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All Propositions Are True

holla1755
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5/3/2017 6:37:59 PM
Posted: 3 years ago
I recently instigated and completed all rounds of the debate "All Propositions Are True," located at http://www.debate.org.... My initial argument in that debate is given in the following paragraph.

Consider the proposition p = "A rectangle is a square." Since some rectangles are squares, a rectangle is a square. Thus, p is true. Since some rectangles are not squares, a rectangle is not a square. Thus, p is not true. So by Conjunction Introduction, p is true and p is not true. But that is a contradiction. Since every proposition follows from a contradiction by the Principle of Explosion, the proposition "all propositions are true" is true. Therefore, all propositions are true.

Is the argument sound? Why or why not?
dylancatlow
Posts: 13,530
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5/3/2017 7:28:47 PM
Posted: 3 years ago
This is only contradictory insofar as "rectangle" is taken to mean one thing only. Where the term rectangle covers multiple, logically distinct shapes, and is understood to do so, it's impossible to speak of what a "rectangle is" except for the common thread tying all of the possible particular rectangles together. One can make statements about what a rectangle is, but to say that a rectangle "is a square" and to stop there means that one can't be talking about the abstract concept of a rectangle encompassing both square and non-square rectangles. There's a logic distinction to be made between particular rectangles and "rectangle" when logic would collapse without the distinction.
Silly_Billy
Posts: 1,253
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5/4/2017 3:10:58 PM
Posted: 3 years ago
At 5/3/2017 6:37:59 PM, holla1755 wrote:
I recently instigated and completed all rounds of the debate "All Propositions Are True," located at http://www.debate.org.... My initial argument in that debate is given in the following paragraph.

Consider the proposition p = "A rectangle is a square." Since some rectangles are squares, a rectangle is a square. Thus, p is true. Since some rectangles are not squares, a rectangle is not a square. Thus, p is not true. So by Conjunction Introduction, p is true and p is not true. But that is a contradiction. Since every proposition follows from a contradiction by the Principle of Explosion, the proposition "all propositions are true" is true. Therefore, all propositions are true.

Is the argument sound? Why or why not?

This argument is not sound. If I make the proposition, " All rectangles are squares", that proposition will never be true which contradicts your proposition that all propositions are true.
keithprosser
Posts: 8,122
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5/4/2017 5:47:50 PM
Posted: 3 years ago
A valid argument is formally correct; a sound argument is formally correct and its premises are true.
http://www.iep.utm.edu...

This is a valid argument:

All Parisians are French
Aristotle is a Parisian
Therefore Aristotle is French.

However it isn't a sound argument because the premise 'Aristotle is a Parisian' is not true.

The premise 'A rectangle is a square' has "complicated truth", because while some rectangles are square, most rectangles are not square, so we can't say that that premise is 'true' for the purpose of 'soundness'. To pin down why 'a rectangle is a square' is not a suitable proposition is tricky because it involves the sematics and grammar of English in addition to the rules of formal logic - I don't think it's worth trying to unpick it.

It seems to me that any false proposition would do just as well. If we take '1+1 does not equal 2' as a premise then a valid argument could show all propositions are true - but it wouldn't be a sound argument.
holla1755
Posts: 11
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5/5/2017 3:04:34 AM
Posted: 3 years ago
At 5/4/2017 3:10:58 PM, Silly_Billy wrote:
This argument is not sound. If I make the proposition, " All rectangles are squares", that proposition will never be true which contradicts your proposition that all propositions are true.

The proposition "All rectangles are squares" both does and does not contradict the proposition "All propositions are true." If all propositions are true, then all rectangles are squares. It's also true that if all propositions are true, then by the Principle of Explosion, it is not the case that all rectangles are squares.

At 5/4/2017 5:34:45 PM, dylancatlow wrote:
I just told you.

I wish you had given a clearer answer. You did not give me a yes or no answer. Either the argument is sound or it is not, and you did not explicitly say which one.

At 5/3/2017 7:28:47 PM, dylancatlow wrote:
This is only contradictory insofar as "rectangle" is taken to mean one thing only.

