'Existential Import'...does such a 'concept' have such?

[Michael Stuart Kelly]

It's also obvious to me that a "set" of "square circles" is more than one, even if the units of the set ("square circles") are imaginary and impossible to exist.

"More than one" still stands as a quantity in my book, despite claims to the contrary. All the double-speak does is to try to muddy the waters to pull the fish out without anyone seeing it.

But this is a Fish-Feriso. The more it sits there, the more it stinks...

Michael

[tjohnson]

LOL, The set of square circles?? This is a contradiction of terms and so it is meaningless. Any discussion about "square circles" is a meaningless waste of time.

[tjohnson]

It seems to me, in set theory, the empty set plays the same role as the number "0" in the integers.

[merjet]

It seems to me, in set theory, the empty set plays the same role as the number "0" in the integers.

The intersection of the sets {1, 3, 5} and {2, 4, 6} is an empty set, not 0. The intersection of {circles} and {squares} seems more like trying to divide by 0.

Here is an informative article about the square of opposition and its history.

[tjohnson]

The intersection of the sets {1, 3, 5} and {2, 4, 6} is an empty set, not 0. The intersection of {circles} and {squares} seems more like trying to divide by 0.

Here is an informative article about the square of opposition and its history.

Yes, well in the case of the union of the empty set and any set;

∀A: A ∪ ∅ = A

This is equivalent to x+0=x for all x.

[BaalChatzaf]

The intersection of the sets {1, 3, 5} and {2, 4, 6} is an empty set, not 0. The intersection of {circles} and {squares} seems more like trying to divide by 0.

Here is an informative article about the square of opposition and its history.

Yes, well in the case of the union of the empty set and any set;

∀A: A ∪ ∅ = A

This is equivalent to x+0=x for all x.

Not equivalent. Cognate is a better word. It is the corresponding relation.

Ba'al Chatzaf

[Roger Bissell]

It's also obvious to me that a "set" of "square circles" is more than one, even if the units of the set ("square circles") are imaginary and impossible to exist.

"More than one" still stands as a quantity in my book, despite claims to the contrary. All the double-speak does is to try to muddy the waters to pull the fish out without anyone seeing it.

But this is a Fish-Feriso. The more it sits there, the more it stinks...

Michael

Michael, Ba'al is right about one thing: an empty set has no members.

How many members does this set have? {deceased authors of Atlas Shrugged} One, right?

OK, how many members does this set have? {living authors of Atlas Shrugged} Are you even ~slightly~ tempted to say "one"? I hope not! The answer is "zero," because this is an empty set.

Now, how many members does this set have? {circles in the Olympic symbol} Five, right?

OK, how many members does this set have? {circles that are also square} Above, you seem to indicate that you would answer this "more than one" or "many" or whatever. But can't you now see that the answer here is the same as for {living authors of Atlas Shrugged}? Zero.

The non-existent does not exist, and so it has no number or quantity. That also applies to the contents of empty sets, sets of things that are non-existent. Such sets have ~no~ members, not one, not more than one, etc.

REB

P.S. -- I know what you mean about the fish-smell, but it is NOT emanating from the nature of empty sets.

[Roger Bissell]

It seems to me, in set theory, the empty set plays the same role as the number "0" in the integers.

The intersection of the sets {1, 3, 5} and {2, 4, 6} is an empty set, not 0. The intersection of {circles} and {squares} seems more like trying to divide by 0.

Here is an informative article about the square of opposition and its history.

Two quick comments, Merlin:

1. The intersection of sets {1, 3, 5} and {2, 4, 6} is an empty set, all right. How is this different from the intersection of {circles} and {squares}? The numbers and shapes all exist. Intersecting their sets produces an empty set in each case. The first example is trying to produce a set of numbers that are both odd and even. How is that any less illogical than trying to produce a set of shapes that are both circular and square? Or trying to divide by zero?

2. The online article you refer to is atrocious, ghastly, horribly mistaken. Here and there are informative bits, but the author makes (or parrots) so many mistakes, it is impossible to know where to begin in a critique. But it's not much worse than the standard textbook accounts.

REB

[merjet]

Two quick comments, Merlin:

1. The intersection of sets {1, 3, 5} and {2, 4, 6} is an empty set, all right. How is this different from the intersection of {circles} and {squares}? The numbers and shapes all exist. Intersecting their sets produces an empty set in each case. The first example is trying to produce a set of numbers that are both odd and even. How is that any less illogical than trying to produce a set of shapes that are both circular and square? Or trying to divide by zero?

I didn't say it was different.

I don't believe the point of an empty set is to begin illogically, but to serve as a sign that some premises are contradictory or absurd. It's use seems something like reductio ad absurdum, which begins with a premise known to be false. So is reductio ad absurdum illogical, too?

2. The online article you refer to is atrocious, ghastly, horribly mistaken. Here and there are informative bits, but the author makes (or parrots) so many mistakes, it is impossible to know where to begin in a critique. But it's not much worse than the standard textbook accounts.

This is not a topic I'm steeped in. However, if the article is as bad as you say, then it should be easy for you to identify a couple of its worst errors.

[Roger Bissell]

Two quick comments, Merlin:

1. The intersection of sets {1, 3, 5} and {2, 4, 6} is an empty set, all right. How is this different from the intersection of {circles} and {squares}? The numbers and shapes all exist. Intersecting their sets produces an empty set in each case. The first example is trying to produce a set of numbers that are both odd and even. How is that any less illogical than trying to produce a set of shapes that are both circular and square? Or trying to divide by zero?

I didn't say it was different.

Ah, I see. I misunderstood what you were saying, i.e., what you were responding to. You first mentioned the intersection of the sets of odd and even numbers as being an empty set. ~Then~ you said that the intersection of the sets of circles and squares was "more like trying to divide by zero." I guess I could have wondered "more like WHAT"? Instead, I simply took your comment to mean "the intersection of the sets of circles and squares is more like trying to divide by zero than the intersection of the sets of odd and even numbers is like trying to divide by zero." But instead you were responding to GS's comment about empty sets being like zero. You were saying that "the intersection of sets of circles and squares is more like trying to divide by zero than it is like zero." Of course, the same thing would be true of the intersection of sets of odd and even numbers, but you didn't ~say~ that, which led me to think that you didn't believe it to be so. Now, I see that you do.

