Ok
Same problem… If you translate a material implication to english it is not “If my name is Sam, then I am a Martian” which again implies some kind of causal relationship…
If you were to translate it it would be “If it is true that my name is sam, then it is true that I am a martian”… it’s a subtle difference, but there is a difference… the truth values of those claims is what is being discussed and not a logical nor causal implication between them.
I looked up Material Implication on Wikipedia and it said this:
Essentially what Uccisore was arguing.
While I never thought a causal implication was at issue here, I’m not sure what difference your rephrasing makes. How does the statement “My name is Sam” being true make the statement “I am a Martian” also true?
To clarify
Normally when you write if P → Q you mean there’s a logical implication or at the very least a causal relationship… but when we’re talking about a material implication, it’s not quite the same thing. When you say “If P is true, then Q is true” All it means is that it IS not the case that P is true, while Q is false… it does NOT say that it CANNOT be the case that P is true, while Q is false… only that it IS not the case.
Basically you are confusing it for a logical implication… Your name being Sam would not make you a Martian… in fact if you wanted to write it in plain everyday english what you are saying is very simply “It is not the case, that my name is sam while my being a martian is false at the same time”
In other words… you are not making a statement about the relationship between the name sam and being a martian… you are making a statement about the state of the truth values of those specific claims “My name is sam” and “I am a Martian”. There is no logical nor causal reason why your name being sam should mean you necessarily have to be a martian given a material implication… all it’s saying is that ~(P^~Q) in our current world happens to be the case…
P being “my name is Sam” and Q being “I am a martian”
Those were not premises.
Those were your conditionals within the one assertion that you stated using them.
Assuming that your assertion was true, you can then follow the truth table concerning implied conclusions.
Your assertion was itself a false premise making the rest of any provided argument meaningless.
Is this another way of stating the so-called Principle of Explosion?
I mean, I start off with:
~P
~Q
and I concluded:
P → Q
which is to say that if I assumed ~P as one of my premises, then given that P is the case, we would have a contradiction, and therefore anything goes. Thus, Q would be the case.
I get what you’re saying: ~(P and ~Q) certainly follows from the premises ~P and ~Q, but it seems that what this is saying is that if you’ve already established ~P then you can’t have P (even if you AND it with something that’s already true like ~Q). So if you do get P, then anything goes and you can allow for Q.
James, here’s my OP:
You see those big, red, bold thing-a-ma-jigs? Those are called “premises”.
Not even a little bit… you are making a very basic error in assuming that a material implication is the same as a logical implication…
No it doesn’t mean “anything goes”… it simply means you were wrong about P being the case… if on the other hand you claim P is true and false at the same time in the same sense… THEN you could argue anything goes or rather logic falls apart.
Also like I pointed out earlier your conclusion is misleading “P → Q” is traditionally reserved for logical implications… which is not a conclusion you can draw from your premises!
So keeping in mind that you did not conclude a logical implication, if it turns out that you were wrong and P does happen to be the case, it would not follow that therefore Q must be the case.
I cannot follow your reasoning here… If you have an argument that goes:
and then you correct one of your premises by saying “woops I was wrong about P”… clearly in this case the result is that your conclusion no longer follows… how does that mean anything goes?
But I didn’t say P was the case.
Ok, so you’re saying that my justification for going from the premises to my conclusion is invalid.
I can’t start with:
~P
~Q
and then say that since “if [false] then [false] = [true]” that means that P → Q.
That’s not what I’m saying. Hear me out. I’m saying I start with these premises:
~P
~Q
At no point do I retrack ~P or realize I was wrong in stating ~P or anything of that sort. I simply say that given ~P, if P were the case at the same time, then we’d have a contradiction. Note that I’m not actually stating P, I’m just saying that if we had P (not that we do have P), then we’d have a contradiction (i.e. we’d have P and ~P at the same time).
This wouldn’t be the formal notation, of course, but I would put this:
P → [contradiction]
Now, if this means we’d have a contradiction, then I believe we can bring in the principle of explosion (can we not?), which states that when you derive a contradiction, you can assert anything. You can assert X, K, L, and any other proposition you want, including Q.
Note that I’m not saying this proves Q. It just means that if you had a contradiction then you could assert Q or any proposition.
This, to me, makes sense out of the dilemma of this thread: saying “If Hitler was a woman, he would have persecuted the blacks” is like saying “If Hitler was a woman, then pigs fly.”
