Again… symbolically P → Q is misleading… usually you distinguish between material implications and logical ones… but let’s assume that’s your symbol for a material implication and move on.
But just to clarify the difference between a logical implication and a material one…
a logical implication is something like “if humans exist, then mammals exist.” That’s a logical implication because of how the term Human is defined (All humans are mammals). In any possible world where humans exist, Mammals necessarily exist.
A Material implication is about how thing HAPPEN to be. So for example a material implication might be “If humans exist, then birds exist” because, as it happens, it is NOT the case that humans exist while birds don’t exist… yet there’s no logical reason why that HAS to be the case.
Everything up to this point you’ve got right. The only thing possibly wrong here is that logically If we had a contradiction NOTHING would follow. It would mean we fucked something up along the way.
that’s like saying if 1+1=3 it would follow that 2+2=6… but according to what system? clearly we’re no longer dealing with traditional math.
In math if you say 1+1=3 you’re just wrong…
Same is true in logic if you say P and ~P…
On the other hand you might mean that since we know for a fact that P is not the case we can claim whatever we like as following from P…
In terms of material implications, yes we can…
In terms of logical or causal implications, no we can’t.
I’m assuming that the notation P → Q is not invalid notation for material implication, just that it isn’t usually used for material implications.
I mean, Uccisore gave the following example:
This would be an example of a material implication if I’m not mistaken. According to Uccisore, it can be denoted as 1 → 2.
But I get your point that using this notation for material implication can lead to ambiguity.
Mmm… I’m not sure it means we necessarily fucked something up. Note the structure of the argument:
premise: ~P
conclusion: P → Q
We’re not actually stating that ~P and P are both true at the same time. We’re stating that given that ~P is true, if P were true, we’d have a contradiction (and Q would follow from P in virtue of the Princicple of Explosion). I know you said earlier that the Principle of Explosion has nothing to do with this, but I have a strong suspicion it has something to do with this. Right now, it’s the only way I can justify rows 3 and 4 of the truth table for conditionals:
In fact, I can even see it in the notation you recommend for material implications. If assuming ~P allows us to say ~(P^~Q), then it also allows us to to say ~(P^R), ~(P^S), ~(P^T), …
Exactly… so long as you have a way of distinguishing one from the other everything’s fine.
Yes…Thats true.
Keep in mind however that the Principle of Explosion does not assume the law of non-contradiction, in fact it demonstrates the need for it…
So GIVEN the law of non-contradiction you CANNOT have P and ~P, which means you cannot use that for anything.
In order to have anything follow from (P^~P) you have to change one of the axioms of standard logic… in other words, you are no longer using standard logic to arrive at anything.
I’m thinking you’re having trouble because the truth table has → in it… but let’s change that bit and see if things make more sense.
Let’s enter “humans are mammals” into those and see how it works out…
Human → Mammal
Human (true) & Mammal (true) = True (you are a human and you are a mammal as well. This does not disprove our premise)
Human (true) & Mammal (false) = False (If you are a human but not a mammal, that means our premise that humans are mammals is false)
Human (false) & Mammal (false) = True (You may be neither human nor mammal, this does not disprove our premise)
Human (false) & Mammal (true) = True (you are not human, yet you are a mammal. Again that does not disprove our premise)
The the only case in which P->Q is false is if P is true and Q is false…
I don’t understand why you would need the principle of explosion to make sense of that…
Well, you seem to be using the word “true” in an unorthodox way. The way I’ve seen “true” used is to saying that something is the case, whereas you seem to be using it to say that something does not disprove what is the case. That’s almost like saying “pigs fly” is true because it does not disprove some other statement like “Snow is white.”
It’s true that 3) above does not disprove our premise, but what I’ve been wanting to know from the begin of this thread (and feel I’ve figured it out at this point) is what makes such a statement inherently true–what makes “Human (false) & Mammal (false)” true in the inherent sense of “true”.
If we interpret the → operator according to material implication, then the sense in which “Human (false) & Mammal (false)” is true is non-problematic; it just means that Human → Mammal is not disproven. But I’m under the impression that the above truth table is more general than that–that it applies to logical implication as well, if not other kinds of implication. So I’ve been looking for a way to understand how “Human (false) & Mammal (false)” can be true in the inherent sense of “true” that logical implications connote.
I think I’ve figured it out with the Principle of Explosion. If you assign “false” to the antecedent of the conditional, that’s equivalent to stating the negation of the antecedent as a premise. For example, if I have P → Q, and I say P=false, then that’s equivalent to stating ~P as though it were a premise. And we all know what follows from that: if ~P, then (P → Q [or anything you like]).
In the case of material implication, which you’re talking about, it isn’t required. I’m just saying that the principles of logic that are at work in the case of material implication can be translated into the principle of explosion.
What 3. says is that if both P and Q are false, then it is not the case that both P is true and Q is false at the same time (which makes sense). Therefore, if P is true, then Q would have to be true (in your words, for the statement to remain true).
