show me how with these two equivalent statements, one is just a definition of the other:
every non empty set of whole positive numbers has a least element.
if a statement is true of the number zero, and if the fact that is is true of some number K inplies it is also true of K+1 for all values of K, then it is true of all whole positive numbers
about implication and causation.
when the great mathematicians were inspired to fix up the hopeless mish mash of aristotelean logic, they found it served them best to just forget about causality altogether when constructing a language of logic. They found they could do all they wanted by DEFINING a->b to mean, by definition, “it is not the case that a is true and b is false”
Now of course, we are not their slaves. we can construct a new language where there will be a concept of causality, where we can write “a causes b”. but as far as I know, nobody has done this yet.
BTW, I see that other posters have already made some of the points I wrote about. Maybe next time I’ll look ahead.
Dave: show me how with these two equivalent statements, one is just a definition of the other:
every non empty set of whole positive numbers has a least element.
if a statement is true of the number zero, and if the fact that is is true of some number K inplies it is also true of K+1 for all values of K, then it is true of all whole positive numbers
To a sufficiently intelligent being, 1 means 2. To a human, it wouldn’t be very illuminating to call it a definitional relation, I agree. See opening chapters of HA.
Where is this from?
thanks Dave, it took some pushing but I think you gave what we needed !
two things that are necessarily true are not necessarily the same truth.
P ) the inner angles of a 3-sides planar shape add up to 180 degrees.
Q) the square root of 2 is an irrational number
P<->Q but P!=Q
Dave:
In the second case as well, if you start with p, you get q [because p implies q] and thus all implied by q. Similarly, if you start with q, you get p, and thus all that follows from p.
The difference between p=q and p<->q is something else. p=q means that p and q are different names for the same thing. It is a statement about objects
p<->q is a statement about TRUTH. It says that statements p and q are either both true or both false.
So that one can say barack obama=current prez of usa, but it is nonsensical to say that barack obama <->current prez of usa, because "barack obama is the name of a person, and not a statement, and thus cannot be true or false, obviously.
Well said, but I was holding out for some kind of analytic/synthetic “solution” to this problem, so I could attack that instead Yes, it’s patently obvious that if P ↔ Q then everything that follows from P follows from Q, if you think of it visually. Here’s a formal proof of the same:
Let P and Q be statements such at P ↔ Q. (1)
Let R be a typical statement such that P → R. (2) [R is represents an arbitrarily chosen member of the set of statements that follow from P]
Then Q → P by (1) above, and so Q → P → R, by (2) above. Therefore Q → R.
QED. (Since R was an arbitrarily chosen member of the set of statements that follow from P, the final sentence implies that every statement that follows from P also follows from Q.)
any beginner’s book on logic that has in its list of symbols the “->” symbol and in its index the phrase “truth tables”.
Those books got it from books that are probably inacessible to most of us. Try working through a few pages of Principia Mathematica.
Nir:
P ) the inner angles of a 3-sides planar shape add up to 180 degrees.
Q) the square root of 2 is an irrational number
P<->Q but P!=Q
To a sufficiently intelligent being, P means Q. Just like to us, 2+2 means 4.
In other words, the English-language distinction between “two propositions imply each other” and “two propositions mean the same” is just a reference to human’s limited capacity for logical figuring.
ok so are we back to ↔ ‘equals’ = ?
No, Dave had it essentially right:
P ↔ Q is a statement about what propositions logically follow from other propositions. The biconditional double-arrow (<–>) is an operator for propositions.
x = y is a mathematical equality. A = B is a definition. So the equals sign (=) can be usefully used an operator for mathematical equality, or a shorthand for “A means B,” where A and B are terms, not propositions. (I recommend against the latter use, precisely because of the confusion it causes as evidenced in this thread. Instead just spell out the definition in English or some other way.)
That I am saying P ↔ Q is equivalent to “P means Q” is only true for a sufficiently intelligent being, so although it’s “true” it’s not useful at all levels of analysis for humans (but it is useful to note at some deep enough levels of analysis).
No. Here’s a proof:
If we assume the axioms of elliptic or hyperbolic geometry [or live in such a world where those axioms are true], then P is false but Q is true. So it is not the case that P means Q, obviously.
But if we assume the axioms of euclidean geometry, then both P and Q are true.
Since it cannot be the case that THE EXACT SAME STATEMENT can be both true in all models [like q is] and true in some models but false in others [like p is] then obviously Pand Q cannot “mean” the same thing, even in a model where both are true. Unless of course you want to give a bizzaro world definition of “meaning”.
