Read the below baring in mind that I only got a B in formal logic, and it was a hard-earned B.
The propositions above would be properly expressed in sentential logic as
“If it IS the case that Hitler was a woman, then it IS the case that Hitler would have persecuted the blacks.”
Baring in mind that sentential logic isn’t examining what Hitler would or would not have done, but merely how certain sentences hang together. The “if it is the case that” can usually be omitted, but here a proper understanding of what’s going on requires it.
And see what I said above- the above is true because it’s not the case that Hitler was a woman, and what we know about ‘either not A or B’. However you want to parse it, you can’t put the labor of discerning what might be the case in the future or another possible world on basic sentential logic- we have other operators for that. The above is not logically equivalent to “If Hitler was a woman, he would have persecuted the blacks”, because this second sentence is not attached to what is in fact the case in the way the above one is. You need that attachment to what is in fact the case for your logic square to work the way you want it to.
So, all sentential logic can say is that the above sentence is true- by virtue of the fact that that’s what you get when the first premise in an ‘if then’ is false. The question of what Hitler would have done in some other circumstance isn’t a subject for sentential logic.
I think you can stick the ‘not’ wherever works for you, because ‘not’ is a basic operator of sentential logic.
Yeah…you might be right about that, on second thought. Still, I don’t think ‘would have’ is what sentential logic 'if’s are about. The truth value of those kind of ifs are a very controversial thing.
But it’s not the way sentences hang together, not in the way we ordinarily use sentences anyway. What the above statement is saying is that there are two cases: 1) that Hitler was a woman, and 2) that Hitler would have persecuted the blacks, and that the latter depends, for its being true, on the former being true. Now you take the rules of conditional statements into account, and they say that if both cases are not true, then the whole statement is true–that is, the second case does depend, for being true, on the first case being true.
But as insightfoul so untacitly stated, there is no correlation between Hitler’s sex and his being a racist (he was a racist, but not because of his sex). That is to say, in reality, the second case does not depend on the first case for its truth. In other words, the rules of logic fail us here. They tell us that it’s true that if Hitler was a woman, he would have persecuted the blacks, but we simply cannot intuitively take that seriously. In reality, one would think, Hitler’s sex had nothing to do with his racist orientations. My point in this thread is that this particular logical rule–that a false antecedent and a false consequent makes for a true conditional statement–is misguided at best.
I’m saying that if logic is to serve a purpose at all, it ought to tell what actually follows in reality, or at least semantically, given certain true assumptions. I’m saying that if all logic is is a system according to which we apply meaningless and arbitrary rules to symbols (i.e. propositions) given certain meaningless and arbitrary operators (i.e. and, or, if/then), then we can make up whatever the hell rules we want. We could say that if the antecedent is true and the consequent is true, then the whole compound proposition is false. Why not? We just have to agree that that’ll be the rule and that we’ll all follow it. That way, we can say statements like “If you drink the poison, you will die,” are unequivocally false. How so? Well, that’s just the rules of logic–and we all know that those rules are just arbitrary and meaningless–we just made them up and agreed to follow them–so don’t worry if it doesn’t make sense in reality, it’s just a logical rule.
Of course, I’m not saying I’m smarter than centuries worth of professional logicians, but that this assignment of binary truth values to propositions is not the only option: we could have a trinary system in which we are allowed to assign an “indeterminate” truth value to some propositions, including compound ones.
Yeah, that one I’m still struggling with. It certainly does seem pretty cut and dry that “Either not A or B” is true given that not-A is true and B is false. Which of course, implies that “If A then B” is also true. Not sure how that will play out, but I’ll think about it some more (maybe).
I’m not exactly sure what you’re saying here, but if I understand you correctly, I think that’s what I’m saying. I’m saying that “Hitler would have persecuted the blacks” depends on “Hitler was a woman” being the case in actuality just to have any truth value at all (let alone true or false). Since “Hitler was a woman” is in fact false, “Hitler would have persecuted the blacks,” has no truth value (in a binary system), and therefore warrants (in a trinary system) the value “indeterminate”.
But you see why that entails a failure on the part of logic–at least in binary systems–that is, if all that sentential logic can say is that the above sentence is true, when in fact there is no truth or falsity to the sentence in reality, then it’s options are rather limited, and it falls short of informing us of actual states of affairs. But if we introduce a trinary system, we get the leverage of re-connecting logic with reality.