I don't believe the point of an empty set is to begin illogically, but to serve as a sign that some premises are contradictory or absurd. It's use seems something like reductio ad absurdum, which begins with a premise known to be false. So is reductio ad absurdum illogical, too?

First, the point of an empty set is to serve as a sign that a given set has no members, pure and simple. There is more than one reason that a given set might be empty. One is that it is the intersection of two incompatible sets, such as circles and squares. This produces a contradictory or absurd result. Another is that it is a file folder, as it were, for something that once existed, but no longer does, such as presently existing dinosaurs or living authors of Atlas Shrugged. So, it's not just a tool for dealing with contradictions and absurdities. It's more generally a tool for dealing with non-existents of whatever kind.

2. The online article you refer to is atrocious, ghastly, horribly mistaken. Here and there are informative bits, but the author makes (or parrots) so many mistakes, it is impossible to know where to begin in a critique. But it's not much worse than the standard textbook accounts.

This is not a topic I'm steeped in. However, if the article is as bad as you say, then it should be easy for you to identify a couple of its worst errors.

It ~is~ easy to identify those errors, but it would ~not~ be easy to suffer through another raft of futile arguments like the foregoing, which is all that it would produce. On the back burner is a book project, viz., a Guide to Logic, which is aimed at college students to help them deal with the flawed material in their logic textbooks. I will include an analysis of the Stanford online material on logic, too. I have a university professor interested in co-writing it with me, so when we get closer to publication, I will announce it here.

However, since you asked me to "identify" two of the worst errors, I will do so, but I'm not going to engage in Battle Royal about it. Feel free, anyone, to disagree and state your reasons, but I don't want to do any more of this interminable wrangling over things that ought to be obvious. (That means you, Ba'al.)

1. The first paragraph and last paragraph (excluding the first sentence) of section 1.2 completely botches the issue of the truth-value of I-statements with empty subject terms.

2. The critique in section 6 of Strawson's defense of the square of opposition is completely botched and fails due to equivocation (failure to identify ambiguity due to different modes of existence of chimera and men). Just a teaser: the final step, if the logic is handled properly, should read: Some non-[real being that is a man] is an imaginary being that is a chimera. The author claims this is false and thus proves Strawson's defense leads to contradictions in extended reasoning. But as I do the five steps he outlines, truth is preserved throughout.

REB

[BaalChatzaf]

However, since you asked me to "identify" two of the worst errors, I will do so, but I'm not going to engage in Battle Royal about it. Feel free, anyone, to disagree and state your reasons, but I don't want to do any more of this interminable wrangling over things that ought to be obvious. (That means you, Ba'al.)

Obvious to whom? I am a mathematician. I am interested in mathematical issues and problems not chimeras. I leave chimeras to the philosophers. If you want to have a set algebra than the empty set is necessary to assure that set intersection is completely defined.

Given set intersection (which I write as ^) and the empty set (which I write as 0') we have

A is included in B if and only if A ^ B = A. We also have 0' ^ B = 0' for all sets B. Hence 0' is included in B for all sets B. This is straightforward, even trivial. I simply do not understand your problem with empty sets. I also do not understand your problem with the simplification of the square of opposition (which is hardly ever used in mathematical discourse). The inclusion A is a subset of B translates to A^B = A. This is equivalent to the categorical assertion all A is B. Since A^B=A holds when A is 0' the removal of existential import is required for algebraic closure in set algebra. Why is this difficult for you to comprehend? It is purely a mathematical requirement. It has no genuine ontological import. It is required for the same reason we have negative numbers (to make + and - well defined and the equation a + x = b always solvable , for any a,b). It is an algebraic requirement, plain and simple. It is not a philosophical issue at all.

Mathematics was liberated from the thrall of metaphysics and philosophy in the 19th century. Since then it has progressed at warp speed, unencumbered by unnecessary bagage.

Ba'al Chatzaf

Edited May 9, 2009 by BaalChatzaf

[tjohnson]

Mathematics was liberated from the thrall of metaphysics and philosophy in the 19th century. Since then it has progressed at warp speed, unencumbered by unnecessary bagage.

Ba'al Chatzaf

Here! Here!

[Brant Gaede]

Mathematics was liberated from the thrall of metaphysics and philosophy in the 19th century. Since then it has progressed at warp speed, unencumbered by unnecessary bagage.

Ba'al Chatzaf

Here! Here!

Isn't it "Hear! Hear!"?

It might be time, GS, to liberate mathematics from gs. Go for the clean sweap by diverting the river of rationality into the stables.

--Brant

[tjohnson]

Isn't it "Hear! Hear!"?

It might be time, GS, to liberate mathematics from gs. Go for the clean sweap by diverting the river of rationality into the stables.

--Brant

Isn't it sweep?

[Brant Gaede]

Isn't it "Hear! Hear!"?

It might be time, GS, to liberate mathematics from gs. Go for the clean sweap by diverting the river of rationality into the stables.

--Brant

Isn't it sweep?

I know that, but I had to leave you something on the table to assuage your embarrassments.

I cannot tell a lie. I did chop down that cherry tree.

--Brant

[thomtg]

Roger,

...

...

The above examples ~terribly~ jumble up my view, if that is what they purport to illustrate (for purposes of reductio ad absurdum?). Let me run through them and give ~my~ interpretations and evaluations of them:

...

Thom, these examples are god-awful distortions of what I advocate, but I'll address them anyway.

...

Thom, I have steeped in this stuff for DECADES. It makes perfect sense to me, and I think that all the major logicians and their texts are out to lunch on this issue! You have done nothing to shake my grasp of this or my confidence that I am correct, not one iota. ...

...

Roger,

Thank you for responding. I wish to acknowledge immediately my admiration of you for the evident fact that after nearly 20 years of steeped study you are still open to fresh engagement with relative newcomers like me, and, perhaps, to be corrected by new evidence and reasoning.

That last post of mine (linked above) was partly an effort at fact finding, to find out what you actually claim about existential import. Now I do believe I understand you. Let me first summarize in my own words your view before giving my interpretation of it.

First, by integrating and reorganizing your pained interpretations we can discern the following ideas:

R1: If a string of real words is grammatical and declarative, it is a sentence conveying a proposition that is either true or false.