But it is also true that if both P and Q are false, then it is not the case that both P is true and W is false at the same time. Therefore, if P is true then W would have to be true.
This may seem odd, but it makes sense if you think about it. We’re only say that if P were true, then W would be true. But is P true? No. We established that P is false when we chose to focus on case 3. above. So if P is true, then we’d have a contradiction, which means, according to the Principle of Explosion, that any proposition you like can follow from P, such as W.
Like I said, I’m only show the connection of how material implication works with the Principle of Explosion, not that the Principle of Explosion is required.
However, as I said in my previous post, I think the above truth table should apply to any kind of conditional, not just material implication but logical implication as well. And what I’m saying is that if we’re treating the conditional as a form of logical implication, then we do need the Principle of Explosion in order to make sense out of 3. above (and 4. for that matter).
I thought of something that might help. In my last post I said I was attempting to translate how the principles of material implication translate into the principle of explosion. Well, I think the form that the principle of explosion takes in the case of material implication could be called the “principle of irrelevance”.
Here’s what I mean:
If we take the 3rd row of the table above:
P (false) & Q (false) = True [~(P^~Q) → (P->Q) ]
then we soon realize that what makes ~(P^~Q) true is that P is false. It is certain not that ~Q is false, for we assigned “false” to Q which makes ~Q true. In other words, what makes ~(P^~Q) true has nothing to do with Q (or ~Q). Q is irrelevant. But since it’s irrelevant, you might as well put anything you want in place of Q, like W, and the statement would remain true. Thus, I call this the Principle of Irrelevance, which is a different form of the Principle of Explosion.
It seems almost intuitive when you think about it… P->Q is just another way of saying ~(P^~Q) in logic…
It’s only when we translate P->Q into spoken english and end up saying “If P, then Q” it seems to suggest that P logically or causally implies Q… and such a statements require justification.
In the case of a logical implication you have to DEFINE it as true… in a causal one, you have to show either that only Q causes P or that P would necessarily cause Q.
But for a material implication all you have to do is show that it is not the case that P is true and Q is false.
Sure it does… a logical implication still translates to ~(P^~Q)… but we’re getting into a tricky area.
Keep in mind that the only difference between a material implication and a logical implication is that a logical implication is defined as true… and that there’s no way to “prove” a definition…
There are entire books dedicated to that distinction and how it “ought” to work… but for now suffice it to say that logical implications are defined as true, but otherwise function like material implications.
What we can and should do, however, is reflect that it’s a logical implication and distinguishing it from a material implication.
So if I state “If my name is Sam, then I am a Martian,” and I mean that as a logical implication, then I’m defining that as true? Is this another way of saying that if I state it, I must mean it?
No If you state “My name is Sam, and I am a Martian” and mean it as a logical implication… you are saying there exists a language such that in that language if you qualify for the term “My name is Sam” you necessarily qualify for the term “I am a Martian”.
Clearly that language isn’t english however
But when I say “If humans exist, then mammals exist” and mean that as a logical implication, I am saying that in the english language, for anything to qualify for the term “human” it must also qualify for the term “mammal”.
Oh, I get it. So you mean “define” as in we define humans as a species of mammal. So if I make a statement like “If it’s a glurb, then it’s a drukit,” this is equivalent to saying “All glurbs are drukits,” or perhaps “Glurbs are a kind of drukit.”
It’s odd to think this could ever be wrong though–I mean, if all we’re doing is defining things, then you would think we could define things however we want. Yet we do have the one row from the conditional truth table that says “if [true], then [false] = false” which says that, yes, sometimes definitions can be false (and of course, this makes sense: if X is a glurb, but it turns out not to be a drukit, then how could the definition hold?). Nevertheless, it’s hard (for me at least) to shake this feeling that something’s odd here.
Oh, and I can now see why you would think the Principle of Explosion would have nothing to do with this.
It wouldn’t… you can’t both define that all glurbs are drukits and also claim that there is a glurb such that it is not a drukit.
If it’s not a drukit then how did it qualify as a glurb?
You can’t disprove a definition… you can prove that someone isn’t using the correct definition or is contradicting himself… but you can’t disprove a definition.
So we’ve covered material implication and logical implication. What about causal implication, or implication that rests on entailment?
I think what I mean by causal implication is pretty obvious (but ask me if it’s not). What I mean by implication that rest on entailment is a conditional statement that is meant to be interpreted such that the antecedent entails the conclusion (i.e. there is some connection between the two). For example: if the victim was killed at 2:00 AM, then the murderer must be Joe Schmoe. In other words, it could not be possible that the victim was killed at 2:00 AM and the murder is not Joe Schmoe.
Note how this differs from Uccisore’s example: if I scratch my head, then a quasar somewhere emits high energy protons. My head scratching and the protons emitted by the quasar have absolutely nothing to do with each other. We’re only able to say if the one is true then the other is true because they both so happen to be true in the same world, but not because the one causes the other, or even that the one entails that the other must be true.
What I’m asking is, does the truth table for conditionals also apply to causal or entailment implications?