But in any case, I thought of this instead:
If my name is Sam, then I’m a Marsian. ← It’s true!
A logical implication is a statement that forms only one particular truth table.
You have to look at your statement and decide if it fits that truth table.
Yours doesn’t.
p = Hitler was a woman
q = persecuted blacks
p → q = Hitler would have persecuted blacks
Truth table for a Logical Implication:
p _ q _ p → q
F _ F _ T
F _ T _ T
T _ F _ F
T _ T _ T
For your statement to be a valid implication, all four lines have to be valid.
Hitler was woman : not persecuted blacks :: T/F Unknown
Hitler was woman : persecuted blacks :: T/F Unknown
Hitler not woman : not persecuted blacks :: F Known
Hitler not woman : persecuted blacks :: T Known
Your statement met only two of the four requirements.
You would have had to have included some assertion connecting being a woman and persecuting blacks.
Not exactly clear on what you’re saying, but it sounds like this: given that my statement is the second one above (which it is), then it has an unknown truth value. “Hitler was woman” is definitely known to be false, but “Hitler would have persecuted the blacks” is unknown. Now, you realize that in order for that to make sense, we would have to be saying that such a statement (with “would” in it) can’t be assigned a truth value, not just that we don’t know in fact. And this may be the case: how does the truth of a statement with “would” in it get decided?
But of course, this is what I’ve been arguing all along–that we allow “unknown” as a third truth value. In a binary system, you’d be right–my statement wouldn’t fit the truth table. But in a trinary system, it could fit the table seeing as how an “unknown” truth value would be perfectly valid.
What about a little box turned sideways so it’s like a 4 sided diamond sitting on one of it’s corners, with an arrow coming out the side of it pointing to the right? It could mean “possibly”.
I had to read this a few times to figure out what was going on, I’ve been revising my position as I go, because these are matters I haven’t thought about for a while. Going back to your very first post is what clued me in, in the end. Let me know what you think about this:
You began this by telling us what we know: we know that Hitler wasn’t a woman, and that he didn’t persecute blacks.
1.) Hitler wasn’t a woman,
2.) Hitler didn’t persecute blacks.
From that we can conclude that
3.) If Hitler wasn’t a woman, then he didn’t persecute blacks.
That should tell you something right there- that the “IF” relation in formal logic is saying nothing at all about causation. It is certainly not saying that Hitler’s not being a woman caused him to not persecute blacks, because that’s not implied in your two premises, and an analytic operator, by definition, contributes no new information.
“Hitler’ WOULD HAVE” persecuted blacks thus comes from no where- no where but an incorrect reading of ‘if’ that assumes causation where, I hope you agree with me based on the above, it does not exist.
“If Hitler was a woman, he would have persecuted Blacks” thus does not follow from the information you gave us. the ‘would have’ clause has different content than what we started with, it is not a simple negation of 2, and it’s not a proper interpretation of what ‘if’ means in formal logic. What the rules of implication actually give us is this
“If Hitler was a woman, then Hitler persecuted blacks”. Not ‘would have’.
I think seeing that 4 is true is a little odd, but it’s not a full break from reality- it’s no harm to our understanding of the universe or logic’s place in it to see that a conditional with a false antecedent and consequent is true. Is 4 true or false on a common-sense understanding? Meh. Hitler was NOT a woman, so common-sense has no horse in this race. If implication tells us it’s true because of the ‘either not a or b’ thing, then that’s fine now that we’re no longer actually making a claim about womanhood or it’s causative influence on racism or whatever, right?
It’s taking me a long time to get to an answer here, because I know intuitively that the problem is with ‘would have’, but I’m so cussed bad at formal logic that sussing it takes a lot of finagling; I apologize for changing my approach with every fresh reply.
A trinary system produces a different truth table (3x3).
Your statement has to fit ALL rows in the truth table for it to be of that type.
An implied statement leaves out one of the presumed premises, which in your case is an unknown.
If Hitler was a woman [and women abuse blacks] then Hitler would have abused blacks.
If Hitler was a woman [and don’t know shit bout women] then Don’t know shit bout Hitler.
I’m glad you think real logic is trinary, but I think you’re confusing conditional statements for syllogisms in the above.