R2: You accept the idea of elliptical statements. E.g., the statement "John Smith is a lottery winner" is ambiguous, because there are many men with such a name. When a newspaper reports him as a winner, the statement is really elliptical to mean "John Aaron Smith of 123 Main Street, Sometown, Somestate, Someday, month, year, etc., is a lottery winner." It is still one simple assertion, asserting one individual having won the lottery.

R3: Whenever the subject term has no referents in the present, the verb tense is elliptical for a compound noun, which must be made explicit. In other words, whenever you see the copula in the present tense, you want to make explicit that the designation of the subject term encompasses only those existents presently existing. And whenever the copula is in the past tense, you want to make explicit that the designation of the subject term encompasses the existents existing only in some past date.

R4: Elliptical modes of existence (a.k.a. domains of discourse) include the real order, the imaginative order, and their subdomains.

R5: Singular statements (of the descriptive kind) must be transmogrified into compound statements.

R6: An affirmation whose designation of the subject is empty must be false.

R7: A denial whose designation of the subject is empty must be true.

R8: Compound-noun statements must be transformed associatively by treating subject terms and predicate terms identically.

Thus,

1. "All dinosaurs are scaly" is R3 elliptical for "All dinosaurs are presently existing scaly creatures"; and since none exists presently (R4), 1 is false.

2. "No dinosaur is scaly" is R3 elliptical for "No dinosaur is a presently existing scaly creature"; and by R6, 2 is true.

3. "All unicorns are horned horses" is elliptical for either

3a. "All unicorns are imaginary horned horses"; and by R4, 3a is true (in the imaginative order).

3b. "All unicorns are real horned horses"; and by R4 and R6 and EAE-1 arg, 3b is false (in the real order).

4. "No unicorn is a horned horse" is elliptical for either

4a. "No unicorn is an imaginary horned horse"; 4a is false (in the imaginative order).

4b. "No unicorn is a real horned horse"; by R7 in the real order, 4b is true (in the real order.

5. "TpKoF is bald" is reinterpreted by R5 to "TpKoF is a presently existing person, AND he is presently bald"; and since the first conjunct is false by R6, the compound statement is false; hence, 5 is false.

6. "TpKoF isn't bald" is reinterpreted by R5 to "TpKoF is a presently existing person, AND he isn't presently bald"; and since the first conjunct is false by R6, the compound statement is false, even though the second conjunct is true by R7; hence 6 is false.

7. "All dinosaurs were scaly" is R3 elliptical for "All dinosaurs were past-existing scaly creatures"; 7 is true.

8. "No dinosaur was scaly" is R3 elliptical for "No dinosaur was a past-existing scaly creature"; 8 is false.

9. "TpKoF was bald" is reinterpreted by R5 to "TpKoF was a past-existing person, AND he was bald"; by R3 and R6, the first conjunct is false, rendering the compound false; hence, 9 is false.

10. "TpKoF wasn't bald" is reinterpreted by R5 to "TpKoF was a past-existing person, AND he wasn't bald"; again, while the second conjunct is true by R3 and R7, since the first conjunct is false, the compound is false; hence, 10 is false.

11. "All squared-circles are triangular"; by R1, R6, R8, and C9 (see scratchpad), 11 is false.

12. "No squared-circle is triangular"; by R1, R7, R8, and C10 (see scratchpad), 11 is true.

---------------

Here now is my interpretation:

Except for R2 and R4, I disagree with R1, partial R3, R5, R6, R7, and R8.

Since I agree with your interpretation of those statements of the imaginative order, we'll skip them. The rest will be discussed in three parts: universals of the real that no longer exist, singulars described to emptiness, universals of emptiness.

Part 1: Among universals of the real that used to exist but exist no longer:

R3 implies that the elliptical phrases "presently existing" and "past-existing" applies to and restricts the subjects that used to exist but exist no longer. If it had been applied simply, I would have no problem with that. What is problematic is its presumption:

P1. The restrictive phrase is confused with the verb tense, which forces the restriction not on the subject but on the predicate. This forced restriction implies R8 being at work.

P1 is seen in the dinosaur statements 1, 2, 7, and 8.

[Given the supposition that dinosaurs had scales,]

1. "All dinosaurs are scaly" becomes by R2 "All dinosaurs (open-ended past, present, and future) are scaly creatures (open-ended past, present, and future)." It becomes by R3 "All presently existing dinosaurs (open-ended present) are scaly creatures (open-ended past, present, and future)." BUT the P1 presumption now enters: "All dinosaurs (open-ended past, present, and future) are presently existing scaly creatures (open-ended present)." The entire transformation changes the predicate--a fallacy of equivocation. What started out and could have been assessed true, is now deemed false.

2. "No dinosaur is scaly" becomes by R2 "No dinosaur (o-eppf) is a scaly creature (o-eppf)." It becomes by R3 "No presently existing dinosaur (open-ended present) is a scaly creature (o-eppf)." BUT the P1 presumption now enters: "No dinosaur (o-eppf) is a presently existing scaly creature (open-ended present)." Again, the entire transformation changes the predicate--another fallacy of equivocation. What started out and could have been assessed false, is now deemed true.

The P1 presumption is also at work in statements 7 and 8, only that it is obscured by the already explicit verb tense.

7. "All dinosaurs were scaly" becomes by R2 "All dinosaurs (o-eppf) were scaly creatures (o-eppf)." It becomes by R3 "All past-existing dinosaurs (open-ended past) were scaly creatures (o-eppf)." But with P1, it becomes "All dinosaurs (o-eppf) were past-existing scaly creatures (open-ended past)." But for the equivocation, everything is true at every stage.

8. "No dinosaur was scaly" becomes by R2 "No dinosaur (o-eppf) was a scaly creature (o-eppf)." It becomes by R3 "No past-existing dinosaur (open-ended past) was a scaly creature (o-eppf)." But with P1, it becomes "No dinosaur (o-eppf) was a past-existing scaly creature (open-ended past)." But for the equivocation, everything is false at every stage.

All of these show that the R3 rule is innocuous by itself and in fact unnecessary even for extinct subjects. They show that verb tense has a separate function (i.e., temporalizing identification) that is independent of referential qualification. What is noxious is P1. If I grant you that "scaly creatures" and "presently existing creatures" aren't synonyms, then you must grant me that "dinosaurs" and "past-existing dinosaurs" aren't synonyms as well.

