Aristotle's term logic is isomorphic to a special (restricted) version of propositional logic. The main difference between Aristotle's term logic and Boole's propositional logic is that universal statements have no existential import.

Aristotle's term logic is isomorphic to a special (restricted) version of propositional logic. The main difference between Aristotle's term logic and Boole's propositional logic is that universal statements have no existential import.

Ba'al Chatzaf

In your dreams, Ba'al. Some universal statements don't have existential import, but many do -- just as many particular statements do have existential import, but some don't. It is a double standard without justification to deny existential import to universal propositions, while granting it to particular propositions.

The whole Boolean approach is a fraud, in my opinion, but especially the bit about universal categorical propositions NOT having existential import, while particular categorical propositions DO. I've read all of B. Russell's blathering in MIND c. 1905, as well as several other interesting essays from then and earlier.

One in particular really expressed the matter fairly well. Check it out: "The Existential Import of Propositions," by W. Blair Neatby, MIND vol. 6, no. 24, Oct. 1897, pp. 542-546.

Neatby basically blasts the whole notion that universal categorical propositions do not have Existential Import by pointing out that a great many of the examples purporting to illustrate it are actually hypothetical propositions expressed as categoricals. He gives this example: "All students arriving late will be penalized." Since there may in fact be no students who arrive late, it is claimed to lack Existential Import. Neatby says, however, that the proposition is more clearly expressed as a hypothetical: if a student arrives late, he will be penalized.

Actually, while Neatby's analysis adequately addresses the issue, I think the matter is even more fundamentally awry, and the clue is in the future tense used to state the school policy. I think it's clear that the statement "All students arriving late will be penalized" is actually a disguised individual proposition with a concrete subject, the school policy, and is better stated as a categorical in present tense: "Our policy is that any student arriving late will be penalized." It's not really a universal proposition at all.

These supposed categorical universal propositions (actually hypotheticals or individuals) are alleged to differ fundamentally from categorical particular propositions. Particular categorical propositions, the Boolean interpretation goes, do have Existential Import. But as Neatby notes, there are just as many problemmatic examples of particular categorical propositions as of universal categorical propositions. For instance, "Some students arriving late will be penalized." There may in fact be no students who arrive late, so this particular categorical proposition lacks Existential Import to the same extent as the parallel universal categorical proposition mentioned in the preceding paragraph. It certainly, Neatby says, "does not necessarily imply that any [student] ever arrived late. It may only be a partial statement of a regulation that provides for the [penalizing] of any [student] who comes late without an excuse signed by his [parent]." [emphasis added]

Here, in my mind, is the clincher from Neatby, pointing out the double standard of the moderns regarding existential import:

Formal logicians have possibly the right to frame whatever convention they find best suited to their purpose. But if they claim the sanction of usage, and consider it urgent to keep in harmony with the actual forms of speech, they are not at liberty to frame a convention that allows a necessary existential import to particular and singular propositions, and denies it to universal propositions. In all three cases, the (even apparent) exceptions are rare, but they are found in all alike. It is difficult to imagine what consideration would entitle us to neglect them in any, without entitling us to neglect them in all. [emphasis added]

Yikes! Well said, Prof. Neatby!

Neatby also discusses examples involving deliberate irony and explicitly denial of existence, which are also included among the examples allegedly justifying the doctrine of Existential Import (i.e., its denial that universal categorical propositions have it).

Neatby decisively refuted the whole Boolean edifice based on Existential Import, and yet the moderns went their merry way as if nothing had been said to seriously challenge their dogma. Aristotle's principles of immediate inference as embodied in the Square of Opposition were tossed out the window, and his wonderful diagram was converted into an "X" of Opposition, gutted of much of its power and usefulness in dealing with proposition about non-existent subjects. And decades later, David Kelley can't find anything seriously wrong with the modern, Boolean approach -- at least, not as recently as the 3rd edition (1998) of his logic text.

I have a standing invitation to anyone here on OL to suggest a genuine universal categorical proposition that they think does not have existential import, a lack which the moderns say exempts such a proposition from the rules of immediate inference via obversion, conversion, and contraposition. We can go through the standard examples like "Sea serpents are scaly," or you can make up one of your own, if you like. But I'd strongly suggest that anyone who is interested in this matter read Neatby's essay, or Veatch's Two Logics or Intentional Logic if they can find them -- something other than the die-hard proponents of the Boolean system that Ba'al thinks have a lock on the truth. Why should revisionist historians have all the fun, anyway? :-)

The whole Boolean approach is a fraud, in my opinion, but especially the bit about universal categorical propositions NOT having existential import, while particular categorical propositions DO. I've read all of B. Russell's blathering in MIND c. 1905, as well as several other interesting essays from then and earlier.

Every electronic computer is or could be designed using Boolean truth tables. Are computers a fraud.

Boolean logic has made the study of logic totally rigorous and mathematically respectable. A boolean algebra is derived from a complemented lattice. This puts propositional logic on a solid foundation. I got this sh*t, when I discussed modern logic with that ignoramus, Leonard Peikoff.

And by the way, there are empty sets. Aristotle's logic could not handle this. By removing existential import one gets the true assertion that the empty set is a subset of every set.

The whole Boolean approach is a fraud, in my opinion, but especially the bit about universal categorical propositions NOT having existential import, while particular categorical propositions DO. I've read all of B. Russell's blathering in MIND c. 1905, as well as several other interesting essays from then and earlier.

Every electronic computer is or could be designed using Boolean truth tables. Are computers a fraud.

Boolean logic has made the study of logic totally rigorous and mathematically respectable. A boolean algebra is derived from a complemented lattice. This puts propositional logic on a solid foundation. I got this sh*t, when I discussed modern logic with that ignoramus, Leonard Peikoff.

And by the way, there are empty sets. Aristotle's logic could not handle this. By removing existential import one gets the true assertion that the empty set is a subset of every set.