Syllogism are of the form:
P → Q
P
Q
Whereas a condition statement is just the “P → Q” above.
There need not be an “implied” or “hidden” clause in “If Hitler was a woman [and women abuse blacks] then Hitler would have abused blacks.” ← This is just the P → Q part. If you want to pull out “and women abuse blacks” then you’re just making a syllogism of the form:
[X is a woman] → [X abuses blacks]
[Hitler is a woman]
[Hitler abuses blacks]
Any syllogism can be converted into a complex conditional statement. For example, the above syllogism (with the P’s and the Q’s) can be rewritten as “[(P → Q) & P] → Q”. I think that’s what you’ve done here: “If Hitler was a woman [and women abuse blacks] then Hitler would have abused blacks.”
What I was saying is that an “implied statement” leaves out one of the conditionals that would normally be required for a deductive statement. In your case, the statement left out involved the connection between being a woman and abusing blacks. If you had put that one in, the statement would have been a “deduction”, not an “implication”. But as it was, it was neither.
Yeah, you got that right. Except I do want to point out that even though logical conditionals have nothing to do with causation per se, they do seem to insinuate a dependence relation–in that the consequent depends on the antecedent.
Technically, you’re right, but I think we can do better. If we could just say that the truth value of the conditional is unknown, that (to me) would fit with the way we think of reality much more squarely.
Yeah, still struggling with this.
No need to apologize, Uccisore, changing one’s approach can be a sign of rationality.
My thoughts so far on the “would” qualifier are as follows: In itself, it seems almost to demand that the truth value of the proposition it’s a part of be determined as part of a conditional. What I mean is, you take the statement “Hitler would have persecuted the blacks,” and even though it is a grammatically well-formed sentence, semantically it seems to hint that it cannot stand by itself, that it requires being the consequent of a conditional just to make any sense. I mean, you say that to someone, and they’ll likely respond “If what?” So the reason we have trouble with it is that we can’t determine its truth value unless we can determine the truth value of some antecedent proposition it is connected to as its consequent. And in some case (such as this one), that the truth value of the antecedent be “true” specifically.
That’s unlike the following conditional:
If my name is Sam, then I’m a Martian.
which again, brings up the same absurdity I tried to exemplify in my OP (although, in some way, not as much), but this time, without the “would” involved, it’s quite easy to see that the antecedent and consequent are both false. You can take either out and treat them as stand-alone propositions, and not only will they be well-formed sentences grammatically, but there won’t be any question as to their truth/falsity.
It is very difficult to understand the area between opposites. If I say I have no friends, it does not mean that I have enemies. If I say that I have no enemies, it does not mean that everybody is a friend to me.
So, how can you understand the state of being where you have neither friends nor enemies?
So, we swing like a pendulum from one end to the other. That is the movement of thought. It is always between these pairs of opposites. It’s very difficult to conceive a state of being where these pairs of opposites do not exist at all.
It’s an unnatural thing that some have accepted as natural. It could be tragic. It’s amazing how many people exist in this state and never questioned it, because if they begin to question, a major part of their existence is at stake. They are that: not different from this movement of thought.
Not all is what it’s perceived to be. In the process of acquiring knowledge and using it to create a state of being or to think of it as a genuine experience of some kind, the whole thing might have begun on an erroneously ascertained premise. And to pursue a conclusion conditional to a bogus antecedent is a waste of effort hindering clarity.
But it doesn't. "If there's smoke, then there's fire" allows for there to be fire without smoke, but not smoke without fire. If anything, the antecedent depends on the consequent.
It would. But then what do we do with “Either not-a or b?”
I think we can make a full break from ‘would’ and ‘if’ by looking at some real world examples of logical if’s. Let’s suppose I roll pairs of dice, and I get the following results.
4, 2
1, 3
3, 3
6, 2
2, 4
Suppose further we take that a number rolled is true, and anything unrolled is false for the given set.
For the first set, “4 and 2” is true. “4 or 3” is true. “5 and 2” is false. “Not 4 or not 2” is false.
“If 4, then 2” is true for the first set. It is ALSO true for the entire series. All it means is that in every instance of 4, 2.
“if 3, 1” is true in the second set, but not for the entire series (thanks to the third set).
