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Cont: Deeper than primes - Continuation 2

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doronshadmi said:
It is wff, as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=11272022&postcount=1395.
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Were that true, you should have no trouble at all showing how it can be developed from the rules for well-formed formulae. Unfortunately for you, IsSuccessorOf is a binary predicate, so it must appear with two terms. You may use function notation instead of infix if you prefer (i.e. IsSuccessorOf(V,E) vs. V IsSuccessorOf E).

No argument (you are not writing to the point...)
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I'm still waiting for a definition. Until you give your personal vocabulary meaning that others can understand, you will not be making any points.

Q IsSuccessorOf W <=> ...?​

Anything?
 
Little 10 Toes said:
Correction: doronshadmi quoted a post with a yes/no question and can't answer it.
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(there is rain outside) OR (there is no rain outside) are both true (or if you prefer, yes) statements, where the OR logical connective prevents the simultaneity of those two true (yes) statements, which means that contradiction is avoided.

Yes/no answer is relevant in case that the statements in question may lead to contradiction, as shown in the case of (there is rain outside) AND (there is no rain outside).

(Q IsSuccessorOf W) OR (Q ~IsSuccessorOf W) are both true (or if you prefer, yes) statements, where the OR logical connective prevents the simultaneity of those two true (yes) statements, which means that contradiction is avoided.

Yes/no answer is relevant in case that the statements in question may lead to contradiction, as shown in the case of (Q IsSuccessorOf W) AND (Q ~IsSuccessorOf W).

Unfortunately, jsfisher or you trapped by the notion that there is one and only one true (or yes) statement here, notate as Q IsSuccessorOf W.

By beaing trapped under this wrong notion, your reasoning gets nowhere.
 
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doronshadmi said:
(there is rain outside) OR (there is no rain outside) are both true (or if you prefer, yes) statements
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Is a true statement. It is a tautology. It's truth is guaranteed without regard to to the definition of the "there is rain" predicate. The two terms of the statement, on the other hand, are not both true. Just one, exactly one of the two terms is true.

...(Q IsSuccessorOf W) OR (Q ~IsSuccessorOf W) are both true (or if you prefer, yes) statements, where the OR logical connective prevents the simultaneity of those two true (yes) statements, which means that contradiction is avoided.
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Is a true statement. It is a tautology. It's truth is guaranteed without regard to to the definition of the "IsSuccessorOf" predicate. The two terms of the statement, on the other hand, are not both true. Just one, exactly one of the two terms is true.

If only we had a definition for the predicate to decide which of the two terms is true for a given pair of parameters. If only.
 
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(Q IsSuccessorOf W) OR (Q ~IsSuccessorOf W)

By using the true statement Q IsSuccessorOf W, W is incomplete (its size is not satisfied).

By using the true statement Q ~IsSuccessorOf W, W is complete (its size is satisfied).

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OK, let's move on.

S is the set of singleton sets, as follows:

S = { {{}}, { {{}, {{}}} }, { {{}, {{}}, {{}, {{}}}} }, ... }

The size of the set of all singleton sets is not satisfied (which means that the term all is not satisfied, in case of S).

Please be aware of the fact that any member of S is a complete (or finite) set, so even if there are infinitely many complete (or finite) sets as members of S, S size is not satisfied (the term all does not hold in case of S).

Now, look at the following:

N is the set of natural numbers, as follows:

N = { {{}}, {{}, {{}}} , {{}, {{}}, {{}, {{}}}}, ... }

Please be aware of the fact that any member of N is a complete (or finite) set, so even if there are infinitely many complete (or finite) sets as members of N, N size is not satisfied (the term all does not hold in case of N).

We have show that what does not hold for set S, also does not hold for set N.

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If this is the case among sets with infinitely many complete (or finite) sets as their members, it is definitely the case among sets with infinitely many incomplete (or infinite) sets as their members.
 
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jsfisher said:
Is a true statement. It is a tautology.
It's truth is guaranteed without regard to the definition of the "there is rain" predicate.
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In other words, Is a true statement is independent of any definition, whatsoever.

jsfisher said:
The two terms of the statement, on the other hand, are not both true. Just one, exactly one of the two terms is true.
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Wrong, ("there is rain" predicate, is true) OR ("there is ~rain" predicate, is true) (no contradiction is involved because they are not simultaneously used because OR is used).
 
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doronshadmi said:
In other words, Is a true statement is independent of any definition, whatsoever.
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Any statement of the form P or ~P, as long as P has a well-formed definition, is a tautology.

The statement itself is of no use in deciding which of the two terms is true. For that, the definition would be required.
 
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jsfisher said:
Any statement of the form P or ~P, as long as P has a well-formed definition, is a tautology.
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EDIT:

In other words, (P or ~P) is a true statement that is independent of any definition, whatsoever.
 
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doronshadmi said:
In other words, (P or ~P) is a true statement is independent of any definition, whatsoever.
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You left out the well-formed part. Also, being a tautology, the important independence is from any instantiation of P.

Finally, because it is a tautology, a trivial tautology, it is useless.
 
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jsfisher said:
instantiation of P.
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So by your two last posts "instantiation" is the same as "definition".

So once again, (P or ~P) is a true statement that is independent of any definition, whatsoever, so it is irrelevant to the discussed subject.

jsfisher said:
You left out the well-formed part.
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I are wrong, "(Q IsSuccessorOf W) OR (Q ~IsSuccessorOf W)" is wff exactly as, for example, "(Q = W) OR (Q ~= W)" is wff, where "IsSuccessorOf" or "=" are forms of relations between Q and W.


(Q IsSuccessorOf W) OR (Q ~IsSuccessorOf W) is useful, as seen in http://www.internationalskeptics.com/forums/showpost.php?p=11274453&postcount=1427.
 
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Since you claim that "instantiation" is not the same as "definition", then please explain your following sentence:

"Also, being a tautology, the important independence is from any instantiation of P"
 
doronshadmi said:
Ok, please explain the difference between "instantiation of P" and "definition of P".
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Oh, that pesky math. So hard to understand, yet so easy to complain about.

A definition of a predicate is an equivalent expression using only previously defined predicates, functions, terms, etc.

For example,
IsSuccessorOf(X,Y) <=> X = {Y}​
would be an acceptable definition for the IsSuccessorOf predicate (shown here in function notation form) since an equivalent expression is provided using already known constructs.

As another example,
IsComplete(X) <=> IsSuccessorOf({X},X)​
would be an acceptable definition only after the meaning of IsSuccessorOf(,) is established.

The instantiation of a predicate is where all the free variables are bound to specific values.
 
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