Do you see these "big bold thing-a-ma-jigs? Those are called “Your False Premise Assertion”.
Let me rephrase that… the notation P->Q is misleading as is stating it as an “if then” type sentence in english. It confuses a material implication with a logical or causal one… since that’s how we usually use those expressions.
The safest way to express your conclusion would be ~(P^~Q) or in english to say “It is not the case that P is true, while Q is false”
Yes, clearly if P and ~P were both true at the same time in the same sense we would have a contradiction… and yes if we were to try to apply a system that has as one of it’s AXIOMS that no contradictions are allowed yet allow one anyway we would be able to break that system.
You could assert anything you like… even if there is no contradiction… you can ALWAYS assert anything you like.
I can assert right now that the moon is made of cheese!
What you cannot do is arrive at Q by applying logic… because the law of noncontradiction prevents you from using a contradiction for anything.
James, give it up.
Mad Man, I’ll get back to you later.
Gib, if you start with a false premise, there is no kind of argument that you can add to it so as to draw any conclusion at all.
They are, they just aren't the rules you thought they were, and they are worded a little tricky in English. It could be that if we were doing this in Arabic there would be no disagreement, because the vernacular would make it just obvious that logical 'if' is merely saying that "In all instances of A, B".
Another thing to keep in mind, related to what we've said to this point, is that "If A then B" is [i]not a rule of logic[/i] any more than "A and B" is. It's a statement that seems like a rule because you are reading causation into it. "If A then B" can't be a rule of logic because [i]maybe it's false[/i].
If A then B
Is a rule of logic. The conclusion cannot be false given the premises. There’s probably a fancy word for the rule the above expresses.
See above. “If Hitler was a woman, then he would have persecuted the blacks,” can be expressed however you like because it has nothing to do with logic until you give me some premises that allegedly show it’s truth. If those premises are
“Hitler wasn’t a woman”
and
“Hitler didn’t persecute the blacks”
Then those premises fail to lead to the intended conclusion because the “would have” comes from no where. If you change the second premise to
“Hitler wouldn’t have persecute the blacks”
then it works, but what does that premise even mean on it’s own? It needs a ‘given what’ to make sense, such that anybody is going to accept the premise as true.
Yeah, I think so. At any rate, saying what would happen seems to be a matter of an evidential relation and not a deductive relation.
Baring in mind that this ‘if’ doesn’t have any more causative influence than the previous 'if’s we’ve discussed, what it adds up to is something like this:
“Given that there are no cases in which “My name is Sam” is true, it is true that In every case in which ‘My name is Sam’ is true, ‘I am a Martian’ is also true.”
How problematic is that?
Well, just bare in mind that the logical "If' Isn't a case of one statement entailing another. It's just a statement, which may or may not entail some other stuff. "If My name is Ryan then I'm an asshole", in formal logic, is entailed by "My name is Ryan and I'm an Asshole", and means nothing additional. The problem is just that our language has two senses of the word 'if', NOT that logic and human thought are clashing.
In English. I don't know, but it's entirely possible that in some other language the formal language operator 'if' is understood perfectly well to mean "In all considered incidents of A, B", their word for "B is brought about every time by A" is some other term entirely, and they would have no idea what you and I are discussing.
I think I’m on board with all of the above.
I think any conflict between the rules of logic and how we think must ultimately be a semantic problem, not a real thing. I think logic is incomplete though, it doesn’t address everything that we want to talk or think about.
James,
I… did… not… start… with… a… false… premise.
I will not follow this line of argument further.
And I hope I didn’t say you could. The closest I came to that was to say that “If P then Q,” but not that P is the case, and therefore not that Q is the case.
As for the rest, see my reply to Uccisore below. I think he’s saying something similar to you:
Not at all. In fact, this is beautiful, Uccisore. It’s like poetry. 
I mean, it clears up a lot.
Just one more question: What’s preventing us from saying “Given that there are no cases in which ‘My name is Sam’ is true, it is true that in every case in which ‘My name is Sam’ is true, ‘I am NOT a Martian’ is also true.”
There is a doubled conditional here. Not only need Hitler be a woman’to satisfy the proposition, but women have to discrimimate.
To mix truth values with logical popositions is like trying to qualify quantified facts.
We have to find out if women did in fact discriminate as described, before we can say anything about Hitler in this regard.