Part 2: For singulars described to emptiness:

The glaring treatment of singulars is in transmogrifying them from a simple assertion into a compound assertion, by use of R5. (Singular statements make use of R3 as well and presumes P1. While this defect has already been described in Part 1, one thing to note about it here is that P1 now distributes across the conjunction.)

Contrast a simple proper name designating rigidly some individual, say, Nicolas Sarkozy. He is the current President of France in 2009 (TcPoF).

D1. "Nicolas Sarkozy is remarried"; this is true.

D2. "Nicolas Sarkozy isn't remarried; this is false.

D3. "Nicolas Sarkozy was divorced"; this is true.

D4. "Nicolas Sarkozy wasn't divorced"; this is false.

D5. "TcPoF is remarried"; this is true.

D6. "TcPoF isn't remarried; this is false.

D7. "TcPoF was divorced"; this is true.

D8. "TcPoF wasn't divorced"; this is false.

A proper name denotes an individual. Without the concrete individual, the name is just a visual-auditory scribble. So, when it is used in a sentence in the attempt to assert a truth, there is no question that the name and the individual have already been linked--linked not inferentially but causally, through causal acts of perception, etc. With that being already done, the assertion is the first and only act needed to identify the fact.

On the other hand, a definite description (i.e., a descriptive phrase uniquely identifying an individual among a population), does require a conceptual act of isolation to locate the individual. An individual concrete must be isolated conceptually before any assertion can be made about him/it. Otherwise, the phrase reverts to just another visual-auditory scribble and there is no subject to be said about anything. This process must eventually reach a causal basis for the linkage between the description to the individual concrete. But however long or tedious this process is, it is prior to an assertion. So, again, the assertion with a singular denoting phrase for a subject stands on its own as a lone act of identification.

The underlying principle from the above is what I will dub in the present context -R1b.

-R1b. A singular statement requires the prior isolation of an individual concrete; otherwise, it isn't a statement.

What happens instead by means of R5, R6, and R7 is the requirement to turn a simple statement into a compound statement, asserting three or more propositions. It goes beyond the mere use of R2. It requires

D5. "TcPoF is remarried" to become:

D5a. "TcPoF is a real organism" and

D5b. "TcPoF is a remarried person" and

D5c. "D5a AND D5b"; so that the whole thing is true.

Similarly, you want

D6. "TcPoF isn't remarried" to become:

D6a. "TcPoF is a real organism" and

D6b. "TcPoF isn't a remarried person" and

D6c. "D6a AND D6b"; so that the whole thing is false.

Note that R5 requires that we add D5a and D5c for D5, and D6a and D6c for D6. Note especially that the predicate for D5a and D6a by the nature of the setup is completely arbitrary. This new middle term didn't have to be "organism"; it could have been "animal" or "human" or "person" or even "president." The point is, it was not in the original sentence. Rule R5 simply and arbitrarily requires an extra identification and an extra conjunction.

What should have been but one, now becomes three or more (by means of R5), and they require independent processes of verification.

AND then, rather than taking -R1b as a principle, the present theory chooses R1, in additional to the already mentioned theses, to interpret singulars described to emptiness.

5. "TpKoF is bald" becomes:

5a. "TpKoF is a real person" and

5b. "TpKoF is a bald person" and

5c. "5a AND 5b"; which happens to be true. And

6. "TpKoF isn't bald" becomes:

6a. "TpKoF is a real person" and

6b. "TpKoF isn't a bald person" and

6c. "6a AND 6b"; which is false true.

If one had accepted the R5 explosion rule as a given, and had one taken -R1b instead of R1, the verification process would have halted on the first recognition--that TpKoF is a mere audio-visual scribble--and every one of the substatements would have yielded neither true nor false. But because the present theory accepts R5(R1), R6 and R7 become its corollaries.

Part 3: On universals of emptiness:

... A squared-circle is traditionally called a repugnant concept; as such, it has no place in any assertion. ...

... In my view, the statements about squared-circles are neither true nor false, because "squared-circle" doesn't mean anything. ...

...

The concept of "squared-circle" has "no place in any assertion"??? It "doesn't mean anything"??? Come on! A squared-circle is just a specific kind of contradictory idea, a "real circle that has an impossible attribute"--and we Randians are comfortable asserting (with Rand) that "contradictions don't exist," aren't we? So why can't we say "contradictions are real things with contradictory attributes"? Surely that is meaningful and false!

...

Okay, you got me. I was hasty and vague. I should have said that "... it has no place in an affirmative assertion. ... the statements asserting squared-circles are neither true nor false..."

Recall the two statements and your interpretation:

11. "All squared-circles are triangular"; by R1, R6, R8, and C9 (see scratchpad), 11 is false.

12. "No squared-circle is triangular"; by R1, R7, R8, and C10 (see scratchpad), 11 is true.

Scratchpad table to transform statements 11 and 12:

C1. "Every circle is a square"; C1 is false.

C2. "Every circle isn't a square"; C2 is true.

C3. "Every circle is a triangle"; C3 is false.

C4. "Every circle isn't a triangle"; C4 is true.

C5. "Every circle is both a square and a triangle"; C5 is false.

C6. "Every circle isn't both a square and a triangle"; C6 is true.

C7. "Every circle that is square is a triangle"; C7 is not considered.

C8. "Every circle that is square isn't a triangle"; C8 is not considered.

C9. "Every circle that is square is a (real) circle that is both square and triangular"; C9 is equivalent to C5.

C10. "Every circle that is square isn't a (real) circle that is both square and triangular"; C10 is equivalent to C6.

(Note: I am taking the liberty of regimenting O-form statements "No S is P" to the more literal version "Every S isn't P" in order to better separate and discern S and P.)

According to your post (linked above), first, 11 is transformed to C9, and 12 to C10. Then from the results, you explicitly equate C9 with C5, and C10 with C6. This is from the assumption of rule R8. Parts of nouns shift from subject to predicate interchangeably.

What is noteworthy is that C7 and C8, the actual original statements, are not identified and acknowledged even as candidates for consideration. Here therefore is the fallacy of diversion, the bait-and-switch fallacy. While the scratchpad table does show correctly that C5 is false and C6 true, they however are not equivalent respectively to C7 and C8.

Since 11 is actually C7, and 12 actually C8, we have to evaluate the subject as a compound noun. But this is precisely what the present theory neglects to do.