You seem to be suggesting that the argument is not sound. Would the argument be sound if "a rectangle" is taken to mean any one rectangle? "A rectangle" in "A rectangle is a square" is, in fact, any one rectangle. This usage is the usage I used and agrees with multiple mathematical sources, three of which I cited in my aforementioned debate. It is also the usage that leads to a contradiction.

It seems that propositions are too dynamic and are heartbreakingly inconsistent. There seems to be a paradox similar to Russell's paradox. As Russell's paradox was related to sets, this paradox is related to propositions. I think Russell's paradox showed that some supposed sets were naively expressed. This proposition paradox shows that some supposed propositions are naively expressed. But unlike Russell's paradox, this proposition paradox seems so true and so embedded in our lives and our language that we should, perhaps must, accept it rather than work around it.

At 5/4/2017 5:47:50 PM, keithprosser wrote:
The premise 'A rectangle is a square' has "complicated truth", because while some rectangles are square, most rectangles are not square, so we can't say that that premise is 'true' for the purpose of 'soundness'. To pin down why 'a rectangle is a square' is not a suitable proposition is tricky because it involves the sematics and grammar of English in addition to the rules of formal logic - I don't think it's worth trying to unpick it.

As a proposition, "A rectangle is a square" is either true or false. It's a suitable proposition as it is used not just in ordinary English, but in contemporary mathematics as well. The three geometry textbooks I cited in my debate are examples that affirm that claim. But I do see that, ultimately, "A rectangle is a square," as I and others use it, may be an unsuitable proposition.
keithprosser
Posts: 8,122
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5/5/2017 5:15:05 AM
Posted: 3 years ago
At 5/5/2017 3:04:34 AM, holla1755 wrote:
As a proposition, "A rectangle is a square" is either true or false. It's a suitable proposition as it is used not just in ordinary English, but in contemporary mathematics as well. The three geometry textbooks I cited in my debate are examples that affirm that claim. But I do see that, ultimately, "A rectangle is a square," as I and others use it, may be an unsuitable proposition.

I'd put the problem down to the fact that statements in natural language can be ambiguous. 'A rectangle is a square' can be interpreted as 'This particular rectangle is a square' (which is true for some choice of rectangle) or as 'all rectangles are squares' which is not true. Without context or clarification it is not possible to know or fix which meaning of 'a rectangle is a square' is in use.

To avoid that problem there are notations involving backward E's and upside down A's that I can't easily reproduce here, but essentially the former meaning would be rendered approximately as E:R in S (ie there exists a rectangle that is square) and the latter meaning A:R in S (All rectangles are squares).

However "All propositions are true" is also the case when the proposition is stated by one's wife... unless you like living in the doghouse.
holla1755
Posts: 11
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5/5/2017 6:53:22 AM
Posted: 3 years ago
At 5/5/2017 5:15:05 AM, keithprosser wrote:
'A rectangle is a square' can be interpreted as 'This particular rectangle is a square' (which is true for some choice of rectangle) or as 'all rectangles are squares' which is not true. Without context or clarification it is not possible to know or fix which meaning of 'a rectangle is a square' is in use.

I suppose you're right.

Some geometry textbooks would assume that "A rectangle is a square" makes reference to an arbitrary square, as that is all the information about the rectangle that is given. A possible problem I see with this approach is that in reading the proposition "A rectangle is a square" in proper order from beginning to end, we start off with an arbitrary rectangle, as that is all the information we are given at some point in our reading. For all we know at that point, it's possible the rectangle is not a square. Then, as we continue our reading, it seems we are forced to contradict ourselves by somehow giving the arbitrary rectangle the property of being a square. In doing so, we no longer have the arbitrary rectangle we started off with. The initial subject of the proposition is no longer of interest to us. It thus seems that the construction of the proposition's meaning is flawed. This may explain why the proposition leads to the contradiction I described in my original post.

To avoid that problem there are notations involving backward E's and upside down A's that I can't easily reproduce here, but essentially the former meaning would be rendered approximately as E:R in S (ie there exists a rectangle that is square) and the latter meaning A:R in S (All rectangles are squares).