The whole Boolean approach is a fraud, in my opinion, but especially the bit about universal categorical propositions NOT having existential import, while particular categorical propositions DO. I've read all of B. Russell's blathering in MIND c. 1905, as well as several other interesting essays from then and earlier.

Every electronic computer is or could be designed using Boolean truth tables. Are computers a fraud?

Boolean logic has made the study of logic totally rigorous and mathematically respectable. A boolean algebra is derived from a complemented lattice. This puts propositional logic on a solid foundation. I got this sh*t, when I discussed modern logic with that ignoramus, Leonard Peikoff.

And by the way, there are empty sets. Aristotle's logic could not handle this. By removing existential import one gets the true assertion that the empty set is a subset of every set.

Ba'al Chatzaf

No. Computers aren't a fraud because of Boolean truth tables any more than candles were a fraud during the period that some people believed they burned because of phlogiston. Just because someone (or a large group of people) cling to an invalid metaphysical interpretation of why calculations work in the real world, that doesn't make the results of those calculations false, any more than the premises "All cows are flying creatures. All flying creatures are mammals" make the conclusion "Therefore, all cows are mammals" false.

As for this bit about the empty set, haven't we been there before? The complement of a set is defined in relation to some larger third set, of which they both are subsets--and together in relation to which they non-overlappingly comprise the total membership of the larger set. For instance, in regard to a set of six apples, the set comprised by two of those apples is the complement of the set comprised by the the other four of those apples. There is no problem understanding the meaning of "complement" here, nor of the union of a set and its complement in relation to a larger whole.

But it is the fact a set and its complement are both subsets of a ~larger~ whole that rules out considering the "empty" set as the complement of the larger whole. To "complement" means to ~add to~ something in order to make a whole. But the six apples ~already~ are a whole six apples, and you cannot meaningfully add ~zero~ apples in order to make the six apples a whole, because they already ~are~ a whole. Zero apples is (are?) NO PART of six apples, and thus not only is zero apples NO SUBSET of six apples, it is NO COMPLEMENT of six apples.

It is a misnomer to speak of the ~union~ of something and nothing, because, for instance, you cannot meaningfully speak of the union of six apples and zero apples. You are not finding the union of ~anything~ with the set of six apples, because the set of six apples is already a set of six apples. I don't see how you can escape the implication that the notation expressing a union of the null set with another set simply means that the operation of set union IS NOT PERFORMED.

Does anything vital to modern science and modern living hinge on the empty set being a subset of every set?

I haven't read Veatch's book but it sounds interesting. I must say though, that I'm having a hard time understanding the difference between a logic of "what is" and the modern logic of relations (predicate calculus).

How do you know what anything IS, except in terms of its relations with other things? It seems to me that the Aristotelian logic can be seen as logic of relations between classes, but those kinds of relations are not the only kind. How, for example, would you express "Everyone falls in love with someone on some enchanted evening" using the traditional syllogism?

It is a misnomer to speak of the ~union~ of something and nothing, because, for instance, you cannot meaningfully speak of the union of six apples and zero apples. You are not finding the union of ~anything~ with the set of six apples, because the set of six apples is already a set of six apples. I don't see how you can escape the implication that the notation expressing a union of the null set with another set simply means that the operation of set union IS NOT PERFORMED.

Nonsense. {a1, a2, a3, a4, a5, a6} U { } = {a1, a2, a3, a4, a5, a6}. I just performed it. Apparently you too suffer from Peikoff Syndrome.

I haven't read Veatch's book but it sounds interesting. I must say though, that I'm having a hard time understanding the difference between a logic of "what is" and the modern logic of relations (predicate calculus).

How do you know what anything IS, except in terms of its relations with other things? It seems to me that the Aristotelian logic can be seen as logic of relations between classes, but those kinds of relations are not the only kind. How, for example, would you express "Everyone falls in love with someone on some enchanted evening" using the traditional syllogism?

You don't. That is why Frege and Peirce invented predicate logic able to handle predicates of order 2 and higher.

Yes, it seems that multiple and nested quantifiers are incapable of being expressed...

Another example: how would express the concept of a mathematical limit without using the predicate calculus? I'm not even sure that Sommer's term logic could do that.

It is a misnomer to speak of the ~union~ of something and nothing, because, for instance, you cannot meaningfully speak of the union of six apples and zero apples. You are not finding the union of ~anything~ with the set of six apples, because the set of six apples is already a set of six apples. I don't see how you can escape the implication that the notation expressing a union of the null set with another set simply means that the operation of set union IS NOT PERFORMED.

Nonsense. {a1, a2, a3, a4, a5, a6} U { } = {a1, a2, a3, a4, a5, a6}. I just performed it. Apparently you too suffer from Peikoff Syndrome.

Ba'al Chatzaf

Nonsense on stilts. If I "unite" with nothing, I am NOT UNITING WITH ANYTHING. I am not performing an operation of union.

It is a misnomer to speak of the ~union~ of something and nothing, because, for instance, you cannot meaningfully speak of the union of six apples and zero apples. You are not finding the union of ~anything~ with the set of six apples, because the set of six apples is already a set of six apples. I don't see how you can escape the implication that the notation expressing a union of the null set with another set simply means that the operation of set union IS NOT PERFORMED.

Nonsense. {a1, a2, a3, a4, a5, a6} U { } = {a1, a2, a3, a4, a5, a6}. I just performed it. Apparently you too suffer from Peikoff Syndrome.

Ba'al Chatzaf

Nonsense on stilts. If I "unite" with nothing, I am NOT UNITING WITH ANYTHING. I am not performing an operation of union.

Apparently you suffer from Russell Syndrome.

REB

The union of sets A and B (symbolized A U B) is the set of all elements x such that x in A or x in B. (inclusive or). Consult any college text book on set theory.

Learn some technical definitions before you spout off.