But suppose we proceed to follow through with the theory as summarized. Since the repugnant concept is without referents in the real order, the subject is empty in designation. Thus, by R6, C7 is false; by R7, C8 is true. And this leads to an internal absurdity when the axiom of identity is regimented in the propositional forms:

C7p: "Every squared-circle is a squared-circle"; by R6 is false.

C8p: "Every squared-circle isn't a squared-circle"; by R7 is true.

---------

In Parts 1, 2, and 3, I have analyzed your view of existential import, showing its dependence on R1-R8 and P1. I agree with R2 and R4. I have shown that R3 is innocuous and superfluous by itself but is untenable with P1. R5 and R8 are shown to be fallacious, and R6 and R7 lead to absurdities. Only one remains, R1. I have partially refuted it with -R1b. The more general refutation to R1 fortunately is already discussed elsewhere on OL (with/by Robert Campbell) and ITOE 19-23 passim. Without R1, R3(P1), R5-R8, this theory of existential import falls apart.

[Roger Bissell]

Thom, your comments are quite interesting and very clearly formulated, which I appreciate greatly, and I will engage them more fully when I have time. I am in Arizona for a weekend jazz gig, and my free time is limited, but I will make some brief comments now.

First, I must take exception to your claim in Part 3 on "universals of emptiness" that I engaged in diversion and "bait and switch." This sounds like an accusation that I knowingly and deliberately misdirected the discussion. I assure you that, whatever fallacy, flaw, or omission there was in my comments, I did ~not~ intentionally leave anything out!

Now, in order to show you how your comments in Part 3 miss the mark, I have to invoke a couple of very important principles about "orders of existence" of "modes of being," that are crucial for propositional meaning and truth/falsity. I refer you to John J. Morrison's "The Existential Import of a Proposition in Aristotelian Logic" in Philosophy and Phenomenological Research, Vol. 15, No. 3 (March 1955), pp. 386-393.

First, for an affirmative proposition to be true, the subject and predicate must belong to the same order of existence, whether actual, possible, or fictional (mentally constructed from things that are actual or possible). Fictional things can only exist "before the mind," but as such, they can still be considered propositionally. Further, for an affirmative proposition to be true, where the subject and predicate must belong to the same order of existence, the existence of the subject must also be identical with that of the predicate.

Just as important, for a negative proposition to be true, either (1) the subject and predicate must not belong to the same order of existence, or (2) the subject and predicate must not be the same identical existence, if they belong to the same order of existence.

This is pretty basic stuff, as I see it, and it has guided me for over 20 years in seeing through the numerous attempts by professional logicians to obfuscate or simply mishandle truth-value of categorical propositions. If it's neglected, all manner of confusion and sophistry are the result. (As Rand would say: for further details, see Modern Logic.)

Now, I want to briefly address the two propositions you say I didn't deal with, and which are fatal to my position about existential import.

C7. "Every circle that is square is a triangle"; C7 is not considered.

C8. "Every circle that is square isn't a triangle"; C8 is not considered.

Invoking rules 6 and 7:

R6: An affirmation whose designation of the subject is empty must be false.

R7: A denial whose designation of the subject is empty must be true.

you say that C7 is false and C8 is true, which you claim leads to absurd consequences, in particular,

C7p: "Every squared-circle is a squared-circle"; by R6 is false.

C8p: "Every squared-circle isn't a squared-circle"; by R7 is true.

Yikes! Do I mean to say that? Of course not!

The problem is not the truth-value of C7 and C8, but the rules R6 and R7, which you attribute to me, as though they were my basis for arguing my claims. They are not!

Bearing in mind the points I made above about consistency of order of existence between subject and predicate, and identity of subject and predicate within the order of existence, the rules I follow should instead be phrased as follows:

REB6: An affirmation whose designation of the subject is empty must be false if the designation of the predicate is implied to be non-empty. Even if the designation of the predicate is implied to be empty, the affirmation will still be false, unless the subject and predicate are identical.

REB7: I will spare you the negative version, because I think you can work it out for yourself.

Now, C7 is indeed false by REB6. (It is by R6, also, but R6 is an inadequate tool for handling the present case. In other words, in stating R6, you simply have not opened up the propositional Swiss Army Knife far enough!) "Every circle that is a square" is empty, and "triangle" is implied to be non-empty (i.e., an actual triangle). So, C7 must be false.

C7p, however, is a horse of a different color! The subject "Every squared-circle" is empty, and the predicate "squared-circle" is ALSO empty. However, the subject and predicate are identical, so by REB6, the affirmation in C7p is TRUE. And indeed, it must be, since C7p is a form of the Law of Identity. (All contradictions are contradictions!)

OK, draw a double line here. I'll get back to you on the rest in several days, Thom. But I think you may be able to see how all this (REB6 and REB7) can also apply to your analysis in Parts 1 and 2.

Best,

REB (the one and only)(except for my two daughters :-)

[Michael Stuart Kelly]

... I have to invoke a couple of very important principles about "orders of existence" of "modes of being," that are crucial for propositional meaning and truth/falsity. I refer you to John J. Morrison's "The Existential Import of a Proposition in Aristotelian Logic" in Philosophy and Phenomenological Research, Vol. 15, No. 3 (March 1955), pp. 386-393.

First, for an affirmative proposition to be true, the subject and predicate must belong to the same order of existence, whether actual, possible, or fictional (mentally constructed from things that are actual or possible). Fictional things can only exist "before the mind," but as such, they can still be considered propositionally. Further, for an affirmative proposition to be true, where the subject and predicate must belong to the same order of existence, the existence of the subject must also be identical with that of the predicate.

Just as important, for a negative proposition to be true, either (1) the subject and predicate must not belong to the same order of existence, or (2) the subject and predicate must not be the same identical existence, if they belong to the same order of existence.

This is pretty basic stuff, as I see it...

Roger,

This may be pretty basic, but I never read it. I have been using this premise by the intellectual seat of my pants, so to speak.

Cool. Now I have good words for it.

Thank you.

This filter easily blasts away a lot of contextual confusion in some of the arguments I have seen, much more easily than the contextual alignments I have been proposing. (They have been correct as far as that goes, but this stuff you just gave is more fundamental.)

