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.
Ok, I think you just blew me out of the water with this one. I cannot deny that there is absolutely no connection between nose scratching and quasar activity, which means that you can even build absurdities (or oddities if you think “absurdity” is too strong a word) out of “if [true] then [true]” statements, which tells me that the rules of logic are not just the rules that rational human thought follows when it is working correctly.
Do you mean this similarly to how, in logic, “and” and “but” are equivalent and that an ordinary English statement such as “A but B” would be translated in formal logic as “A and B”? I mean, are you saying that “If Hitler was a woman, then he would have persecuted the blacks,” would be translated in formal logic as “If Hitler was a woman, then he did persecute the blacks”?
What do you mean by that? Do you mean that the truth of a “would” statement can never be determined with absolute certainty?
That’s not quite what I meant. What I meant was that logic is the rules that thought must follow if it wants to guarantee it will be right. At least, that’s what I assumed when I started this thread (and now I’m second guessing myself). For example, it’s true that people think inductively, and this is a very rational way of figuring things out, but you should be able to reason with a rational thinker and get him to admit that inductive reasoning doesn’t always get it right. If all cows I’ve seen have four legs, I will inductively conclude that all cows have four legs, but it shouldn’t take much to imagine that I could be wrong, that there might be a cow out there with three legs and I just haven’t come across her.
Well, I think you’re right about the original argument I made in the OP. I was a bit careless in translating the negation “Hitler would have persecuted the blacks,” as “Hitler didn’t persecute the blacks,” but that’s why I switched to the statement “If my name is Sam, then I’m a Martian.” At this point, I’m fairly certain that I can start with the premises “My name is not Sam,” and “I am not a Martian,” and conclude that “If my name is Sam, then I am a Martian”.
You see, Uccisore, even though I think you’re right, and you clearly blew me out of the water with your example of nose scratching and quasar activity, I’m still struggling with this. What’s bugging me is that logic shouldn’t be allowing me to draw conclusions like “If my name is Sam, then I am a Martian.” ← This seems nonsensical to me. Yet my name is clearly not Sam, and last I checked my birth certificate, I wasn’t a Martian, so I have to conclude that it’s true.
But I think maybe I’ve solved the problem. Tell me what you think:
I think maybe I’m wrong to think that the rules of logic are supposed to be the rules that rational human thought follows when it is working correctly (as defined above–i.e. being able to guarantee it will be right). But there has to be some relation to rational human thought–otherwise, what are we doing with logic? What purpose does it serve? In other words, if we discover the rules of logic by examining how rational human thought works, then we shouldn’t be able to apply the rules of logic to get statements that clearly clash with rational human thought. I think it’s pretty clear that if I tell someone that if my name is Sam, then I’m a Martian, their immediate reaction won’t be “Yep. Makes sense to me. If you’re name is Sam, then obviously your a Martian.” I expect that a rational person would say “Why would your name being Sam entail that you’re a Martian?” And if this is the way a rational person would respond, then how is that, having applied the rules of logic to what he’s responding to, those rules were derived from examining how rational thought works?
But now I’m wondering if this is the way it works: yes, the rules of logic are derived from examining how rational human thought works, but that doesn’t mean those rules of logic just are the rules that rational human thought follow. What happens is that when logicians want to understand the rules of something like a conditional statement, they ask “When an ordinary rational human being utters a conditional statement, what are the conditions under which we’d say he is correct?” Well, those conditions would be that when the antecedent is true and the consequent is true, then the whole conditional is true. (That’s at least one of the conditions). Note that the conditional statement would have to be something that insinuates an actual connection between the antecedent and the consequent. Your nose scratching/quasar activity example would not apply in this case. If we’re starting with ordinary rational human thought, then “If I scratched my nose, then billions of light-years away, a quasar emitted some high-energy particles,” would not be considered a rational statement unless the speaker had some reason to suppose there was a connection between nose scratching and quasar activity. So in trying to determine the logical rules of conditional statement, logicians would have to start with conditional statements that actually are rational according to rational human thought–something like “If I drink poison, then I’ll die.” They’d ask: what are the conditions that make this correct, and at least two conditions: 1) the antecedent is true, and 2) the consequent is true, must be met. They could go further and say that another condition that must be met is that there is an actually connection between the antecedent and the consequent (I would think that’s part of what it means for the conditional statement to be rational), but this condition is technically not relevant to logic (more on this below).
