says that PC does not apply to the vague because it is not the case, with regard to every predicate “P” and every vague subject-term “S,” that “S is P and S is not-P” is false; sometimes such propositions are true. For example,

 

“Some philosophers own dogs and some philosophers do not own dogs.”

 

“Some philosophers” is a vague subject-term.

 

So some instances of “Some philosophers are P and some philosophers are not-P” are true.

 

So Peirce’s claim that PC does not apply to the vague does not imply that vague propositions are both true and false.[2]

 

 

[10.4.3.] Modal Propositions.[3]

 

Yet another connection between generality and necessity, and between vagueness and possibility, is revealed by the following:

 

... that which characterizes and defines an assertion of Possibility is its emancipation from the Principle of Contradiction, while it remains subject to the Principle of Excluded Third; while that which characterizes and defines an assertion of Necessity is that it remains subject to the Principle of Contradiction, but throws off the yoke of the Principle of Excluded Third ... (MS 678:34, late 1910)

 

PEM does not apply to “the general”…   this includes “assertions of necessity” (propositions expressing necessity):               “S is necessarily P”             “S must be P”             “S has to be P”   Peirce said that PEM does not apply to assertions of necessity because it is not the case, with regard to every such assertion “S is necessarily P,” that “S is necessarily P or S is necessarily not-P” is true; sometimes such propositions are false:   e.g., “The Secretary of State is necessarily male or the Secretary of State is necessarily non-male.”   Peirce’s claim that PEM does not apply to assertions of necessity does NOT imply that such assertions are NEITHER true nor false.

PC does not apply to “the vague”…   this includes “assertions of possibility” (propositions expressing possibility):   “S is possibly P” “S may be P” “S can be P”   Peirce says that PC does not apply to assertions of possibility because it is not the case, with regard to every such assertion “S is possibly P,” that “S is possibly P and S is possibly non-P” is false; sometimes such propositions are true:     e.g., “The President is possibly from Texas, and the President is possibly not from Texas.”   Peirce’s claim that PC does not apply to assertions of possibly does NOT imply that such assertions are BOTH true and false.

 

 

[10.4.4.] “Does Not Apply To” vs. “Is False With Regard To”

 

On Peirce’s view, there is an important difference between saying that PEM or PC does not apply to a proposition, and saying that PEM or PC is false with regard to a proposition:

 

... I do not say that the Principle of Contradiction is false of Indefinites [i.e., vagues]. It could not be so without applying to them which is precisely what I deny of it. An argument against what I say, namely, that the Principle of Contradiction does not apply to “A man” because “A man is tall” and “A man is not tall,” can only amount to saying that that man that is tall is not, while tall, not tall. That is true; and that is what I mean by refusing to say that the Principle of Contradiction is false of “A man” but when it is said of that man that is tall, that he is not not-tall, this is said of the existing man, which is not Indefinite, but is, on the contrary, a certain man and no other. (MS 641:24 2/3 - 3/4, 1909)

 

 

Peirce’s “doesn’t apply to” / “is false with regard to” distinction:

 

“does not apply to”   If a logical principle does not apply to x, then that principle says nothing about x.

“is false with regard to”   If a logical principle is false with regard to x, then what it says about x is false.   For a principle to be false with regard to x, it must apply to x, i.e., it must say something about x.

 

 

The distinction as applied to PC:

·         PC is not false with regard to propositions with vague (indefinite) subject-terms, since that principle can only be false with regard to propositions to which it applies, and it applies only to propositions with definite (non-vague) subject-terms.

·         To say that PC is false with regard to “S is P” is to imply (i) that “S” is definite and (ii) that “S is P” is both true and false. Peirce DOES NOT say this about propositions with vague subject-terms (e.g., “Some philosophers own dogs”).

·         But Peirce’s position is that PC does not apply to propositions with vague subject-terms. It doesn’t say anything about them. Rather, it states a necessary condition for definiteness: if S is definite, then S is not both P and not-P.

 

Although Peirce did not mention PEM in the passage quoted above, it is reasonable to think that his position was analogous with regard to PEM...[4]

 

The distinction as applied to PEM:

·         PEM is not false with regard to propositions with general subject-terms, since that principle can only be false with regard to propositions to which it applies, and it applies only to propositions with individual (non-general) subject-terms.

·         To say that PEM is false with regard to “S is P” is to imply (i) that “S” is individual and (ii) that “S is P” is neither true nor false. Peirce DOES NOT say this about propositions with general subject-terms (e.g., “All Georgians are residents of Carroll County.”)

·         But Peirce’s position is that PEM does not apply to propositions with general subject-terms. It doesn’t say anything about them. Rather, it states a necessary condition for individuality: if S is individual, then either S is P or S is not-P.

 

As we will see in the next set of notes, the distinction between a logical principle not applying to a proposition and being false with regard to a proposition is essential for a correct understanding of Peirce’s later synechism and his work in triadic logic.

 

 

Stopping point for Wednesday November 7. For next time, finish reading “Issues of Pragmaticism” (EP 2:354-59).