Michael

[Roger Bissell]

... I have to invoke a couple of very important principles about "orders of existence" of "modes of being," that are crucial for propositional meaning and truth/falsity. I refer you to John J. Morrison's "The Existential Import of a Proposition in Aristotelian Logic" in Philosophy and Phenomenological Research, Vol. 15, No. 3 (March 1955), pp. 386-393.

First, for an affirmative proposition to be true, the subject and predicate must belong to the same order of existence, whether actual, possible, or fictional (mentally constructed from things that are actual or possible). Fictional things can only exist "before the mind," but as such, they can still be considered propositionally. Further, for an affirmative proposition to be true, where the subject and predicate must belong to the same order of existence, the existence of the subject must also be identical with that of the predicate.

Just as important, for a negative proposition to be true, either (1) the subject and predicate must not belong to the same order of existence, or (2) the subject and predicate must not be the same identical existence, if they belong to the same order of existence.

This is pretty basic stuff, as I see it...

Roger,

This may be pretty basic, but I never read it. I have been using this premise by the intellectual seat of my pants, so to speak.

Cool. Now I have good words for it.

Thank you.

This filter easily blasts away a lot of contextual confusion in some of the arguments I have seen, much more easily than the contextual alignments I have been proposing. (They have been correct as far as that goes, but this stuff you just gave is more fundamental.)

Michael

You're welcome, Michael! All I can say is that your (our) "instincts" are good, and thanks to Morrison's fairly clear way of laying it all out, we now have a better tool for attacking such problems than in the past.

I'm the first one to admit that I'm just an amateur, while the guys I'm taking on are heavyweights, high rollers, etc. in the world of logic -- guys like Russell, Copi, and...yes...Kelley. But just as a string of academic letters following your name is no guarantee that you have a lock on the truth (50 million modern philosophers can't be wrong?), neither is the absence of such a string a sign that you having nothing valid to offer. :-)

REB

[Roger Bissell]

MY REPLY TO Thom's PART 2

Thom, in regard to your Part 2, “Singulars described to emptiness” (? I don’t know what that means), you give a group of statements which you say shows another weakness of my analysis or premises or both:

5. "TpKoF is bald" becomes:

5a. "TpKoF is a real person" and

5b. "TpKoF is a bald person" and

5c. "5a AND 5b"; which happens to be true. And

6. "TpKoF isn't bald" becomes:

6a. "TpKoF is a real person" and

6b. "TpKoF isn't a bald person" and

6c. "6a AND 6b"; which is false true.

First, I have no earthly idea how you arrived at the truth values for 5c and 6c. In particular, I am mystified as to how 6 c can be “false true,” whatever that is! I hope you can clarify it for us, if you see a need to do so after reading the following comments.

Now, as per my comments about your Part 3, I want to use the rules/principles that guide ~my~ thinking, rather than R6 and R7, which you have ascribed to me. Here, again, is what I use to guide my analysis of propositions with empty, possible, and actual referents:

REB6: An affirmation whose designation of the subject is empty

must be false

if the designation of the predicate is implied to be non-empty. E.g., a unicorn is a real creature. Even if the designation of the predicate is also implied to be empty, the affirmation

must be false

, unless the subject and predicate are identical. E.g., a unicorn is an imaginary island. (The positive form of this is: an affirmation containing an empty subject term

must be true

if the predicate is also an empty term,

and

the subject and predicate are identical—i.e., if the non-existent thing designated by the subject is also the non-existent thing, or one of the non-existent things, designated by the predicate. E.g., a square-circle is an impossible figure. A square-circle is a square-circle. A unicorn is a mythological beast. A unicorn is a unicorn.)

REB7: A denial whose designation of the subject is empty

must be true

if the subject and predicate do not belong to the same order of existence. E.g., a unicorn is not a real creature. Even if the subject and predicate belong to the same order of existence, the denial

must be true

, if the subject and predicate are not the same identical existence. E.g., a unicorn is not an imaginary island. (The negative form: a denial containing an empty subject term

must be false

if the predicate is also an empty term,

and

the subject and predicate are identical. E.g., a unicorn is not a mythological beast. A unicorn is not a unicorn. A square-circle is not a square-circle. A square-circle is not an impossible figure.)

OK, let’s apply these (actually, I have already, in giving examples) to the King of France issue.

5. The present King of France is bald.

First, we must express this so that subject and predicate are in the same ~category~.

5d. The present King of France is a person who is bald.

This alone brings nearly enough clarity to solve all these supposed dilemmas. The copula “is” not only connects subject and predicate, but also says that they are identical, the same thing. An entity cannot be the same thing as an attribute, but an entity can be the same thing as an entity, viz., what is designated by “present King of France” is a person (real or fictitious), and that person may be the same thing as one of the persons designated by “bald person.”

If we like, for clarity, we can spell out the fact that the predicate “person who is bald” is implied to be an actual person:

5e. The present King of France is a real person who is bald.

Now, the application of REB6 is straightforward. The predicate “person who is bald” is implied to be an actual person, while “present King of France” is empty. Thus, 5e and 5 are false.

Note that:

6. The present King of France is not bald.

is ALSO false. Expressing it with subject and predicate in the same category:

6d The present King of France is a person who is not bald.

Or (spelling out the order of existence of the predicate term):

6e. The present King of France is a real person who is not bald.

We see that again, the predicate “a person who is bald” designates a real person, while “present King of France” is empty. Applying REB6, we see that 6e and 6 must be false.

Suppose we had interpreted the denial in 6 in this way:

6f. The present King of France is not a real person who is bald.

6f is the contradictory of 5e, and thus just as 5e must be false (by REB6), 6f must be true, and so it is. Applying REB7: 6f is a denial. “The present King of France” is empty. “A real person who is bald” is implied to be a real person. The subject and predicate belong to different orders of existence. We are denying that a non-existent is a real person. This must be true.

If you want the actual contradictory of 5, 6f is how you should state it. If you do so, you will see at once that opposite truth value in contradiction is upheld. (If a proposition is true, its contradiction is false, and conversely.) However, this is not how most people interpret the contradictory of 5. They think that if you simply negate the copula (as in 6), you’ve done all the work necessary, when in fact you aren’t even half done yet! Worse, when people try to interpret 5 and 6 as contradictories, what they are really intending to assert are 5d and 6d (or 5e and 6e), which are NOT contradictories, but contraries. And while contradictories cannot both be false, but contraries CAN. 6f says that the subject is ~some~ unspecified thing other than some particular real person who is bald – but that something may be a cherry tree, an asteroid, or some particular real person who is NOT bald. 6e on the other hand says that the subject most specifically is some particular real person who is NOT bald. It's no wonder that people go round and round about the truth value of empty singular and empty universal propositions!