I am aware of universal and existential quantifications. I do not dispute that mathematics can be expressed successfully using those quantifications. But there are more propositions than just explicit universal and existential quantifications. So far in this discussion, I am interested in some of those other propositions.
Silly_Billy
Posts: 1,253
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5/5/2017 10:57:08 AM
Posted: 3 years ago
At 5/5/2017 6:53:22 AM, holla1755 wrote:
At 5/5/2017 5:15:05 AM, keithprosser wrote:
To avoid that problem there are notations involving backward E's and upside down A's that I can't easily reproduce here, but essentially the former meaning would be rendered approximately as E:R in S (ie there exists a rectangle that is square) and the latter meaning A:R in S (All rectangles are squares).

I am aware of universal and existential quantifications. I do not dispute that mathematics can be expressed successfully using those quantifications. But there are more propositions than just explicit universal and existential quantifications. So far in this discussion, I am interested in some of those other propositions.

It seems to me that what you are really pointing out is that language is not a precise enough instrument to convey the true meaning of a message which leads to propositional paradoxes that would not exist if language had been precise enough.

The problem in my opinion is translation and interpretation. When I think of the word "chair", the thought in my head related to that word (such as the mental image that that words gives to me) will not be the same as the one that you have. Therefor when I speak the word "chair", the meaning of the word is slightly altered when it is received by you simply because your interpretation of the word is not identical to the thought I had when I said it.
holla1755
Posts: 11
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5/5/2017 2:44:04 PM
Posted: 3 years ago
At 5/5/2017 3:04:34 AM, holla1755 wrote:
The proposition "All rectangles are squares" both does and does not contradict the proposition "All propositions are true."

Better yet, consider the following proposition, (A).

(A) The proposition "Not all propositions are true" contradicts the proposition "All propositions are true."

Is (A) true or false? (A) is both true and false. Since "Not all propositions are true" is the negation of "All propositions are true," (A) is true. Since "Not all propositions are true" is implied by "All propositions are true," (A) is false. By Conjunction Introduction, (A) is true and (A) is false. So, a contradiction exists. Therefore, by the Principle of Explosion, all propositions are true.
TwoMan
Posts: 167
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5/5/2017 7:53:35 PM
Posted: 3 years ago
The proposition "Not all propositions are true" is true.
The proposition "All propositions are true" is false. This proposition, by itself, does not imply that the other is true. The fact that it is false does. There is no contradiction.
holla1755
Posts: 11
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5/5/2017 9:48:17 PM
Posted: 3 years ago
At 5/5/2017 7:53:35 PM, TwoMan wrote:
The proposition "Not all propositions are true" is true.
The proposition "All propositions are true" is false.

You've assumed, explicitly, the opposite of what I'm arguing for. In criticizing my argument, you've committed the fallacy of begging the question.
TwoMan
Posts: 167
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5/6/2017 12:23:47 AM
Posted: 3 years ago
At 5/5/2017 9:48:17 PM, holla1755 wrote:
At 5/5/2017 7:53:35 PM, TwoMan wrote:
The proposition "Not all propositions are true" is true.
The proposition "All propositions are true" is false.

You've assumed, explicitly, the opposite of what I'm arguing for. In criticizing my argument, you've committed the fallacy of begging the question.

I would say that your argument, in fact, does that. By saying "Not all propositions are true" is implied by "All propositions are true" uses circular reasoning to imply a conclusion within the premise.
holla1755
Posts: 11
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5/6/2017 1:40:46 AM
Posted: 3 years ago
At 5/6/2017 12:23:47 AM, TwoMan wrote:
By saying "Not all propositions are true" is implied by "All propositions are true" uses circular reasoning to imply a conclusion within the premise.

"'Not all propositions are true' is implied by 'All propositions are true'" can be proven, and I will prove it below.

Proof.
Suppose all propositions are true.
Since "Not all propositions are true" is a proposition, and all propositions are true by the supposition, "Not all propositions are true" is true.
In other words, not all propositions are true.
By discharging the supposition through conditional introduction, "All propositions are true implies not all propositions are true."
In other words, "'Not all propositions are true' is implied by 'All propositions are true.'" QED

I'm unsure what you mean by saying that circular reasoning is used "to imply a conclusion within the premise." I'm unsure of what conclusion you are referring to and of what premise you are referring to. In all valid proofs, the conclusions are implied within the premises.
TwoMan
Posts: 167
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5/6/2017 2:44:55 AM
Posted: 3 years ago
Perhaps you should get a second opinion because I'm not grasping the purpose of this line of thinking. If you want to make a claim with no logical foundation ("Suppose all propositions are true") then you can always twist your argument to appear correct. I don't see the point.
holla1755
Posts: 11
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5/6/2017 3:01:19 AM
Posted: 3 years ago
At 5/6/2017 2:44:55 AM, TwoMan wrote:
Perhaps you should get a second opinion because I'm not grasping the purpose of this line of thinking.

