When Mises says they “are contained”, I doubt he meant they are part of the meaning of the starting axioms. He meant they are consequences. And if he did mean they are part of the meaning, then I disagree.
As for the the Sufficiently Inteeligent Being, if he was really intelligent, he would not say "You already told me that, but rather “You already gave me enough info to deduce that”.
Quine is way beyond me, just flies right by, so Im not competent to say anything about that last paragraph.
I have two beginner’s books on logic and it never defined conditionals like that, they just hinted that for some reason, causation was not included.
Which brings me to a question, why was aristotelian logic wrong -all logic is logic so at most the old logic was just not useful or something.
And two, why exactly are conditionals inappropriate for causation? If I say that if it’s raining then my shirt is wet, the truth conditions of the conditional are not unreasonably applied to reality (in my opinion), for the conditional is true in most cases except when the antecedent is false and the consequent is true -a rather reasonable set of conditions if one wants to model the real world. Indeed it is true that the conditional is little too loose -since there may be other conditions attached to that condition but still, just because there are other causes doesn’t mean that one secondary cause is incorrect -it may mean that we should just have a really, long conditional.
There’s way too much mysticism surrounding language and logic in this thread. Given discrete states (or “symbols”) which can be readily distinguished, we can reduce any language to the universal language, that is, the language accepted by a Universal Turing Machine.
Logic in the digital era is mundane. There is nothing of the mysterious or transcendental in it.
Someone suggested the biconditional is not an operator. This is simply incorrect. The bi-conditional is, in fact, a short-hand for two conditionals joined in conjunction:
A<->B == (A->B) ^ (B->A)
Read this as “A if and only if B is defined as the conjunction of A implies B and B implies A”.
This shows that the bi-conditional is an operator or function, it takes two logic values as input and returns a logic value as output.
Identity or definition, on the other hand, is not an operator. It simply expresses a relationship. For example, the above statement is a definition, not a function. It takes no inputs and generates no outputs, it simply expresses a relationship that defines the bi-conditional operator in terms of the more basic conditional and conjunction operators.
Clayton -
I explained this twice before, please refer to my earlier posts.
No, the problem is that conditionals apply to variables which are only correlated. Correlation is not a causal relation because one cause can have many effects. That’s why I used the rain/wet-hair/wet-shirt illustration. One cause (rain) has two effects (wet hair and wet shirt). Even though wet hair and wet shirt are not causally related, we can place them in a conditional relationship… (wet hair)->(wet shirt) and this relationship is perfectly valid. So, you can have implication without causation. This is why you cannot conclude causation from dependence in probability theory… the fact that two variables are dependent does not mean they are causally related since they may both share a common cause.
Clayton -
They should have mentioned somehwere along the line that A->B is the same as “not A or B”.
Now you’ve aroused my interest, what does implication have to do with the definition of conditionals? -A or B is derivable from A–>B but yet this is connected with A–>B being defined as whenever A is false and B is true…(if I understand you right).
And second, why did the mathematical community make a rule about logic? Shouldn’t logicians be the ones deciding what goes in their field, that is, if logic and math are separate fields with different ends?
Now your apetite is whetted, you should look around for an introductory text on logic that resonates with you. Some books may be too terse, some too verbose, some with too much philosophical discussion for your taste, some with too little. There are so many out there, by excellent authors, that you are sure to find something. Maybe you could go to a Border’s book store and get a feel for what they have to offer. Or maybe some large university library or bookstore in your area.
I think what happened was that the mathematicians did such a good job that the logicians accepted their work as the standard.
I did read your earlier posts,Clayton, but still I can’t help but think that cause and effect are correlated with each other and as such they should be expressible in conditionals just as any other correlation, though perhaps one could not tell cause and effect just from the conditional itself.
@fakename: Yes, cause and effect are correlated, of course. The problem is that the existence of a correlation does not establish causality. That A is true whenever B is true does not prove that A causes B or vice-versa. It only establishes that A and B are not causally independent.
Clayton -
So saying A–>B is not enough to say A causes B but only enough to say A is correlated with B because the truth conditions of the conditional are too broad so that you are bound to pick up correlations but not necessarily causation?
But admitting that cause and effect are correlated, doesn’t this mean that conditionals at least tenously establish a cause and effect relationship?