It is so damned annoying when people who are technically ignorant of a subject pontificate upon it.

Nonsense on stilts. If I "unite" with nothing, I am NOT UNITING WITH ANYTHING. I am not performing an operation of union

This is akin to Roger's much discussed assertions about zero in this thread. One such assertion -- bizarre in my view -- is that 0+5 makes sense but 5+0 doesn't (link). So, Roger, what do you make of

Nonsense on stilts. If I "unite" with nothing, I am NOT UNITING WITH ANYTHING. I am not performing an operation of union

This is akin to Roger's much discussed assertions about zero in this thread. One such assertion -- bizarre in my view -- is that 0+5 makes sense but 5+0 doesn't (link). So, Roger, what do you make of

Seems pretty straightforward, whether in this form or as distributed.

In this form, the intersection of the first two sets is the "empty set." (They have no common members.) The "union" of the empty set and the third set is simply the third set, {a1, a2, a3, a4, a5, a6}, because there is nothing to unite the third set with, so nothing is united with it.

As distributed, the union of the first and third sets is {a1, a2, a3, a4, a5, a6, a7, a8, a9} and the union of the second and third sets is {a1, a2, a3, a4, a5, a6, a10, a11, a12}. The intersection of these unions (the members in common between the two unions) is again simply the third set.

As for 0 + 5 versus 5 + 0, I don't say the latter doesn't make sense. I'm only saying there is no operation being performed. The meaning of "5 + 0" is "5 with no operation of addition being performed upon it by any number," which is simply "5."

The former, "0 + 5," again is a case of no operation being performed. It simply means "5 not being added to any number," which is again simply "5." They both make perfect sense to me, as interpreted.

Similarly, the union of an empty set with a non-empty set is the same as the union of that non-empty set with the empty set. There is no such thing, for there is no such operation being performed in either case. So, in the former case, the meaning of the result of the supposed operation is: that non-empty set not being united with any set." In the latter, the meaning of the result of the supposed operation is: that non-empty set with no operation of set union being performed on it by any set.

In each case, numbers and sets, it is as if the number and non-empty set had been stated with a single symbol. As Kant might have said (in a much different context, where he was denying existence as a predicate), ~nothing has been added to~ (or ~united with~) the number or non-empty set. And he wouldn't have meant "nothing" like "some empty kind of thing," but "nothing" like "the absence of anything." You don't add non-existence to existence, and you don't unite non-existence with existence. Trying to do so is a fruitless, meaningless gesture. You get what you started with, which means you didn't do anything except wave your arms or flap your jaw.

As for 0 + 5 versus 5 + 0, I don't say the latter doesn't make sense. I'm only saying there is no operation being performed. The meaning of "5 + 0" is "5 with no operation of addition being performed upon it by any number," which is simply "5."

The former, "0 + 5," again is a case of no operation being performed. It simply means "5 not being added to any number," which is again simply "5." They both make perfect sense to me, as interpreted.

Okay, maybe I should have said "make sense qua operation." In any case months ago you said differently about one of them: "In the example you gave, after five additions, you arrive at 0 + 5. You can add 5 to 0 perfectly well. You can add five chairs to an empty room. But you cannot add zero chairs to a room with five chairs in it. You cannot add 0 to 5" (link).

As for 0 + 5 versus 5 + 0, I don't say the latter doesn't make sense. I'm only saying there is no operation being performed. The meaning of "5 + 0" is "5 with no operation of addition being performed upon it by any number," which is simply "5."

The former, "0 + 5," again is a case of no operation being performed. It simply means "5 not being added to any number," which is again simply "5." They both make perfect sense to me, as interpreted.

Okay, maybe I should have said "make sense qua operation." In any case months ago you said differently about one of them: "In the example you gave, after five additions, you arrive at 0 + 5. You can add 5 to 0 perfectly well. You can add five chairs to an empty room. But you cannot add zero chairs to a room with five chairs in it. You cannot add 0 to 5" (link).

That is what happens when one equates the identity element of addition to Nothing At All.

As for 0 + 5 versus 5 + 0, I don't say the latter doesn't make sense. I'm only saying there is no operation being performed. The meaning of "5 + 0" is "5 with no operation of addition being performed upon it by any number," which is simply "5."

The former, "0 + 5," again is a case of no operation being performed. It simply means "5 not being added to any number," which is again simply "5." They both make perfect sense to me, as interpreted.

Okay, maybe I should have said "make sense qua operation." In any case months ago you said differently about one of them: "In the example you gave, after five additions, you arrive at 0 + 5. You can add 5 to 0 perfectly well. You can add five chairs to an empty room. But you cannot add zero chairs to a room with five chairs in it. You cannot add 0 to 5" (link).

That's correct, Merlin. My present understanding on this point is not the same as it was two years ago. I think it's clearer and more consistent now.

All I'm saying here is that the ontological meaning of the identity element in addition is that no operation of addition is being performed. But not adding anything or to anything is only functionally and verbally equivalent to adding nothing or to nothing. Ontologically, they are as different as...existence and non-existence!

That's correct, Merlin. My present understanding on this point is not the same as it was two years ago. I think it's clearer and more consistent now.

All I'm saying here is that the ontological meaning of the identity element in addition is that no operation of addition is being performed. But not adding anything or to anything is only functionally and verbally equivalent to adding nothing or to nothing. Ontologically, they are as different as...existence and non-existence!

There is the conflict -- criticizing formalisms for ontological reasons. Formalisms are (usually) intended to be logical, not ontological. It's an emphasis of form over content.

Suppose you want to learn how much in coins you have in your pockets. You reach in one pocket and extract 43 cents. You reach in another pocket and come up empty. It is not that you performed no operation; it is that you still only have 43 cents.

Syllogisms with propositions such as 'All S is P' are the same way. We can grasp that a syllogism is formally correct or incorrect w/o having to check if the terms have referents.