Logicians would have to go further, of course–they would have to fill in the entire truth table. They might next ask: what are the conditions under which we’d say the statement is false. And of course, that’s a no brainer: when the antecedent is true, and the consequent false, we’d say the statement is false. Now the last two rows of the table are a little more puzzling: what would a rational human being say about the truth of a conditional statement when the antecedent is false. If my name is Sam, then… I still think that ordinary human thought wouldn’t necessarily have anything to say on the matter–and that’s where it would pull out the indeterminate value. But as logicians working in a binary system, they want a definitive answer: what woud be the value–“true” or “false”–if the antecedent is false and the consequent is [whatever]? But here, they don’t have to appeal to orindary rational human thought. As you point out, the either/or version of the conditional entails that “if [false] then [false]” is true, and so logicians get their answer that way. How they get their answer for the “if [false] then [true]” case is beyond me, but I’m guessing it’s something similar to the deduction from the either/or case.
That’s how logicians would determine the rules of logic. But when logicians apply the rules of logic back onto human thought, it doesn’t quite work the same way. When trying to decide what the rule of logic are, logicians assume a statement to be true (or false) and then ask what are the conditions under which this is the case. But when applying the rules of logic (after they’ve been derived), it’s the other way around: they assume the conditions are met (assign truth values to the propositions) and ask whether that makes the overall statement true or false.
What I failed to understand from the beginning is that when going from ordinary rational human thought to formal logic, we not only need to ignore the semantics of the propositions, but the semantics of the operators as well. In the statement A → B, it doesn’t matter what A or B mean–they are meaningless symbols and that’s ok, the logic still works–but the operator → is also meaningless. It doesn’t mean what “if/then” ordinarily means in rational human thought. In ordinary rational human thought, “if/then” does connote a connection between antecedent and consequent. When someone says “if A then B,” he means to say that A being the case makes it such that B is also the case. But just as we throw out the semantics when we turn ordinary English statements into formal logical notation, we must also throw out the semantics of the operators. If there’s any semantics left in the operators, it is merely the rules laid out in the truth tables for those operators. I said earlier that the rules of logic can’t just be meaningless and arbitrary–well, it turns out they can be meaningless, but not arbitrary. They are still derived by examining rational human thought, plus deductions from other logical statements and rules. What this means is that whereas with ordinary rational human thought, the truth of the statements is not the only condition under which we say a compound statement is true–i.e. we say, in the case of conditionals for example, that the antecedent being the case makes it such that the consequence is also the case–it is the only condition when it comes to formal logic. This in turn means that the rules of formal logic, when applied back onto rational human thought, end up being a lot more generalizable than they would be if they had to take the original meaning of the operators into account. They can be applied to statements that would otherwise seem absurd or nonsense to ordinary rational human thought.
It’s this generalizability that leads me to conclude that the rules of logic are not the rules of ordinary rational human thought–they may be a subset of the rules followed by ordinary rational human thought, but we can’t say that all we need to predict or describe how ordinary rational human thought is the rules of logic. This generalizability of the rules of logic is why, sometimes, the rules of logic clash with ordinary rational human thought.
What do you think, Uccisore? Does this resolve the issue?
Gibb may I suggest that by rational thought you may mean th w content of thought, and by the logic is meant the form of it? The content has to do with conditionals such as if and only if something is true something else is also true, whereas
Formally something may be true if another thing also could be true.
The first type is deductive it eliminates all probable to the only possible by excluding all but the only possible.
In this case ,Hitler would only be a killer of blacks, if women were killers of blacks AND Hitler was a woman. The double conditional implies the prior and sequential premises to imply that the prior premise is necessary foe the latter to be even possibler(contingent )
Once thats established formally , then, the qualifiers can fill in the content -which is deductive, and reductive to the effect of excluding all probables ,such as, Hitler is a killer of blacks if he is not a woman. The law is contradiction becomes spurious, and the law of identify excludes that possibility. That is the content.
What you are concerned about as having to do with rational thought pertains to the actual truth value, and not the logical validity. That’s the way I see it as triple staged, necessary argument, contingent argument, and truth value. Sorry if I am redundant, but fond this the best way to go at it
only
