[10.5.] Vague Belief.

 

Character IV. By all odds, the most distinctive character of the Critical Common-sensist, in contrast to the old Scotch philosopher, lies in his insistence that the acritically indubitable is invariably vague. (EP 2:350, CP 5.446)[1]

 

Recall that in his letter to Calderoni, Peirce wrote that CCS corrects “six errors which I find in the Scotch doctrine of common sense.” He went on to say that “the most important” of those errors is that the philosophers of the Common Sense school

 

failed to remark the extreme vagueness of our indubitable beliefs. For example, everybody’s actions show that it is impossible to doubt that there is an element of order in the world; but the moment we attempt to define that orderliness we find room for doubt. (EP 2:541 n.10, CP 8.208, emphasis added)

 

So one illustration of what Peirce means by “vague” is the belief that “there is an element of order in the world.” It only becomes dubitable for us once we begin to say more precisely what it means.

 

Elsewhere Peirce gives as an example of the sort of vague indubitable belief recognized by CCS: “fire burns” (5.498)

 

 

[10.5.1.] Peirce’s “Logic of Vagueness.”

 

On Peirce’s view, logicians have not paid sufficient attention to vagueness:

 

Logicians have been at fault in giving Vagueness the go-by, so far as not even to analyze it. The present writer has done his best to work out the [logic][2] of the subject, but can here only give a definition or two with some proposals respecting terminology. (EP 2:350, CP 5.446)

 

He goes on to explain part of what elsewhere he calls “the logic of vagueness” (5.506, R 291, c.1905, not in EP), his account of the various sorts of indeterminacy which can affect the meaning of a sign.

 

The logic of vagueness (LOV) consists, in part, of Peirce’s views on the meaning of the subject-term (or terms) of a proposition.

 

On Peirce’s view, a propositional subject-term (or terms), which refers to the proposition’s object (or objects), can be determinate or indeterminate, and there are two ways for an object-term to be indeterminate:

 

vague object-indeterminacy (indefiniteness) e.g.,   “A man I could mention is conceited.” “Some man is conceited.” “There is an x such that x is a man and x is conceited,” i.e.

object-determinacy (a.k.a. singularity, definite individuality) e.g.,   “George W. Bush is president”

general object-indeterminacy (universality, non-individuality) e.g.,   “Man is mortal”, i.e. “All men are mortal,” i.e. “For all x, if x is a man, then x is mortal”, i.e.

 

 

An object determinate proposition is one with a subject-term that picks out a definite individual to which the predicate is supposed to apply. Peirce often called these singular subjects (e.g., 5.152ff., 1903).

·         In the proposition “George W. Bush is President,” “George W. Bush” is determinate. Its object (that to which it refers) is a definite individual, namely, George W. Bush.

 

An object indeterminate proposition is one with a subject-term that does not pick out a definite individual.[3] There are two types of object-indeterminacy: generality (or universality) and vagueness (or indefiniteness) (EP 2:350-3, 5.447-9, 1905).

 

 

[10.4.1.] The Right of Interpretation.

 

Peirce explains the difference between object-general propositions and object-vague propositions in terms of exchanges made by two interlocutors (i.e., two people who are having a conversation with each other).

 

His explanation involves what “right” each sort of object term, when used in a proposition, “extends” to one’s listener:

 

A sign ... that is in any respect objectively indeterminate (i.e., whose object is undetermined by the sign itself) is objectively general in so far as it extends to the interpreter the privilege of carrying its determination further. Example: “Man is mortal.” To the question, What man? the reply is that the proposition explicitly leaves it to you to apply its assertion to what man or men you will. A sign that is objectively indeterminate in any respect is objectively vague in so far as it reserves further determination to be made in some other conceivable sign, or at least does not appoint the interpreter as its deputy in this office. Example: “A man whom I could mention seems to be a little conceited.” The suggestion here is that the man in view is the person addressed; but the utterer does not authorize such an interpretation or any other application of what she says. She can still say, if she likes, that she does not mean the person addressed. (EP 2:350-51, CP 5.447)

 

 

[10.5.2.] Excluded Middle and Contradiction.