That's correct, Merlin. My present understanding on this point is not the same as it was two years ago. I think it's clearer and more consistent now.

All I'm saying here is that the ontological meaning of the identity element in addition is that no operation of addition is being performed. But not adding anything or to anything is only functionally and verbally equivalent to adding nothing or to nothing. Ontologically, they are as different as...existence and non-existence!

There is the conflict -- criticizing formalisms for ontological reasons. Formalisms are (usually) intended to be logical, not ontological. It's an emphasis of form over content.

I'm not criticizing the formalism of "x + 0 = x." The formalism is just fine. It's the ~standard ontological interpretation~ of that formalism I'm saying is not correct. I'm saying that when you attempt to perform the operation of addition with zero, nothing happens--and that is not because you are ADDING NOTHING (0) to x, but because you are _NOT ADDING ANYTHING_ to x.

Let me put it another way, to those who like to remind me that zero is the "additive identity." The ~reason~ that zero is the additive identity is that "x + 0 = x" is the equivalent of saying "x = x." If you start with x and you perform no operation of addition on x, then x remains what it is, instead of becoming something different!

"0 + x = x" is the equivalent of saying "x = x," because if you don't start without anything, you are in fact not starting with anything until you introduce x, at which point you are ~starting with~ x. Again, there is no operation of addition being performed, and x again remains what it is, instead of becoming something different by being added to something.

Suppose you want to learn how much in coins you have in your pockets. You reach in one pocket and extract 43 cents. You reach in another pocket and come up empty. It is not that you performed no operation; it is that you still only have 43 cents.

Oh, Kant rare! Take a closer look at how you are interpreting your process--or, rather, PROCESSES--there. You are package-dealing the process of reaching in and pulling out whatever is there with the process of ACTUALLY COUNTING THE COINS--and that with the additional process of TRYING TO ADD actual coins with non-existent coins.

Let's walk through it: "OK, I'm reaching in my left pocket, and I feel some coins in there, so I grab and pull them out. Then I add up the face value of all the coins from that pocket and find that there are 43 cents in total. Now, I'm reaching in my right pocket to see if there are any coins whose value I might add to the 43 cents from my left pocket. Aw shucks, no coins. Hmmm, that means there are no coins, no cents to add to the 43 cents from my left pocket. So, I won't perform any operation of addition, since I don't have anything to add."

Actually, that might be a good comment to end with, except I ~do~ have something to add to Merlin's closing comment. :-)

Syllogisms with propositions such as 'All S is P' are the same way. We can grasp that a syllogism is formally correct or incorrect w/o having to check if the terms have referents.

Well, because of the additive identity of zero, it's certainly true that the equation "x + 0 = x" is necessarily correct, regardless of what units are referred to by "x"--AND EVEN IF X DOES NOT REFER TO ANY UNITS AT ALL! If x = 0 (i.e., if there are no units referred to by "x"), then it is indeed true that "0 + 0 = 0"--NOT because you are ADDING ZERO to zero, however, but because you are NOT ADDING ANYTHING TO zero. In other words, when x = 0, "0 + 0 = 0," BECAUSE "0 = 0." Again, the true ontological interpretation of zero as the additive identity is clear to see.

Mathematical and logical formalisms that WORK in the real world didn't just fall out of somebody's overactive imagination or some other sphincter. They were ABSTRACTED FROM REAL CONNECTIONS in the world.

In the case of addition, we know that when you don't increase a given quantity by some specific quantity, that given quantity remains unchanged. And when you don't have any specific quantity to start with, then there isn't anything to be changed by the introduction of some specific quantity. In all other cases, the result is something different from either of the two quantities on the left side of the equation, because the first is ~something~ that is ~increased~ by ~something else~. You count the units specified by the first number, then continue the sequence by counting the additional units specified by the second number.

When either number is zero, there is no counting beyond the units specified by the non-zero number, so there is no operation of addition being performed, and the additive identity prevails, because when there is only one number involved, that number is itself. All other addition facts are connections abstracted from reality that must be learned by examination, study, and memorization, and checked and re-checked if necessary -- but the additive identity is a fundamental, axiomatic connection in reality that is grasped by realizing that to deny it leads to a contradiction.

Something analogous happens in logically valid syllogisms using false statements or even nonsense terms. Yes, the middle term acts as the linking term between the terms that are the subject and predicate of the conclusion. But the logical link in a syllogism does not always point to a real connection between concepts, just as the plus sign for addition does not always point to a real operation of addition being performed.

Just as the operation of addition is abstracted from an actual process of counting the total number of units referred to by two or more numerals, so is the operation of syllogistic inference abstracted from an actual process of identifying the causal connection between two facts. And because the formal structure of causal connections is universal and abstractable from reality, it can be used with ~any~ propositional content to generate a formal connection. And when the premises are true, then the formal connection also corresponds to reality. But when the premises are false, then it's, as Rand would say, "deuces wild" -- or as our computer mavens might say, "garbage in, garbage out."

All cows are mammals.

All mammals are creatures that fly.

Therefore, all cows are creatures that fly.

All bats are mammals.

All mammals are creatures that fly.

Therefore, all bats are creatures that fly.

In each case, the middle term, "mammals," purports to be the CAUSE or REASON in reality of the fact that all cows, or all bats, are creatures that fly. In the first case, the conclusion is false. In the second case, it's true -- but it's only true BY ACCIDENT. GIGO. Some garbage ~just happens~ to be true. But as the conclusion of a syllogism with false premises, it's not knowledge, just a true proposition you know to be true for reasons independent of the unsound syllogism.

I mention all of this, not to argue that syllogism are just like addition, but to point out that valid formalisms have a source in the real world, and that empty or false ~uses~ of those formalisms have to be ~understood~ and not casually manipulated as though they were arbitrary, logical tinkertoys.