I'm here to get opinions. And I'm glad you've given yours. I thank you for it.

If you want to make a claim with no logical foundation ("Suppose all propositions are true") then you can always twist your argument to appear correct.

Some proofs suppose propositions that are never true or impossible. Supposing propositions that are never true or impossible is an acceptable practice in contemporary mathematics and philosophy. For example, some proofs of the irrationality of the square root of 2 assume the proposition "The square root of 2 is rational," even though the proposition is never true or impossible.
Tay_Jordan
Posts: 48
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5/8/2017 4:57:32 AM
Posted: 3 years ago
0th order logic isn't equipped to parse "a rectangle is a square" as an atomic sentence. If you move up to 1st order logic, the quantifiers make the problem here obvious.

You're either saying Ex[Rx + Sx] (There exists an x such that x is both a rectangle and a square)

Or you're saying Ax[Rx > Sx] (in all cases of x, if x is a rectangle then x is a square)

If you're saying the former, then there's no contradiction is stating that there exists another x such that x is not both a rectangle and a square, and therefor you cannot derive whatever you want. If you're saying the later, you're just plainly wrong, and so even though you can derive anything in theory, since you're starting from an unsound premise, your conclusions derived from it are necessarily divorced from reality.
holla1755
Posts: 11
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5/8/2017 5:25:51 AM
Posted: 3 years ago
At 5/8/2017 4:57:32 AM, Tay_Jordan wrote:

You're either saying Ex[Rx + Sx] (There exists an x such that x is both a rectangle and a square)

Or you're saying Ax[Rx > Sx] (in all cases of x, if x is a rectangle then x is a square)

"There exists a thing such that it is a rectangle and it is a square" and "Some rectangles are squares" are logically equivalent, but they are not equal. Similarly, "There exists a thing such that it is a rectangle and it is not a square" and "Some rectangles are not squares" are logically equivalent, but they are not equal. It appears that not all propositions represented in the English language can be perfectly represented in first-order logic.

It appears that some propositions represented in the English language are ambiguous and inconsistent, yet meaningful and acceptable to use.

On another note, let d be a proposition.

d if and only if d is true.

That boldface claim suggests that all propositions are true. It suggests that the proposition's existence depends wholly on its truth.
Tay_Jordan
Posts: 48
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5/8/2017 5:40:48 AM
Posted: 3 years ago
At 5/8/2017 5:25:51 AM, holla1755 wrote:
At 5/8/2017 4:57:32 AM, Tay_Jordan wrote:

You're either saying Ex[Rx + Sx] (There exists an x such that x is both a rectangle and a square)

Or you're saying Ax[Rx > Sx] (in all cases of x, if x is a rectangle then x is a square)

"There exists a thing such that it is a rectangle and it is a square" and "Some rectangles are squares" are logically equivalent, but they are not equal. Similarly, "There exists a thing such that it is a rectangle and it is not a square" and "Some rectangles are not squares" are logically equivalent, but they are not equal. It appears that not all propositions represented in the English language can be perfectly represented in first-order logic.

It appears that some propositions represented in the English language are ambiguous and inconsistent, yet meaningful and acceptable to use.

On another note, let d be a proposition.

d if and only if d is true.

That boldface claim suggests that all propositions are true. It suggests that the proposition's existence depends wholly on its truth.

You haven't said anything meaningful here. What is the relevant difference between these "logically equivalent but not equal" sentences? Also, in your OP you said P = "A rectangle is a square," not "some rectangles are squares."

"Some rectangles are squares" is equal to Ex[Rx + Sx] because the existential quantifier means there is at least one case of x wherein x has the following qualities/functions.