This reminds me of:
that “The cat is on the mat” does not mean that “the mat” is an attribute of “the cat,” nor that “on-the-mat” is the genus to which “the cat” belongs, nor yet that “the-cat” equals “on-the-mat”
Dave:
That’s not how it works.
In any context, there are unstated starting premises/axioms. So in any context, we’re not technically saying just P ↔ Q. When we say P ↔ Q, we’re actually implicitly saying something of the following form:
Given Axiom 1, Axiom 2, …, and Axiom N, and defining the statements P and Q as “…” and “…” respectively. Then, given all that, P ↔ Q.
that “The cat is on the mat” does not mean that “the mat” is an attribute of “the cat,” nor that “on-the-mat” is the genus to which “the cat” belongs, nor yet that “the-cat” equals “on-the-mat”
Yup. “Is” is a tricky word. “The” is a tricky word. Words are devious little tricksters. Hopefully they will hurry up and perfect cortex scanning so we can start developing less ambiguous, visual languages.
I agree, but how does that refute my proof?
It seems absurd to me to make such a statement as “given the following axioms, then P means the same thing as Q.” It either means the same thing or it doesn’t. The meaning of a statement does NOT depend on what other assumptions you make about your world. [It’s truth value may, but that is not the same as its meaning, obviously].
It’s meaning depends only on the concepts mentioned in the statement itself. Any meaning of “meaning” that does not meet this pretty basic requirement does not conform to common use.
To give a very simple example, consider the statement “George Washington is dead.”
That statement means the same thing today as it did in 1776. The meaning of a statement is of course independent of the passage of time.
And yet, in 1776, that statement was logically equivalent to “2+2=4”, and today it isn’t.
Are you saying that in 1776 George Washington is dead meant 2+2=4 and today it doesn’t mean that anymore?
If we say, P ↔ Q, we are actually saying (TASIB*) that “this set of axioms and P” means “this set of axioms and Q.” It’s just that that’s cumbersome to write when most proofs already assume an axiom system that need not be overtly referred to at that point in the presentation (actually not just that; it’s also become convention because I don’t think this aspect of logic is fully-understood).
*To A Sufficiently Intelligent Being
And I know my definition of meaning doesn’t correspond to common usage: as I say, it’s TASIB, not for humans. For humans, it makes sense for explanatory purposes (in most cases) to use the conventions that we do. See my reply to Nir.
EDIT: No actually, I don’t think I’ve ascertained what you mean. Why was “George Washington is dead” logically equivalent to “2+2=4” in the year 1776? Re: George Washington, you’re highlighting an ambiguity of language, not a counterexample.
from wikipedia:
The use of the operator is stipulated by logicians, and, as a result, can yield some unexpected truths. For example, any material conditional statement with a false antecedent is true. So the statement “2 is odd implies Paris is in America” is true. Similarly, any material conditional with a true consequent is true. So the statement, “If pigs fly, then Paris is in France” is true.
So that any two true statements are implied by each other, and are therefore equivalent.
You implicitly accepted this when you admitted that nir’s two statements in an earlier post were equivalent.
I’m going to drop this discussion. I still stand by my statement, which I consider self evident and common usage, that the meaning of something does not depend on any axioms assumed. Only its truth may depend on them, but never its meaning.
I see no way of convincing you that my version of meaning is the usual one assumed by mankind since the dawn of time, nor do I see any possible thing you can say to shake me of my conviction. So we’ll have to agree to disagree.
I’ll try to be as clear as I can.
Any two statements that follow from the same set of starting premises are indeed equivalent, TASIB. As Mises says in the opening sections of Human Action, all statements that follow from a set of axioms are already contained in those axioms.
So supposing P <–>Q in a given axiomatic system, a sufficiently intelligent being would hear the axioms, then hear statement P, followed by statement Q. When you told him statement Q, he’d say, “Hey bro, you already told me that!” Make sense?
What may be confusing is the issue of whether statements about the “real world” are to be regarded as the same as mathematical/logical statements. I contend that they are (see the Quine thread from yesterday-ish for that argument).
RE: definition of the word “mean,” as I mentioned above, this applies to a sufficiently intelligent being and is not a convenient term when talking to humans at most levels of analysis. However, it is useful - and I contend far more elucidating - at some levels of deep/fundamental analysis. Some of the discussion in this thread has necessitated that level of analysis, hence my use of the term where applicable.