I'm not criticizing the formalism of "x + 0 = x." The formalism is just fine. It's the ~standard ontological interpretation~ of that formalism I'm saying is not correct. I'm saying that when you attempt to perform the operation of addition with zero, nothing happens--and that is not because you are ADDING NOTHING (0) to x, but because you are _NOT ADDING ANYTHING_ to x.

You may think there is a deep ontological difference here, but it strikes me as merely semantics.

Oh, Kant rare! Take a closer look at how you are interpreting your process--or, rather, PROCESSES--there. You are package-dealing the process of reaching in and pulling out whatever is there with the process of ACTUALLY COUNTING THE COINS--and that with the additional process of TRYING TO ADD actual coins with non-existent coins.

Let's walk through it: "OK, I'm reaching in my left pocket, and I feel some coins in there, so I grab and pull them out. Then I add up the face value of all the coins from that pocket and find that there are 43 cents in total. Now, I'm reaching in my right pocket to see if there are any coins whose value I might add to the 43 cents from my left pocket. Aw shucks, no coins. Hmmm, that means there are no coins, no cents to add to the 43 cents from my left pocket. So, I won't perform any operation of addition, since I don't have anything to add."

No, I'm not package-dealing; I'm applying a formalism. Indeed, your second paragraph "package-deals" two operations -- reaching into your pocket and abstract addition. It and the first quote appear to confound reaching into your pocket and finding nothing with not reaching into your pocket at all.

Oh, Kant rare! Take a closer look at how you are interpreting your process--or, rather, PROCESSES--there. You are package-dealing the process of reaching in and pulling out whatever is there with the process of ACTUALLY COUNTING THE COINS--and that with the additional process of TRYING TO ADD actual coins with non-existent coins.

Start with an empty cup. Now toss six pennies into the cup. How many pennies in the cup? Six.

I have just added six coins to a set of non-existent coins.

Oh, Kant rare! Take a closer look at how you are interpreting your process--or, rather, PROCESSES--there. You are package-dealing the process of reaching in and pulling out whatever is there with the process of ACTUALLY COUNTING THE COINS--and that with the additional process of TRYING TO ADD actual coins with non-existent coins.

Start with an empty cup. Now toss six pennies into the cup. How many pennies in the cup? Six.

I have just added six coins to a set of non-existent coins.

I am afraid you are no mathematician, Roger.

Ba'al Chatzaf

It seems to me that ~anywhere~ and ~everywhere~ that someone ~might~ place six coins would be "a set of non-existent coins." The universe, by this view, is through-and-through empty sets, waiting for the ~possible~ addition of six coins, ten thousand coins, three horses, twelve bogus mathematical-theoretical arguments for the existence of empty sets, whatever.

This all smacks of idealism, to me, Ba'al. I don't know if you have heard of, or recall, J. S. Mill's notion that matter is the permanent possibility of sensation. But it sounds a lot like how your view implies the universe is like: the permanent possibility of non-empty sets.

A set is not like a room or a cup. A set is a collection of objects. The room or cup or area on a table or on the ground where the pennies are placed is not a set before the pennies are placed there. It is just a room, cup, etc.

If I showed someone my empty china cabinet and said what do you think of my set of china, he would rightly say: "What set of china?" According to you, I should say, "You must be blind. It's an empty set of china." To which he would rightly say: "Oh, yeah, right next to the empty set of sane ideas that are coming out of your mouth right now."

I don't like that the empty set is a subset of every set, just as I don't like that X raised to the power zero is one. However, the truth of both is required to preserve the total logic of a system. Denial that X raised to the power zero equals one leads to contradiction, so that's that.

Likewise for the empty set. The subsets of {a1,a2} can be confirmed by checking the unions:

{a1, a2} U {a1, a2} = {a1, a2}

{a1, a2} U {a1} = {a1, a2}

{a1, a2} U {a2} = {a1, a2}

{a1, a2} U { } = {a1, a2}

Therefore, all four ( {a1, a2}, {a1}, {a2} and { } ) are subsets of {a1, a2}.

It isn't pretty, but it appears unavoidable.

I don't see anything wrong with saying, "It's an empty set of china." An empty china cabinet IS an empty set of china.

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## BaalChatzaf

Aristotle's term logic is isomorphic to a special (restricted) version of propositional logic. The main difference between Aristotle's term logic and Boole's propositional logic is that universal statements have no existential import.

Ba'al Chatzaf

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## Roger Bissell

AuthorIn your dreams, Ba'al. Some universal statements don't have existential import, but many

do-- just as many particular statements do have existential import, but somedon't. It is a double standard without justification to deny existential import to universal propositions, while granting it to particular propositions.The whole Boolean approach is a fraud, in my opinion, but especially the bit about universal categorical propositions NOT having existential import, while particular categorical propositions DO. I've read all of B. Russell's blathering in MIND c. 1905, as well as several other interesting essays from then and earlier.

One in particular really expressed the matter fairly well. Check it out: "The Existential Import of Propositions," by W. Blair Neatby, MIND vol. 6, no. 24, Oct. 1897, pp. 542-546.

Neatby basically blasts the whole notion that universal categorical propositions do not have Existential Import by pointing out that a great many of the examples purporting to illustrate it are actually hypothetical propositions expressed as categoricals. He gives this example: "All students arriving late will be penalized." Since there may in fact be no students who arrive late, it is claimed to lack Existential Import. Neatby says, however, that the proposition is more clearly expressed as a hypothetical: if a student arrives late, he will be penalized.