There are a lot of examples you could create, but what’s important is that in this model, the first number rolled has absolutely no influence over the second number rolled, and despite that, the logical “if” is absolutely correct. There are common-sense understandings of “if” that would be false by virtue of the fact that we know the numbers are being determined by random rolls- “if 1 than 3” is false because thanks to what we know of dice, it’s preposterous to assume that every pair that has a 1 in it will have a 3. But that isn’t the logical if- and the logical if does have it’s place in real world examples.
To bring it back to your “Hitler would have”, you can see each set of numbers as a premise. Based on the premises, “If 4 then 2” is true. So let me roll the dice again and see…
4, 3
“If 4 then 2” tells us nothing at all about premises that aren’t in the argument in the logical sense- it was making an observation, not stating a rule. Your sense of “would have” states a rule. Logical ‘if’ doesn’t need to know anything about the above numbers to assert “If 4 then 2” is false (even though it was true before that last premise was added). Your ‘would have’ if needs to know that these are merely dice with no causative influence on each other in order to draw any conclusions.
So the two if’s are commenting on different kinds of things, which as it happens, can sometimes coincide in confusing ways.
Oops. You got me there. But still, it insinuates a connection between antecedent and consequent. If you have the antecedent, then you must have the consequent.
I was thinking about that and I think I’ve solved it. In a binary system, all propositions (including compound ones) must be either true or false. When you have “if [false] then [false],” it isn’t so clear what the overall proposition should be, but the either/or version of it tells you that it has to be false.
But what I overlooked is that I’m thinking in terms of a trinary system, and in a trinary system, the either/or version would also be indeterminate:
If I’m right that:
then the same would hold when the “would” proposition exists in the either/or proposition:
Either [It is not the case that {Hitler was a woman}] or [Hitler would have persecuted the blacks].
Because an either/or statement is only true when only one of its propositions is true and the other is false (i.e. they can’t both be true as they can in a simple “or” statement), then since that pesky “would” in [Hitler would have persecuted the blacks] make that proposition indeterminate, we can’t say whether the whole either/or statement is true or false, and thus it too is indeterminate.
I only barely understand what you’re saying here (I’m slow).
I understand that the rules of logic have nothing to do with the causal connections of things in the world, and I don’t think logic should ever have to touch causation.
All I’m trying to sort out in this thread is how to prevent the rules of logic from allowing us to state absurdities. If in the statement “If Hitler was a woman, he would have persecuted the blacks,” we take the antecedent and the consequent to be false, the rules of logic tell me that the statement overall is true–and to me that seems absurd. I think we’ve established that “would” is the culprit here, and your example above nicely illustrates the cases in which the if/then rules apply. But what I’m suggesting in this thread is that if the rules of logic have nothing to say about statements of the kind I started with (particularly, “would” statements), then that means there is room for development in logic.
I’ve always assumed that logic is developed from studying the nature of rational thought–it is essentially the rules that thought follows when it is working the way it is supposed to work (i.e. figuring things out correctly). Logic and its rules are not discovered by looking under rocks and finding them, or looking through our telescopes and discovering modus ponens or DeMorgan’s Law floating around the vicinity of Saturn. They are discovered by examining how we think.
Of course, we have to start out simple. It was Aristotle who got the ball rolling (wasn’t it?). He not only developed the simple rules of syllogisms and such, but found that you could extract quantifiers from propositions and use them as operators that determined truth values. So whereas without quantifiers, the two propositions:
“All men are mortal.” = X
“Some men are mortal.” = Y
would have to be denoted with different variables (X and Y), with quantifiers, you only need one variable:
(Ax)P
(Ex)P
The “not” operator is similar:
“All men are mortal.” = X
“Not all men are mortal.” = Y
becomes:
P
~P
Later in the development of logic, we discovered modal operators:
“Possibly all men are mortal.” = X
“Necessarily all men are mortal.” = Y
becomes:
[diamond]P
[square]P
What I’m saying here is that there is still room for further development in logic, and perhaps “would” ought to be treated as another operator:
“Hitler would have persecuted the blacks.” = X
“Hitler did persecute the blacks.” = Y
becomes:
[would]P
[did]P
And of course, the whole point of a logical operator is to help determine the truth of a proposition. In this case, the rule is:
If P=T, then [did]P=T and [would]P = indeterminate
If P=F, then [did]P=T and [would]P = indeterminate
It’s rather simple, obviously, but it comes down to this: the would operator converts the truth value of the proposition (whatever it is) to indeterminate. Of course, you realize that I need a trinary system for this, which is why I’m pushing for it. And I think I have right to do this–I mean, if I’m correct in thinking that logic and its rules are discovered by examining the nature of rational thought, then it would seem that there ought to be indeterminate truth values, for thought doesn’t always give us propositions that are either self-evidently true or self-evidently false. “This statement is false” is not obviously true or false. That means that thought has, in its storehouse of truth values, an “indeterminate” value. Similarly for “If Hitler was a woman, he would have persecuted the blacks.”