Now, this ought to give a clear example of what I mean by the “ambiguity” of propositions that don’t spell out the intended order of existence of the predicate term. Denials containing empty terms are radically ambiguous! Unless there is some special reason dictating otherwise, however, I think that – as against the approach taken by most people analyzing such cases – the proper way to interpret them is as contraries, not contradictories, of the corresponding assertions. That seems to get at what people really mean when they think they are negating a singular proposition with empty subject.

The most important conclusion to draw from this, however, is that the same analysis that applies, Thom, to your Part 3 on empty universals also applies to your Part 2 on empty singulars. As we’ll see, it also applies to your Part 1 on things that once existed but don’t presently exist. Hopefully I will get time to work on that part of your comments in the early part of this week.

[thomtg]

MY REPLY TO Thom's PART 2

[...]

The most important conclusion to draw from this, however, is that the same analysis that applies, Thom, to your Part 3 on empty universals also applies to your Part 2 on empty singulars. As well see, it also applies to your Part 1 on things that once existed but dont presently exist. Hopefully I will get time to work on that part of your comments in the early part of this week.[Emphasis mine, Thom]

Roger, the week has gone by, and its weekend is nearly over. I assume then that my Part 1 in Post #66 has proven insuperable to overcome in your conception of existential import. Let me finish it off with some comments to your two rejoinders to my Parts 3 and 2, respectively, in Post #67 and Post #70.

Your rejoinder to Part 3 (Post #67): On universals of emptiness:

[...]

First, I must take exception to your claim in Part 3 on "universals of emptiness" that I engaged in diversion and "bait and switch." This sounds like an accusation that I knowingly and deliberately misdirected the discussion. I assure you that, whatever fallacy, flaw, or omission there was in my comments, I did ~not~ intentionally leave anything out!

[...]

I apologize for inadvertently ascribing ill intention on your exposition. I meant to highlight only the actual switch in the direction of analysis. My identification of the particular fallacy was erroneous. (There is still a fallacy, but now I don't know what to call it.)

(It is by R6, also, but R6 is an inadequate tool for handling the present case. In other words, in stating R6, you simply have not opened up the propositional Swiss Army Knife far enough!)

I thinly induced your principles, Roger, from only the limited jumble of "god-awful" examples and distortions I was able to generate in Post #31 earlier in our discussion. I did not fully realize that I should have been much more outlandish with the examples in order to capture the full Rube Golberg. :angel:

Nevertheless, I see your point of view, and I am perfectly content to substitute in REB6 and REB7 to the list of principles that characterize your theory of existential import: R1-R8, P1, which I enumerated at the beginning of Post #66. Here they are, in my own words (from your second version in Post #70):

REB6: An affirmation whose designation of the subject is empty must be false, except when the predicate is either identical to the subject or an empty superclass thereof; which then makes it vacuously true.

REB7: A denial whose designation of the subject is empty must be true, except when the predicate is either identical to the subject or an empty superclass thereof; which then makes it vacuously false.

The problem with the reply to my Part 3, Roger, is that it re-affirms my very point that there is a switch in meaning. What used to be a triple set of presumed equivalences is now a quadruple set. The guidelines even when emended prescribe equivocations for three sets.

Here are your answers, first in Post #32:

[...]

In regard to squared-circles, you have needlessly complicated the example. You are saying that circles that are square are (or are not) also triangular. But fine, let's work with that.

If I assert: "all squared-circles are real circles that are square and triangular," that is not intelligible and meaningful? Seems to me it jolly well ~is~ intelligible and meaningful! And false, to boot! For that is what "all squared-circles are triangular" ~really means~. That is what it really ~intends~ to assert. Which is clearly an incorrect assertion of an identification of reality.

"No squared-circle is triangular" similarly really means "no squared-circle is a real circle that is square and triangular." Here we have an intelligible, meaningful, TRUE proposition. It is a FACT that there are no real circles that are square and triangular! This is a correct assertion of an identification of reality!

[...]

With the above, I reorganized them in Post #66 as:

11. "All squared-circles are triangular"; by R1, R6, R8, and C9 (see scratchpad), 11 is false.

12. "No squared-circle is triangular"; by R1, R7, R8, and C10 (see scratchpad), 11 is true.

[...]

Scratchpad table to transform statements 11 and 12:

C1. "Every circle is a square"; C1 is false.

C2. "Every circle isn't a square"; C2 is true.

C3. "Every circle is a triangle"; C3 is false.

C4. "Every circle isn't a triangle"; C4 is true.

C5. "Every circle is both a square and a triangle"; C5 is false.

C6. "Every circle isn't both a square and a triangle"; C6 is true.

C7. "Every circle that is square is a triangle"; C7 is not considered.

C8. "Every circle that is square isn't a triangle"; C8 is not considered.

C9. "Every circle that is square is a (real) circle that is both square and triangular"; C9 is equivalent to C5.

C10. "Every circle that is square isn't a (real) circle that is both square and triangular"; C10 is equivalent to C6.

(Note: I am taking the liberty of regimenting O-form statements "No S is P" to the more literal version "Every S isn't P" in order to better separate and discern S and P.)

According to your post [Post #32], first, 11 is transformed to C9, and 12 to C10. Then from the results, you explicitly equate C9 with C5, and C10 with C6. This is from the assumption of rule R8. Parts of nouns shift from subject to predicate interchangeably.

[...]

Then in your rejoinder, you merely affirm that the set {C7, C8} is equivalent to {11, 12}, which I agree. However, you haven't denied that your Post #32 asserts {11, 12} equivalently to {C9, C10}, and yonder {C5, C6}.

Now I grant you that the odd Cs (C5, C7, C9) are all false, and the even Cs (C6, C8, C10) are all true; but I draw a triple line against the respective triplets being equivalent in meaning as you claimed. I drew the scratchpad to make that clear. My essential point from Post #66 is still unchallenged:

While the scratchpad table does show correctly that C5 is false and C6 true, they however are not equivalent respectively to C7 and C8.