So the proposition should instead say,
If it can be found that all women discriminate blacks, then, if Hitler was a woman, then Hitler would discriminate if he was a woman. The IP is neither logically exclusive or factually inclusive. Therefore, it mixes logic with truth ascerning. This is the opposite of reductive discerning, as it seeks to defer inferences away from complete ideas, and it disassociates from common sense interpretation. It diffuses meaning by coflating language and logic.
If there are no cases where my name is Sam, then on every case where my name is Sam. …you could qualify the contradiction …by adding : except on Mars.
If and only if there were martians named Sam .
This confusion stems from you using the terms “if then” and P->Q… if only you would express things as ~(P^~Q) things would be much clearer.
I’m sure he is… but I’ll check it out.
The problem I think you are trying to point out is the “seeming” paradox of ending up saying (P->Q) ^ (P->~Q) or rather P->(Q^~Q)
Which if you would have payed attention to my previous posts, would be much easier to understand if you would just step away from those expressions, that are seemingly confusing you.
Material implications are NOT usually expressed as P->Q…
It’s so much easier to understand if you simply say
~P (you don’t need the second premise)
Therefore ~(P^~Q) ^ ~(P^Q)
That looks far less objectionable doesn’t it?
yet it is the EXACT same as what you are trying to say.
… if Hitler were a woman, she would have persecuted blacks. 
There are two sources of confusion here: conditionals and logical vs. common-sense English. English speakers don’t use material conditionals when they talk, they use indicatives or counterfactuals. So it sounds strange to convert material implications into English language, because you’re using the exact same syntax of a counterfactual conditional here. Counterfactuals assert a positive truth (if I hadn’t broken my leg, I could have been a great dancer) whereas the vast majority of material conditionals are entirely vacuous (if I hadn’t broken my leg, Rome would never have been built/the moon is made of cheese/fire engines are blue). Vacuous truths are… well, vacuous. Uninteresting in any context. en.wikipedia.org/wiki/Vacuous_truth
The only reason it’s important to be clear on the difference is that you might be tempted to use a vacuous truth as a positive claim, which is not valid. If Hitler were a woman, she’d have allied with Canada in order to invade the US. So we can’t trust Canadians, they’re closet fascists.
So to summarize this whole thing
What you can say in english, to accurately express a material implication of the sort you are attempting would be:
Given that my name is not Sam, the propositions “my name is sam and I am a martian” and “my name is sam and I am not a martian” are both false, so are any other proposition that state, among other things, that my name is sam.
The only reason a material Implication is sometimes expressed by saying “If P is true, then Q is true” is because when you have a premise like ~(P^~Q) it follows that if P were the case Q would have to be the case in order for that statement to remain true…
If Hitler had been a woman ‘‘he’’ would have been a ‘‘she’’… No??
I got your point, Mad Man. Keep in mind that my question to Uccisore was a loaded one. I’m anticipating the answer but I want to hear it from him. It’s one of the ways I confirm my own suspicions.
I can see how that works.
You seem to know a lot about this stuff, Mad Man, so I’m going to ask you the follow in order to clear up my confusion (not because I think I actually found some holes in logic):
What you said above makes sense if we had the premise ~(P^~Q). But you also said above that:
So if our only premise is ~P, we should be able to arrive at:
~(P^~Q) ^ ~(P^Q)
which says to me: since we’ve established that ~P is the case, then it’s not the case that [P is the case plus anything else like Q or ~Q].
But now I’m going to put this into a if/then form. I know you don’t like that but if material implication says that ~(P ^ ~Q) is equivalent to P → Q, I should be able to derive statements in the if/then form from the material implication form without violating any rules of logic.
So ~(P^~Q) ^ ~(P^Q) can be made into:
(P → Q) ^ (P → ~Q)
Now, at first this seems like a paradox–but here’s where I need you’re help (remember, I’m not trying to be a smart ass; I just honestly don’t get this). You’re right that the material implication form is way less problematic, and I could understand P → Q in the sense of a material implication if P → Q were the only thing I derived. But since I also derived P → ~Q, then it seems this is saying: If P is the case, then Q would have to be the case and ~Q would have to be the case as well in order for the statement ~(P ^ Q) ^ ~(P ^ ~Q) to remain true.
But I’m going to take a guess: if we have both ~(P^~Q) and ~(P^Q), are we simply saying that P cannot be the case? If so, then certain things from my previous posts follow–namely, that if P were the case, then we’d have a contradiction (i.e. we’d have P and ~P), and so anything you want could follow from P, including Q or even ~Q.