What this gives me in the end is the further development in logic I was looking for. As I’ve been arguing all along, logic is in the service of making sense out of language and our understanding of reality. This does the trick for me. It helps me avoid certain absurdities in our statements by expanding the cases that logic can apply to.
Now what’s left is:
If my name is Sam, then I’m a Martian.
There is no “would” there, but we still get an absurdity (as far as I’m concerned). At first, I thought that since we’re working within the context of a trinary system, there might be room to fudge the rules of if/then. I mean, if the if/then truth table was determined only within a binary system, perhaps the if/then truth table in a trinary system is up for grabs, and maybe that means I can assert that in a trinary system, “If [false], then [false] = indeterminate”… but then that pesky either/or issue crops up again. We’re not dealing with “woulds” here, so it’s pretty clear cut, even in a trinary system, that “Either not-[false] or [false]” is true, which of course means that “If [false], then [false]” is true.
So I’m back to square one… puzzling over either/or
I don’t think so, and I think this is the key of our disagreement, which I tried to express with my dice example. If
1.) I scratched my nose
is true and
2.) Billions of light-years away, a quasar emitted some high-energy particles
is true, then
3.) If 1 then 2
is true. There is not a bit of prediction or obligation or any of that in it.
“A and B” entails “If A then B” in other words, regardless of the content or relationship between A and B.
I think you made sense there, but I’m also not convinced we need indeterminate as an option to make sense of this (anymore). I think eliminating ‘would’ or ‘must’ from our logical understanding of ‘if’ is enough.
But they don’t do that, because ‘he would have’ isn’t a part of any premises we can confirm, and logic can’t speak about it. All you get is “If Hitler was a woman, then he persecuted blacks”, which is odd, but no absurdity.
Some kinds of ‘would’ can be explored with modal logic, which is indeed a development of logic. But most forms of would can’t be examined by logic because they are inductive, not deductive.
Logic of the kind we’re talking about is just a subset of all the ways people figure stuff out rationally. Induction isn’t logical, memory isn’t logical, empiricism isn’t logical, etc. They might be logical in the common-sense usage of the word, like how ‘if’ implies causation in the common-sense definition of the word.
I don’t think you need a new operator for that. I mean,
1.) If A then B
2.) B + C
2.) there is indeterminate, there’s not some way to express that in logic now? Or do we just say that everything that isn’t entailed by the premises is false? I don’t remember what the standard is.
That second statement only has an indeterminate truth value given a set of premises you gave that are insufficient to determine if it's true or not, and that's nothing new- most premises have an indetermined truth value given some other, random premise. I seriously still think that what happened here is that you drew a conclusion that didn't follow from a pair of premises, and that's it. It's not a flaw in logic that there's no logical operator to describe the relationship between "If A then B" and "If ~A then C"...because there is no such relationship. In the end, that's the argument you made- it just so happens that your C uses a lot of the same words as B and it took a bit to realize it was a different premise.
To make a related point, ‘would’ doesn’t function any different than any other non-operator word:
1.) I would have eaten a million tacos.
2.) The stars were right.
Given that 1 is true and 2 is true, then “If 2 then 1” is true, irrespective of the word ‘would’. That it’s difficult or impossible to figure out how one could possibly determine that 1 is true is irrelevant. In formal logic, stipulating it is enough.
I don’t understand the problem with the “If My name is Sam, then I am a Martian” thing.
Gib, you stated an assertion. You cannot valid an assertion from its own conditionals. Your assertion was either true or not. The truth of the individual conditionals has nothing to do with it.