In fact, it would seem that you have transgressed your guidelines to move from one order of existence to another, from the realm of impossible objects (circle that is square) to the realm of real objects (circle). This is the direct effect of R8, which you indirectly affirm in the reformulated REB6-7 rules.

And speaking of REB6-7 and R8, I want to be clear that I take the word "identical" precisely to mean having two separate things be similar to the highest degree. (See Merlin Jetton's Objectivity article "Pursuing Similarity", p. 42.) If we are on the same page with this meaning, then as I said before in Post #31, we are moving the issue beyond merely the theory of existential import, to the theory of propositions.

Your rejoinder to Part 2 (Post #70): On singulars described to emptiness:

Okay, I see two typos in my presentation in this part in Post #66. While I first recapitulated your interpretation correctly (from your Post #32), I then made the typos in its expansion.

[...]

5. "TpKoF is bald" is reinterpreted by R5 to "TpKoF is a presently existing person, AND he is presently bald"; and since the first conjunct is false by R6, the compound statement is false; hence, 5 is false.

6. "TpKoF isn't bald" is reinterpreted by R5 to "TpKoF is a presently existing person, AND he isn't presently bald"; and since the first conjunct is false by R6, the compound statement is false, even though the second conjunct is true by R7; hence 6 is false.

[...]

5. "TpKoF is bald" becomes:

5a. "TpKoF is a real person" and

5b. "TpKoF is a bald person" and

5c. "5a AND 5b"; which happens to be true. And [It should have been "false" as above.]

6. "TpKoF isn't bald" becomes:

6a. "TpKoF is a real person" and

6b. "TpKoF isn't a bald person" and

6c. "6a AND 6b"; which is false true. [It should have been just "false."]

[...]

Though typographically incorrect, I correctly attributed your R6-7 (or REB6-7) for each expansion. My point about the expansions was not so much on the truth and falsity of the statements, which I had already highlighted in the recapitulations, as on the expansions themselves being required. I wanted to draw attention to your reliance on R5, which goes beyond R2. It is the need to transmogrify one assertion into three. And you affirmed the same approach again in Post #70.

Why must you make singular statements of the descriptive kind special? Notice that R5 is orthogonal to REB6-7. R5 is the real issue here while REB6-7 are merely tangential, which is why I even used "The current President of France" ("TcPoF") to illustrate the same fallacy in using R5. It is what I mentioned in Post #31 as your translation of Bertrand Russell's theory of descriptions.

---

To conclude: Although you may have saved REB6-7 in the Part 3 by adding an exception clause, that does not change the fact that you make exceptions to empty designations, as opposed to designations of different orders of existence. REB6-7 is orthogonal to R4, which I agreed from the start. So, in a sense, your first rejoinder has not landed on the real targets, which in Part 3 are my comments against R8. And in Part 2, the comments against your R5 are the targets, which are independent of your defense of REB6-7, and they have not been sunk.

[Roger Bissell]

MY REPLY TO Thom's PART 2

[...]

The most important conclusion to draw from this, however, is that the same analysis that applies, Thom, to your Part 3 on empty universals also applies to your Part 2 on empty singulars. As well see, it also applies to your Part 1 on things that once existed but dont presently exist. Hopefully I will get time to work on that part of your comments in the early part of this week.[Emphasis mine, Thom]

Roger, the week has gone by, and its weekend is nearly over. I assume then that my Part 1 in Post #66 has proven insuperable to overcome in your conception of existential import. Let me finish it off with some comments to your two rejoinders to my Parts 3 and 2, respectively, in Post #67 and Post #70.

Same to you, with the tongue bit. Your assumption is unwarranted. I work for a living, sometimes double-shifts, usually also on my "days off" (which are nominally Tuesday and Wednesday), and I have just been too busy to work down my list of priorities to epistemology/logic correspondence. Even now, I do not have time to deal with your comments in Part 1.

As for your replies (not included below) to my critique of your Parts 2 and 3, I can't really follow them well enough to understand your point, let alone how to reply. I'll try again when I have time.

Your rejoinder to Part 3 (Post #67): On universals of emptiness:

[...]

First, I must take exception to your claim in Part 3 on "universals of emptiness" that I engaged in diversion and "bait and switch." This sounds like an accusation that I knowingly and deliberately misdirected the discussion. I assure you that, whatever fallacy, flaw, or omission there was in my comments, I did ~not~ intentionally leave anything out!

[...]

I apologize for inadvertently ascribing ill intention on your exposition. I meant to highlight only the actual switch in the direction of analysis. My identification of the particular fallacy was erroneous. (There is still a fallacy, but now I don't know what to call it.)

Well, I take you at your word that you ascription of ill intent to me was inadvertent. I will just point out that the phrase "bait and switch" is well known in American culture and the English language to refer to a ~deliberate~ attempt to deceive or cheat another person by offering one thing and then substituting another. This is not a fallacy, as far as I know, but simply unethical behavior.

I think the fallacy you were looking for was ~equivocation~, something along the lines of the fallacy of the middle term.

(It is by R6, also, but R6 is an inadequate tool for handling the present case. In other words, in stating R6, you simply have not opened up the propositional Swiss Army Knife far enough!)

I thinly induced your principles, Roger, from only the limited jumble of "god-awful" examples and distortions I was able to generate in Post #31 earlier in our discussion. I did not fully realize that I should have been much more outlandish with the examples in order to capture the full Rube Golberg. :angel:

Do you mean Rube Goldberg? He is my inspiration. I'm hoping to follow in his footsteps and win a Pulitzer Prize for designing a new, needlessly convoluted, overly complex method for analyzing the actual meaning of propositions and assessing their truth value. Yes, the more outlandish your examples, the better.

Seriously, I find it curious that you induced ~my~ principles from ~your~ examples and distortions. That is a new wrinkle on Objectivist methodology that I will probably not be adopting any time soon.

REB

[BaalChatzaf]

1. "All dinosaurs are scaly" is R3 elliptical for "All dinosaurs are presently existing scaly creatures"; and since none exists presently (R4), 1 is false.

Wrong! It means that if x is a dinosaur then x is scaly.

All A are B translates to (x)(x in A -> x in B). In case A is empty the formula is true.

Ba'al Chatzaf

Edited May 22, 2009 by BaalChatzaf