Actually, while Neatby's analysis adequately addresses the issue, I think the matter is even more fundamentally awry, and the clue is in the future tense used to state the school policy. I think it's clear that the statement "All students arriving late will be penalized" is actually a disguised

individualproposition with a concrete subject, the school policy, and is better stated as a categorical inpresenttense: "Our policy is that any student arriving late will be penalized." It's not really a universal proposition at all.These supposed categorical universal propositions (actually hypotheticals or individuals) are alleged to differ fundamentally from categorical particular propositions. Particular categorical propositions, the Boolean interpretation goes,

dohave Existential Import. But as Neatby notes, there are just as many problemmatic examples of particular categorical propositions as of universal categorical propositions. For instance, "Some students arriving late will be penalized." There may in fact be no students who arrive late, so this particular categorical proposition lacks Existential Importto the same extentas the parallel universal categorical proposition mentioned in the preceding paragraph. It certainly, Neatby says, "does not necessarily imply that any [student] ever arrived late. It may only be a partial statement of a regulation that provides for the [penalizing] of any [student] who comes late without an excuse signed by his [parent]." [emphasis added]Here, in my mind, is the clincher from Neatby, pointing out the double standard of the moderns regarding existential import:

Yikes! Well said, Prof. Neatby!

Neatby also discusses examples involving deliberate irony and explicitly denial of existence, which are also included among the examples allegedly justifying the doctrine of Existential Import (i.e., its denial that universal categorical propositions have it).

Neatby decisively refuted the whole Boolean edifice based on Existential Import, and yet the moderns went their merry way as if nothing had been said to seriously challenge their dogma. Aristotle's principles of immediate inference as embodied in the Square of Opposition were tossed out the window, and his wonderful diagram was converted into an "X" of Opposition, gutted of much of its power and usefulness in dealing with proposition about non-existent subjects. And decades later, David Kelley can't find anything seriously wrong with the modern, Boolean approach -- at least, not as recently as the 3rd edition (1998) of his logic text.

I have a standing invitation to anyone here on OL to suggest a genuine universal categorical proposition that they think does not have existential import, a lack which the moderns say exempts such a proposition from the rules of immediate inference via obversion, conversion, and contraposition. We can go through the standard examples like "Sea serpents are scaly," or you can make up one of your own, if you like. But I'd strongly suggest that anyone who is interested in this matter read Neatby's essay, or Veatch's

Two LogicsorIntentional Logicif they can find them -- something other than the die-hard proponents of the Boolean system that Ba'al thinks have a lock on the truth. Why should revisionist historians have all the fun, anyway? :-)REB

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## BaalChatzaf

Every electronic computer is or could be designed using Boolean truth tables. Are computers a fraud.

Boolean logic has made the study of logic totally rigorous and mathematically respectable. A boolean algebra is derived from a complemented lattice. This puts propositional logic on a solid foundation. I got this sh*t, when I discussed modern logic with that ignoramus, Leonard Peikoff.

And by the way, there are empty sets. Aristotle's logic could not handle this. By removing existential import one gets the true assertion that the empty set is a subset of every set.

Ba'al Chatzaf

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## merjet

It's not a winner take all competition.

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## BaalChatzaf

Aristotelian logic has been shown to be a very restricted subset of a much broader discipline. Modern first order logic

swallowed up its Aristotelian predecessor and spit out the rind.

Ba'al Chatzaf

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## Roger Bissell

AuthorNo. Computers aren't a fraud because of Boolean truth tables any more than candles were a fraud during the period that some people believed they burned because of phlogiston. Just because someone (or a large group of people) cling to an invalid metaphysical interpretation of why calculations work in the real world, that doesn't make the results of those calculations false, any more than the premises "All cows are flying creatures. All flying creatures are mammals" make the conclusion "Therefore, all cows are mammals" false.

As for this bit about the empty set, haven't we been there before? The complement of a set is defined in relation to some larger third set, of which they both are subsets--and together in relation to which they non-overlappingly comprise the total membership of the larger set. For instance, in regard to a set of six apples, the set comprised by two of those apples is the complement of the set comprised by the the other four of those apples. There is no problem understanding the meaning of "complement" here, nor of the union of a set and its complement in relation to a larger whole.

But it is the fact a set and its complement are both subsets of a ~larger~ whole that rules out considering the "empty" set as the complement of the larger whole. To "complement" means to ~add to~ something in order to make a whole. But the six apples ~already~ are a whole six apples, and you cannot meaningfully add ~zero~ apples in order to make the six apples a whole, because they already ~are~ a whole. Zero apples is (are?) NO PART of six apples, and thus not only is zero apples NO SUBSET of six apples, it is NO COMPLEMENT of six apples.

It is a misnomer to speak of the ~union~ of something and nothing, because, for instance, you cannot meaningfully speak of the union of six apples and zero apples. You are not finding the union of ~anything~ with the set of six apples, because the set of six apples is already a set of six apples. I don't see how you can escape the implication that the notation expressing a union of the null set with another set simply means that the operation of set union IS NOT PERFORMED.

Does anything vital to modern science and modern living hinge on the empty set being a subset of every set?

REB

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## Davy

I haven't read Veatch's book but it sounds interesting. I must say though, that I'm having a hard time understanding the difference between a logic of "what is" and the modern logic of relations (predicate calculus).

How do you know what anything IS, except in terms of its relations with other things? It seems to me that the Aristotelian logic can be seen as logic of relations between classes, but those kinds of relations are not the only kind. How, for example, would you express "Everyone falls in love with someone on some enchanted evening" using the traditional syllogism?

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## BaalChatzaf

Nonsense. {a1, a2, a3, a4, a5, a6} U { } = {a1, a2, a3, a4, a5, a6}. I just performed it. Apparently you too suffer from Peikoff Syndrome.

Ba'al Chatzaf

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## BaalChatzaf

You don't. That is why Frege and Peirce invented predicate logic able to handle predicates of order 2 and higher.

Ba'al Chataf

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## Davy

Yes, it seems that multiple and nested quantifiers are incapable of being expressed...

Another example: how would express the concept of a mathematical limit without using the predicate calculus? I'm not even sure that Sommer's term logic could do that.

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## Roger Bissell

AuthorNonsense on stilts. If I "unite" with nothing, I am NOT UNITING WITH ANYTHING. I am not performing an operation of union.

