So the importance of the subject matter gives an especial motive-force to my curiosity. From another thread I left off defending the idea that identity was equal to the biconditional and that since the biconditional could be simplified to the conditional, then therefore the identity was equal to the conditional; either any one of them, or both of them.
Is this logically unsound and if not, then why are we in disagreement?
Edit: I wrote an explanation, but actually nevermind. I think you’ll first need to give a simple example with animals or geometric shapes or something, because you may have been exposed to some very odd ideas about logic.
a–>b & b–>a
using simplification one gets either a–>b or b–>a. That’s what I mean. In the meantime a<–>b is equivalent to a=b so therefore a=b–>a–b v b–>a
This may be helpful.
“http://en.wikipedia.org/wiki/Geocentric_model”
is just as relevant.
Not to down on aristotle or anything, but I am sure there is something more relevant.
Physically, car is both red, green, brown, non-existent, made to fit squid creatures, one-wheeled, 3000fett long, etc… until other entities in the universe interact with it.
fakename,
“OR” has a special meaning in logic. It’s not a casual “either one or the other is true.” It means rather, “one or the other is true (and they might both be true).” In casual English, “either this or that is true” usually excludes the bolded possibility.
fakename,
The biconditional is not equal to the indentity relation.
In the first case, everything that follows from P must also follow from Q, because they are merely two ways of saying the same thing.
In the second case, some of what follows from P does not follow from Q, and vice versa.
The biconditional assets a truth-functional relation, not an identity.
Why in the second case, does some of what follows from P not follow from Q?
After all if you have a biconditional, then you just split it up into the two inverted conditionals. Then assuming you have the antecedent symbol for the first conditional you should be able to get the consequent symbol for that first conditional. But the consequent of that first conditional would be the antecedent of the second conditional so assuming you have P you could get Q.
It is interesting that you say that the bicond. asserts a truth functional relation and not an identity. I guess the ID doesn’t assert a truth functional relation, but then why is that?
I know about the OR but what is it that I dont’ know about it that makes it so important here?
Lee, I have some faith that you are one of the experts in this field so in that spirit, could you offer an exemplar?
Let P be “I am wet” and let Q be “it is raining”. “I am wet” implies “it is raining” and “it is raining” implies “I am wet”. Yet, “I am wet” is not just another way of saying “it is raining.”
Clayton -
unfortunately that can’t be the exemplar we hope for since neither the fact that there is precipitation somewhere imply that some person somewhere is wet, nor is the inverse logically necessary.
plus if we try to firm it up we find that even though, rain that falls on me implies that i wetten, my wettening does not imply that rain falls on me.
@nir: It was illustrative to help you reason through what Lee was saying. Yes, I understand that in this physical universe, there are many possible causes for being wet. The point is that implication is neither causation or identification. If B is always correlative with A, then A->B even though A may not cause B. For example, let P be “My hair is wet” and Q be “my shirt is wet”. P->Q. Yet my wet hair is not the cause of my wet shirt. The rain is the cause of my wet hair and my wet shirt. Whenever it rains, both my shirt and my hair get wet. “Double implication” if you want to call it that is a type of correlation. It’s basically saying “the truth functions of P and Q are identical”, but that does not imply that P and Q themselves are identical. In the rain illustration, we could say, “my hair is wet” implies “my shirt is wet” and “my shirt is wet” implies “my hair is wet”, that is, “my hair is wet if and only if my shirt is wet” (and vice-versa). Yet wet hair and wet shirt are obviously different things. They share a common cause (the rain) which is what accounts for the identity of their truth functions.
Clayton -
im trying to stay within the realms of logical necessity, i.e. the formal relations between things, and not the empirical ‘likely’ relations with things (which would be the common sense kind of ‘implication’)
it is an empirical correlation that ‘often’ P(Johns hair is wet) implies Q (Johns shirt is wet). but P does not formally imply Q in the case of what we are considering.
| P(wet hair) | Q(wet shirt) | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
P → Q is not always true under the domain of wet hairs and wet shirts, precisely because it is a logical possibility for your hair to be wet whilst your shirt is dry, which invalidates the hypothesis that P-> Q
so can we pick a P and a Q for which P necessarily implies Q and Q necessarily implies P and yet P is not identical with Q.
I’m not saying that it can be done, or that it can’t be done . Others have said it can be done without them yet having given an example. That is all this thread requires for the matter to be settled 
Thanks clayton, that clears up some issues (disregard my other post since that one was rendered irrelevant and I don’t have the authority to delete it).
So to summarize, the identity will always be true, and the biconditional will always have two functions with the same truth conditions, but of course, will not always be true?
But two more questions spring to mind: I never understood why exactly the conditional was ill-suited for expressing causality, the author’s I’ve read seem united in this skepticism but I don’t understand why. I ask this because the original post was based on P and Q being in fact, identical things. If that is true, then wouldn’t the logic be enough to express real world causation?
And lastly, the ID can’t be the same as the bicon. because they have two different truth conditions?
Hmm, I don’t understand your question here.
Because we distinguish between causation and correlation. We say that a car’s motor causes the vehicle to move forward and this is something stronger than “the car’s motor is always running when the car moves forward” because other things might also be always true whenever the car is moving forward… for example, your hair might always be blowing around whenever the car is moving forward, if it is a convertible. Yet your blowing hair is not what is causing the car to move forward. So, the fact that two things are correlated (always true together and maybe also always false together) is not sufficient to establish causation.
Causality is its own philosophical can-of-worms.
No. Causality is a very sophisticated subject.
No, it’s because identity means there is a formal relationship where bi-conditional does not.
I’m a computer engineer, so I think about it in analogy to computer circuits. Consider an electrical signal named A. We can create a copy of that signal by attaching a wire to A and naming that wire B. B is identical to A because it is just a copy of A.
But bi-conditional is itself a logic function. Let’s start with the conditional and define a function called IF():
F F T
F T T
T F F
T T T
Now, the bi-conditional is also a logic function and we can define it similarly, IFONLYIF():
F F T
F T F
T F F
T T T
This is the truth table of what is called an “XNOR gate” in electronic circuits, or the negated exclusive-or function. IFONLYIF(a,a) is always true, no matter the value of a, so if b is a copy of a (identical to a), then IFONLYIF(a,b) would always be true. That is, the biconditional of two identical variables is always true. That’s the relationship that makes identity seem like it’s the same thing as bi-conditional. Yet, in electronic circuit terms, an XNOR gate is not the same thing as copying a signal. A gate requires transistors, copying a signal is simply connecting two wires together.
Clayton -
This runs up against the (supposed) analytic/synthetic distinction. In the so-called “analytic” realm, biconditional is the same as identity, which is again the same as definition.
Example in the real number line (an “analytic” realm):
X is greater than 0 <—> X is positive
But of course “X is greater than 0” is the definition of “X is positive” (that is, it’s just another way of saying the same).
It would bring matters to the fore, Lee, if you could give an example where
P ↔ Q
yet “some of what follows from P does not follow from Q, and vice versa.”
This one is a big mistake.
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.
no. it is raining does not imply i am wet, because i could be warm and dry indoors. i am wet does not imply it is raining, because i might have gone for a swim