Of course there are nuances of natural languages that can't be captured in propositional calculus (like how the word "but" implies some degree of surprise, but can only be expressed as "+" in logic), but these nuances don't change the truth-function of a sentence.
Philosophy101
Posts: 2,065
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5/8/2017 5:54:03 AM
Posted: 3 years ago
At 5/3/2017 6:37:59 PM, holla1755 wrote:
I recently instigated and completed all rounds of the debate "All Propositions Are True," located at http://www.debate.org.... My initial argument in that debate is given in the following paragraph.

Consider the proposition p = "A rectangle is a square." Since some rectangles are squares, a rectangle is a square. Thus, p is true. S

This is where the argument goes wrong. If some rectangles are squares it also means some are not squares. Not all rectangles are squares (all squares are rectangles). Thus all you have proven is that it is sometimes true a rectangle is a square.

since some rectangles are not squares, a rectangle is not a square.

This is also sometimes true because on the corollary sometimes rectangles are not squares (against sometimes they are).

Thus, p is not true. So by Conjunction Introduction, p is true and p is not true. But that is a contradiction. Since every proposition follows from a contradiction by the Principle of Explosion, the proposition "all propositions are true" is true. Therefore, all propositions are true.

Your conjunction is:
some p's are q's
Some p's are ~q's
Therefore q then p and ~p
Therefore each thing is all things (I do not know the notation for that.

The first leap is unwarranted by logic and the second really doesn't make sense.

Is the argument sound? Why or why not?
holla1755
Posts: 11
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5/8/2017 6:10:50 AM
Posted: 3 years ago
At 5/8/2017 5:40:48 AM, Tay_Jordan wrote:
What is the relevant difference between these "logically equivalent but not equal" sentences?

Those propositions always have the same truth value. That's what's meant by "logically equivalent." But they are not equal. Technically, they express different things and thus do not mean the same thing. Some of those propositions involve a conjunction and some of them do not. Some of them use the word "some" and some of them do not. Some of them use the clause "There exists" and some of them do not. Every little difference counts.

Also, in your OP you said P = "A rectangle is a square," not "some rectangles are squares."

I don't believe I've ever equated p with "Some rectangles are squares." "Some rectangles are squares" is always true, but p is sometimes false. "Some rectangles are squares" and p do not mean the same thing.

"Some rectangles are squares" is equal to Ex[Rx + Sx] because the existential quantifier means there is at least one case of x wherein x has the following qualities/functions.

They are not equal, but they are logically equivalent. This difference should not be disregarded.

Of course there are nuances of natural languages that can't be captured in propositional calculus (like how the word "but" implies some degree of surprise, but can only be expressed as "+" in logic), but these nuances don't change the truth-function of a sentence.

They may not change the "truth-function" of a proposition. But they do change the meaning.
Tay_Jordan
Posts: 48
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5/8/2017 6:44:27 AM
Posted: 3 years ago
At 5/8/2017 6:10:50 AM, holla1755 wrote:
At 5/8/2017 5:40:48 AM, Tay_Jordan wrote:
What is the relevant difference between these "logically equivalent but not equal" sentences?

Those propositions always have the same truth value. That's what's meant by "logically equivalent." But they are not equal. Technically, they express different things and thus do not mean the same thing. Some of those propositions involve a conjunction and some of them do not. Some of them use the word "some" and some of them do not. Some of them use the clause "There exists" and some of them do not. Every little difference counts.

Also, in your OP you said P = "A rectangle is a square," not "some rectangles are squares."

I don't believe I've ever equated p with "Some rectangles are squares." "Some rectangles are squares" is always true, but p is sometimes false. "Some rectangles are squares" and p do not mean the same thing.

"Some rectangles are squares" is equal to Ex[Rx + Sx] because the existential quantifier means there is at least one case of x wherein x has the following qualities/functions.

They are not equal, but they are logically equivalent. This difference should not be disregarded.

Of course there are nuances of natural languages that can't be captured in propositional calculus (like how the word "but" implies some degree of surprise, but can only be expressed as "+" in logic), but these nuances don't change the truth-function of a sentence.

They may not change the "truth-function" of a proposition. But they do change the meaning.

I don't think you have a firm grasp of propositional calculus.

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