Apparently you suffer from Russell Syndrome.

REB

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## BaalChatzaf

The union of sets A and B (symbolized A U B) is the set of all elements x such that x in A or x in B. (inclusive or). Consult any college text book on set theory.

Learn some technical definitions before you spout off.

It is so damned annoying when people who are technically ignorant of a subject pontificate upon it.

Ba'al Chatzaf

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## merjet

This is akin to Roger's much discussed assertions about zero in this thread. One such assertion -- bizarre in my view -- is that 0+5 makes sense but 5+0 doesn't (link). So, Roger, what do you make of

{{{a7, a8, a9} ∩ {a10, a11, a12}} U {a1, a2, a3, a4, a5, a6}} ?

Edited by Merlin Jetton## Link to comment

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## Roger Bissell

AuthorSeems pretty straightforward, whether in this form or as distributed.

In this form, the intersection of the first two sets is the "empty set." (They have no common members.) The "union" of the empty set and the third set is simply the third set, {a1, a2, a3, a4, a5, a6}, because there is nothing to unite the third set with, so nothing is united with it.

As distributed, the union of the first and third sets is {a1, a2, a3, a4, a5, a6, a7, a8, a9} and the union of the second and third sets is {a1, a2, a3, a4, a5, a6, a10, a11, a12}. The intersection of these unions (the members in common between the two unions) is again simply the third set.

As for 0 + 5 versus 5 + 0, I don't say the latter doesn't make sense. I'm only saying there is no operation being performed. The meaning of "5 + 0" is "5 with no operation of addition being performed upon it by any number," which is simply "5."

The former, "0 + 5," again is a case of no operation being performed. It simply means "5 not being added to any number," which is again simply "5." They both make perfect sense to me, as interpreted.

Similarly, the union of an empty set with a non-empty set is the same as the union of that non-empty set with the empty set. There is no such thing, for there is no such operation being performed in either case. So, in the former case, the meaning of the result of the supposed operation is: that non-empty set not being united with any set." In the latter, the meaning of the result of the supposed operation is: that non-empty set with no operation of set union being performed on it by any set.

In each case, numbers and sets, it is as if the number and non-empty set had been stated with a single symbol. As Kant might have said (in a much different context, where he was denying existence as a predicate), ~nothing has been added to~ (or ~united with~) the number or non-empty set. And he wouldn't have meant "nothing" like "some empty kind of thing," but "nothing" like "the absence of anything." You don't add non-existence to existence, and you don't unite non-existence with existence. Trying to do so is a fruitless, meaningless gesture. You get what you started with, which means you didn't do anything except wave your arms or flap your jaw.

REB

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## BaalChatzaf

There are none so blind as those who WILL not see.

This poor man does not believe in the inclusive or.

Ba'al Chatzaf

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## merjet

Okay, maybe I should have said "make sense qua operation." In any case months ago you said differently about one of them: "In the example you gave, after five additions, you arrive at 0 + 5. You can add 5 to 0 perfectly well. You can add five chairs to an empty room. But you cannot add zero chairs to a room with five chairs in it. You cannot add 0 to 5" (link).

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## BaalChatzaf

That is what happens when one equates the identity element of addition to Nothing At All.

See http://en.wikipedia.org/wiki/Monoid

Ba'al Chatzaf

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## Roger Bissell

AuthorThat's correct, Merlin. My present understanding on this point is not the same as it was two years ago. I think it's clearer and more consistent now.

All I'm saying here is that the ontological meaning of the identity element in addition is that no operation of addition is being performed. But not adding anything or to anything is only functionally and verbally equivalent to adding nothing or to nothing. Ontologically, they are as different as...existence and non-existence!

REB

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## merjet

There is the conflict -- criticizing formalisms for ontological reasons. Formalisms are (usually) intended to be

logical, notontological. It's an emphasis offormovercontent.Suppose you want to learn how much in coins you have in your pockets. You reach in one pocket and extract 43 cents. You reach in another pocket and come up empty. It is not that you performed no operation; it is that you still only have 43 cents.

Syllogisms with propositions such as 'All S is P' are the same way. We can grasp that a syllogism is

formallycorrect or incorrect w/o having to check if the terms have referents.Edited by Merlin Jetton## Link to comment

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## Roger Bissell

AuthorI'm not criticizing the formalism of "x + 0 = x." The formalism is just fine. It's the ~standard ontological interpretation~ of that formalism I'm saying is not correct. I'm saying that when you attempt to perform the operation of addition with zero, nothing happens--and that is not because you are ADDING NOTHING (0) to x, but because you are _NOT ADDING ANYTHING_ to x.

Let me put it another way, to those who like to remind me that zero is the "additive identity." The ~reason~ that zero is the additive identity is that "x + 0 = x" is the equivalent of saying "x = x." If you start with x and you perform no operation of addition on x, then x remains what it is, instead of becoming something different!

"0 + x = x" is the equivalent of saying "x = x," because if you don't start without anything, you are in fact not starting with anything until you introduce x, at which point you are ~starting with~ x. Again, there is no operation of addition being performed, and x again remains what it is, instead of becoming something different by being added to something.

Oh, Kant rare! Take a closer look at how you are interpreting your process--or, rather, PROCESSES--there. You are package-dealing the process of reaching in and pulling out whatever is there with the process of ACTUALLY COUNTING THE COINS--and that with the additional process of TRYING TO ADD actual coins with non-existent coins.

Let's walk through it: "OK, I'm reaching in my left pocket, and I feel some coins in there, so I grab and pull them out. Then I add up the face value of all the coins from that pocket and find that there are 43 cents in total. Now, I'm reaching in my right pocket to see if there are any coins whose value I might add to the 43 cents from my left pocket. Aw shucks, no coins. Hmmm, that means there are no coins, no cents to add to the 43 cents from my left pocket. So, I won't perform any operation of addition, since I don't have anything to add."