 

Peirce goes on to define generality and vagueness in terms of two principles of logic:

 

Perhaps a more scientific pair of definitions would be 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)

 

This will take a lot of work to untangle...

 

 

[10.5.2.1.] Contemporary Understandings of the Law of Excluded Middle.

 

By “law of excluded middle,” contemporary philosophers mean:

 

Law of Excluded Middle (LEM)

·         Every instance of “p or not-p” is true.

·         Either p or not-p.

·         p Ú ~p

 

The idea behind this principle is to “exclude” a “middle” option between “p” and “not-p”. The idea is that, no matter what proposition “p” stands for, it will be true either than p or that not-p.

 

To deny or reject LEM is to imply that there is an instance of “p or not-p” that is not true.

 

One plausible candidate for a value of “p” that yields an instance of “p or not-p” that is not true is a proposition that predicates baldness of a person who is neither clearly bald nor clearly non-bald, e.g. John McCain (Republican Senator from Arizona, currently a candidate for the GOP presidential nomination).

 

Let’s let “p” stand for “John McCain is bald.” This yields the following instance of “p or not-p”:

 

“John McCain is bald, or John McCain is not bald.”

 

Suppose that this disjunction (an either-or proposition) is not true.

 

If this is the case, then “John McCain is bald” is neither true nor false. If it were true, then the disjunction would be true; and if it were false, the disjunction would be true; so if the disjunction is not true, then “John McCain is bald” is neither true nor false.

 

So to deny or reject LEM is to deny or reject another principle with is closely associated with LEM:

 

Principle of Bivalence (df.): every meaningful proposition is true or else false.

 

 

[10.5.2.2.] Contemporary Understandings of the Law of Non-Contradiction.

 

By “principle of contradiction,” contemporary philosophers mean:

 

Law of Non-Contradiction (LNC)

^·          Every instance of “p and not-p” is false.

^·          Not both p and not-p.

^·          ~(p · ~p)

 

To deny LNC would be to imply that there is an instance of “p and not-p” that is true. This is normally viewed as a bad thing, because in classical logic a proposition implies anything whatsoever:

 

1. p and not­-p.

2. p. [from 1]

3. p or q. [from 2] [where “q” stands for any proposition you like]

4. not-p [from 1]

5. q [from 3 and 4]

 

So if there is a contradiction that’s true, then every single proposition there is, is true.

 

1. John McCain is bald and John McCain is not bald.

2. So, John McCain is bald.

3. So, either John McCain is bald or pigs can fly.

4. But John McCain is not bald.

5. So, pigs can fly.[4]

 

 

[10.5.2.3.] Peirce’s Understanding of the Principles of Excluded Middle and Contradiction.

 

The logical laws explained in the previous two sections are not what Peirce is referring to when he speaks of the “principles of excluded middle and contradiction.”[5]

 

As he understands them, the principles are as follows:

 

Principle of Excluded Middle (PEM)

Material mode: for any property and for any individual, either that individual possesses that property or that individual does not possess that property.

Formal mode: for any pair of contradictory predicates “P” and “not-P” and for any individual (non-general) subject-term “S”, either “S is P” or “S is not-P” is true.[6] [7]

 

Principle of Contradiction (PC)

Material mode: for any property and for any definite (non-vague) subject, it is not the case both that the subject possesses that property and that the subject does not possess that property.

Formal mode: for any pair of contradictory predicates “P” and “not-P” and for any definite subject-term “S”, “S is P” and “S is not-P” are not both true.[8]

 

When he says that “anything is general in so far as the principle of excluded middle does not apply to it,” Peirce means that an objectively-general proposition and its internal negation (the result of negating the predicate rather than the entire proposition) can both be false:

 

It is possible for “All men are mortal” and “All men are not moral” to both be false.

 

And when he says that “anything ... is vague in so far as the principle of contradiction does not apply to it,” Peirce means that an objectively-vague proposition and its internal negation can both be true:

 

It is possible for “Some man is conceited” and “Some man is not conceited” to both be true.

 

 

Stopping point for Monday November 5. For next time, no new reading, but it will be especially helpful if you have a copy of Wednesday’s online lecture notes with you when you come to class.