When one says “we are getting hotter because we are walking closer to the sun” and someone else says “we are getting hotter because we are hiking up a hill” they express both ideas in the same form but materially, one is giving a cause and the other a correlation.
But in logic the form alone counts.
The identity means for two symbols A and B that you may replace them with each other. The biconditional means for A and B that A follows from B and vice versa. The difference is, that you have to give the identity some meaning depending on the subject you are speaking about. In formal logic, the identity can be defined as “A and B are identic if A implies B and B implies A” (because everything that follows from A follows from B and vice versa it makes no difference: They always the same truth value and formal logic only deals with this truth values). In logic, the identity yields true if A and B are identical, that is, they always have the same truth value.
But in other fields you are able to use the “structure” of A and B to define the identity. for example you can define that two points in space are identical if all their coordinates are identical (though you’d have to define what identical for coordinates means).
So the true difference between identity and biconditional is that biconditional is a symbol used in logic to connect truth values and the identity has to be defined so that you can explain how you are able to replace things.
Example: While it is “The Eiffel Tower is bigger than my house, so my house is smaller than the Eiffel Tower and vice versa” and it makes sense to identify those two statements, it makes no sense to say “2+2 implies 4” or vice versa though it is “2+2=4”.
Hmm, not sure what you mean. Two events can have a common cause and be, thereby, correlated while not being in a causal relationship.
Let P be “My hair is wet”
Let Q be “my shirt is wet”
Let R be “it is raining”
P->Q this is true because whenever my hair is wet, it is raining, which causes my shirt to also be wet.
But wet hair is not what causes my shirt to be wet. The rain causes both my shirt and my hair to be wet. So, this means that causality is more complex than logical implication. However, we can say that if P causes Q then P also implies Q. Let’s define the symbol => to mean “causes”:
(P=>Q) → (P->Q)
But as you know, it is a fallacy (affirming the consequent) to conclude that B->A from A->B alone. I think this is the mistake you are making, you are concluding that P=>Q follows from P->Q since P->Q follows from P=>Q. This is the fallacy of affirming the consequent.
To go back to the rain analogy:
R=>P
R=>Q
But it is false that P=>Q, Q=>P, P=>R or Q=>R even though all of these variables imply one another (whenever it rains, my shirt is wet and my hair is wet and whenever it is not raining my hair and shirt are dry so all of these variables have identical truth functions, that is, P<->Q, Q<->R and R<->P).
Well, this is what the whole scientific method of constructing and testing hypotheses is all about. Since we can’t conclude causality from correlation but all we can really observe are correlations, we have to construct hypothetical causal constructs and then perform experiments and/or make observations to see if these hypothetical constructs hold up to empirical testing. If they do hold up, then we have more confidence in them than we had before we tested them since we know they are not false under the conditions tested but then, we still cannot actually affirm our hypothesized construct to be true since we will have made the mistake of affirming the consequent.
In symbols:
Let A=>B be my scientific theory that A causes B
Let E be “the state of affairs after the experiment”
The logical structure of the experiment testing my hypothesized scientific theory is:
A=>B → E
If we observe E after performing an experiment in which we hypothesized that A would “cause” B, then we can conclude that the evidence is consistent with our hypothesis. Yet, we still cannot reverse the conditional and conclude from the evidence that A does in fact cause B. Causation itself can never be observed because it is a process or relationship, not an event and only events can be observed. That’s the conundrum.
Clayton -
P.S. I’m really regurgitating this lecture from Dr. Hoppe which is so excellent I recently watched it a second time through. I recommend it for you, I think it will really make you think these issues through.
clayton can’t you find a clearer way of explaining yourself without resort to examples which are plainly false?( this confused me)
Let P be “My hair is wet”
Let Q be “my shirt is wet”
Let R be “it is raining”
because it is possible for Q to be false, whilst P is true. therefore it is not the case that P->Q,
once you start of saying that it is the case that P->Q you lose me altogether…
@nir: I’m using P, Q and R as illustration of a logical principle. Forget the real world, imagine that I live in a world where there is no indoors, only outdoors and the moment it rains I am soaking wet from head to toe and the moment it stops raining, I am dry from head to toe. In such a world, whenever it rains, it would be the case that my hair and shirt would become wet from the rain. The rain caused my hair and shirt to get wet. Since the only time my hair is wet is when it rains and since, whenever it rains, my shirt also gets wet, it is the case that my hair being wet implies my shirt being wet. Yet, my wet hair does not cause my wet shirt.