Actually, that might be a good comment to end with, except I ~do~ have something to add to Merlin's closing comment. :-)

Well, because of the additive identity of zero, it's certainly true that the equation "x + 0 = x" is necessarily correct, regardless of what units are referred to by "x"--AND EVEN IF X DOES NOT REFER TO ANY UNITS AT ALL! If x = 0 (i.e., if there are no units referred to by "x"), then it is indeed true that "0 + 0 = 0"--NOT because you are ADDING ZERO to zero, however, but because you are NOT ADDING ANYTHING TO zero. In other words, when x = 0, "0 + 0 = 0," BECAUSE "0 = 0." Again, the true ontological interpretation of zero as the additive identity is clear to see.

Mathematical and logical formalisms that WORK in the real world didn't just fall out of somebody's overactive imagination or some other sphincter. They were ABSTRACTED FROM REAL CONNECTIONS in the world.

In the case of addition, we know that when you don't increase a given quantity by some specific quantity, that given quantity remains unchanged. And when you don't have any specific quantity to start with, then there isn't anything to be changed by the introduction of some specific quantity. In all other cases, the result is something different from either of the two quantities on the left side of the equation, because the first is ~something~ that is ~increased~ by ~something else~. You count the units specified by the first number, then continue the sequence by counting the additional units specified by the second number.

When either number is zero, there is no counting beyond the units specified by the non-zero number, so there is no operation of addition being performed, and the additive identity prevails, because when there is only one number involved, that number is itself. All other addition facts are connections abstracted from reality that must be learned by examination, study, and memorization, and checked and re-checked if necessary -- but the additive identity is a fundamental, axiomatic connection in reality that is grasped by realizing that to deny it leads to a contradiction.

Something analogous happens in logically valid syllogisms using false statements or even nonsense terms. Yes, the middle term acts as the linking term between the terms that are the subject and predicate of the conclusion. But the logical link in a syllogism does not always point to a real connection between concepts, just as the plus sign for addition does not always point to a real operation of addition being performed.

Just as the operation of addition is abstracted from an actual process of counting the total number of units referred to by two or more numerals, so is the operation of syllogistic inference abstracted from an actual process of identifying the causal connection between two facts. And because the formal structure of causal connections is universal and abstractable from reality, it can be used with ~any~ propositional content to generate a formal connection. And when the premises are true, then the formal connection also corresponds to reality. But when the premises are false, then it's, as Rand would say, "deuces wild" -- or as our computer mavens might say, "garbage in, garbage out."

All cows are mammals.

All mammals are creatures that fly.

Therefore, all cows are creatures that fly.

All bats are mammals.

All mammals are creatures that fly.

Therefore, all bats are creatures that fly.

In each case, the middle term, "mammals," purports to be the CAUSE or REASON in reality of the fact that all cows, or all bats, are creatures that fly. In the first case, the conclusion is false. In the second case, it's true -- but it's only true BY ACCIDENT. GIGO. Some garbage ~just happens~ to be true. But as the conclusion of a syllogism with false premises, it's not knowledge, just a true proposition you know to be true for reasons independent of the unsound syllogism.

I mention all of this, not to argue that syllogism are just like addition, but to point out that valid formalisms have a source in the real world, and that empty or false ~uses~ of those formalisms have to be ~understood~ and not casually manipulated as though they were arbitrary, logical tinkertoys.

REB

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## merjet

You may think there is a deep ontological difference here, but it strikes me as merely semantics.

No, I'm not package-dealing; I'm applying a formalism. Indeed, your second paragraph "package-deals" two operations -- reaching into your pocket and abstract addition. It and the first quote appear to confound reaching into your pocket and finding nothing with not reaching into your pocket at all.

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## BaalChatzaf

Start with an empty cup. Now toss six pennies into the cup. How many pennies in the cup? Six.

I have just added six coins to a set of non-existent coins.

I am afraid you are no mathematician, Roger.

Ba'al Chatzaf

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## Roger Bissell

AuthorIt seems to me that ~anywhere~ and ~everywhere~ that someone ~might~ place six coins would be "a set of non-existent coins." The universe, by this view, is through-and-through empty sets, waiting for the ~possible~ addition of six coins, ten thousand coins, three horses, twelve bogus mathematical-theoretical arguments for the existence of empty sets, whatever.

This all smacks of idealism, to me, Ba'al. I don't know if you have heard of, or recall, J. S. Mill's notion that matter is the permanent possibility of sensation. But it sounds a lot like how your view implies the universe is like: the permanent possibility of non-empty sets.

A set is not like a room or a cup. A set is a collection of objects. The room or cup or area on a table or on the ground where the pennies are placed is not a set before the pennies are placed there. It is just a room, cup, etc.

If I showed someone my empty china cabinet and said what do you think of my set of china, he would rightly say: "What set of china?" According to you, I should say, "You must be blind. It's an empty set of china." To which he would rightly say: "Oh, yeah, right next to the empty set of sane ideas that are coming out of your mouth right now."

REB

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## Jon Letendre

I don't like that the empty set is a subset of every set, just as I don't like that X raised to the power zero is one. However, the truth of both is required to preserve the total logic of a system. Denial that X raised to the power zero equals one leads to contradiction, so that's that.

Likewise for the empty set. The subsets of {a1,a2} can be confirmed by checking the unions:

{a1, a2} U {a1, a2} = {a1, a2}

{a1, a2} U {a1} = {a1, a2}

{a1, a2} U {a2} = {a1, a2}

{a1, a2} U { } = {a1, a2}

Therefore, all four ( {a1, a2}, {a1}, {a2} and { } ) are subsets of {a1, a2}.

It isn't pretty, but it appears unavoidable.

I don't see anything wrong with saying, "It's an empty set of china." An empty china cabinet IS an empty set of china.

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