On the TFL (Mis)Representation of Singular Propositions as General

The following is a valid argument:

1. Pittacus is a good man
2. Pittacus is a wise man
—–
3. Some wise man is a good man.

That this argument is valid I take to be a datum, a given, a non-negotiable point. The question is whether traditional formal logic (TFL) is equipped to account for the validity of this argument. As I have already shown, it is quite easy to explain the validity of arguments like the above in modern predicate logic (MPL). In MPL, the logical form of the above argument is

Gp
Wp
—
(Ex)(Wx & Gx).

In order to evaluate the argument within TFL, it must be put into syllogistic form, otherwise the rules of the syllogism cannot be applied to it. Thus,

Every Pittacus is a wise man
Some Pittacus is a good man
—–
Some wise man is a good man.

This has the form:

Every P is a W
Some P is a G
—–
Some W is a G.

It is easy to prove that this form is valid by using a Venn diagram (not to be confused with an Euler diagram), or by applying the syllogistic rules. You will notice that I have rigged the argument so that those who deny that universal propositions have existential import will be satisfied that it is valid. Note also that the Venn diagram test would not work if the argument were given the following form:

Every P is a W
Every P is a G
—–
Some W is a G.

You can verify for yourself that if you diagram the premises you will not thereby have diagrammed the conclusion.

But is it logically acceptable to attach a quantifier to a singular term? How could a proper name have a sign of logical quantity prefixed to it? 'Pittacus' denotes or names exactly one individual. 'Every Pittacus' denotes the very same individual. So we should expect 'Every Pittacus is wise' and 'Pittacus is wise' to exhibit the same logical behavior. But they behave differently under negation.

The negation of 'Pittacus is wise' is 'Pittacus is not wise.' So, given that 'Pittacus' and 'every Pittacus' denote the same individual, we should expect that the negation of 'Every Pittacus is wise' will be 'Every Pittacus is not wise.' But that is not the negation (contradictory) of 'Every Pittacus is wise'; it is its contrary. So 'Pittacus is wise' and 'Every Pittacus is wise' behave differently under negation, which shows that their logical form is different. My argument can be put as follows:

a. Genuinely singular sentences have contradictories but not contraries.
b. Sentences like 'Every Pittacus is wise' have both contradictories and contraries.
Therefore
c. Sentences like 'Every Pittacus is wise' are not genuinely singular.
d. 'Pittacus is wise' is genuinely singular.
Therefore
e. The TFL representation of singular sentences as quantified sentences does not capture their logical form, and this is an inadequacy of TFL, and a point in favor of MPL.

MPL 1, TFL 0.

Posted

July 22, 2011

Logica Docens, Scholasticism New and Old

Bill Vallicella

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5 responses to “On the TFL (Mis)Representation of Singular Propositions as General”

Edward the nominalist

July 22, 2011

>>But is it logically acceptable to attach a quantifier to a singular term? How could a proper name have a sign of logical quantity prefixed to it? ‘Pittacus’ denotes or names exactly one individual. ‘Every Pittacus’ denotes the very same individual. So we should expect ‘Every Pittacus is wise’ and ‘Pittacus is wise’ to exhibit the same logical behavior. But they behave differently under negation.
This is an absolutely standard objection to TFL. Have you been re-reading Sommers, by any chance?
>>The negation of ‘Pittacus is wise’ is ‘Pittacus is not wise.’ So, given that ‘Pittacus’ and ‘every Pittacus’ denote the same individual, we should expect that the negation of ‘Every Pittacus is wise’ will be ‘Every Pittacus is not wise.’ But that is not the negation (contradictory) of ‘Every Pittacus is wise’; it is its contrary. So ‘Pittacus is wise’ and ‘Every Pittacus is wise’ behave differently under negation, which shows that their logical form is different.
As I have argued here before, ‘Pittacus is wise’ and ‘Pittacus is not wise’ are in fact contraries. For the first implies that someone (Pittacus) is wise. The second implies that someone (Pittacus again) is not wise. Both imply the existence of Pittacus (or at least – to silence impudent quibblers – that someone is Pittacus). Thus they are contraries. Both are false when no one is Pittacus. I handled a very similar objection by Greg Frost-Arnold in a post here http://ocham.blogspot.com/2011/06/is-atlantis-west-of-london.html .
For the record, the scholastics distinguished between two forms of negation. Indefinite or predicate negation, which is logically affirmative, and extinctive or true negation which is logically negative. Thus ‘Socrates is non-wise’ is logically affirmative, and is the contrary of ‘Socrates is wise’, and requires the existence of Socrates. And ‘Not: Socrates is wise’ is negative, and is the contradictory, and this has two conditions of truth (causae veritatis) – the first when ‘Socrates is non-wise’ is true, the second when Socrates does not exist. For singular propositions, the distinction is clearer in Latin, as determined by the placing of the negation operator ‘non’. In English and possibly other European languages, it is less clear. Perhaps MPC is a merely accidental result of modern grammar.
>>MPL 1, TFL 0.
I think not. Indeed, the biggest own-goal of MPC is the way it requires all meaningful singular terms to have a referent (i.e. an existing referent).

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Edward the nominalist

July 22, 2011

Addressing your argument above, the blatantly false premiss is
>>a. Genuinely singular sentences have contradictories but not contraries.
Clearly any affirmative singular sentence has a contradictory – namely the denial that the sentence is true, and a contrary – namely the assertion that the predicate is not true of the subject. It is a distinction that is grammatically ambiguous in English, although not in Latin. ‘Socrates is not wise’ can mean that ‘Socrates is wise’ is false, or simply that Socrates is non-wise.

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Bill Vallicella

July 22, 2011

Ed,
I am not currently re-reading Sommers, but I have read him, and I don’t claim any originality in the above. I think I found the objection in Geach.
>>As I have argued here before, ‘Pittacus is wise’ and ‘Pittacus is not wise’ are in fact contraries. For the first implies that someone (Pittacus) is wise. The second implies that someone (Pittacus again) is not wise. Both imply the existence of Pittacus (or at least – to silence impudent quibblers – that someone is Pittacus). Thus they are contraries. Both are false when no one is Pittacus.<< Well, you do have Sommers on your side.

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Edward the nominalist

July 22, 2011

>>Well, you do have Sommers on your side.
And the Truth. The thing is, you really have a problem otherwise. If ‘Socrates is wise’ and ‘Socrates is not wise’ are contradictories, and if ‘Socrates is not wise’ implies ‘someone (Socrates) is not wise’, as standard MPC holds, you are committed to the thesis that the sentence is not meaningful when Socrates ceases to exist (or if he never existed because Plato made him up). Which (on my definition) is Direct Reference.
So you have this horrible choice.
1. Direct reference
2. Traditional Logic.

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Eric “One Brow” Hogue

July 22, 2011

I agree the statement “Every Pittacus is wise” is not genuinely sngular. It refers to the members of the class whose members are Pittacus, which is not innately singular. I could refer to “every blogger at Maverick Philosopher” or “every blogger at M-Phi”. The form is the same, even though one class has one element the other about twenty. “Every Pittacus is wise” refers to a single individual by happenstance, while “Pittacus is wise” refers to a single individual by the construction of the sentence.

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On the TFL (Mis)Representation of Singular Propositions as General

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5 responses to “On the TFL (Mis)Representation of Singular Propositions as General”

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