I’ll try one more illustration (again, it will be in physical language for the sake of illustration but please do not mistake me to be claiming that logic applies directly to the empirical world). Hopefully I don’t muddy the waters even further.
Imagine that there is a box with a switch, a red light and a green light. If you opened the box, you would find that the switch directly connects electricity to both the red and green lights but you cannot open the box, you can only turn the switch and see what happens. Whenever you turn the switch, both the red light and green light both always come on. You have observed this many times (let’s say, thousands of times). Now, let’s say I cover the green light with my hand. When you flip the switch and you see the red light, I ask you “Is the green light on?” You cannot observe the green light because I have covered it with my hand yet you can confidently say, “Yes, the green light is on.” When I lift my hand, it will turn out that you were correct to conclude this.
Why?
The reason is that the red light being on implies that the green light is also on because the states of the red light and green light are correlated. But if we open up the box, we will find that the red light does not cause the green light to come on, instead, the switch causes both the red light and the green light to come on.
In symbols (again using my ‘=>’ symbol for “causes”):
Let S be “the switch is on”
Let G be “the green light is on”
Let R be “the red light is on”
When we flip the switch, we observe the following relationships:
S<->G
S<->R
R<->G
That is, either all variables are true or all variables are false, together.
Yet, the story for causality is not so simple. If we open the box, we find that:
S=>G
S=>R
… and the following are false:
G=>S
R=>S
G=>R
R=>G
So, implication holds in situations in which causation is false. All these variables are totally correlated, that is, they have identical truth tables. Yet, it is the switch that causes the lights to come on, not the other way around nor the lights that cause each other to come on.
Please understand that this is an idealized thought-experiment. The box, switch, wires and lights are noise-free, error-free, they never break down, and the electrical supply is never interrupted. I am not making claims about the empirical world nor saying that this is how we do science. It is an idealization for the sake of illustrating a logical principle. I hope this is a better illustration.
Clayton -
Dave:
As for the the Sufficiently Inteeligent Being, if he was really intelligent, he would not say "You already told me that, but rather “You already gave me enough info to deduce that”.
These are two points on the same continuum. Again, as Mises implies, a sufficiently intelligent being does not need to “deduce” things that are simple enough for it to see immediately, just as we do not need to “deduce” that 2+2=4.
EDIT: A more elucidating answer to the above is: You’re right, the computer may well say, “You already gave me enough info do deduce that,” but only because it would likely know that humans are not as smart as it is. It would be “talking down” to us, as a parent does to a child, so that we could understand.
Say you had two computers. One works out the logical consequences of a statement at the speed of light, or faster. The other plodded along, taking days and weeks to arrive at the most obvious little conclusions.
Does the fast computer not need to deduce? Of course it does. The programs the two computers are running are identical. One has swifter hardware, but it is still DOING THE EXACT SAME THING as the slow one, deducing.
Think of it this way:
If you see two dogs in one half of your visual field and two more dogs in the other half, you do not have to deduce that you see four dogs in your visual field. To your mind, seeing two dogs in one half of your visual field and two dogs in the other half means the same thing as seeing four dogs in your visual field.
I hope this example shows that there’s no reason to get caught up on the word “mean.”
However, what I believe you’re really caught up on is the idea that we think visually/sensually. I don’t expect you to agree with this, but if we are to agree to disagree I hope you understand that this is our point of disagreement, not some issue of logic.
Re: the computer example
This is actually a good illustration of what I mean, and suggests better wording: Perhaps a good way to think of the human language distinction between “X follows from Y” and “X means Y” is the following.
I use “X follows from Y” when seeing such implication requires conscious processing, and use “X means Y” when seeing such requires no conscious processing. (In your computer example the distinction becomes moot because computers don’t have anything that corresponds to a conscious/unconscious division (that I know of).)
OK. That clarifies your position. Until now I thought it pretty wild.
But I still disagree. Because the unconscious is a thick blanket that covers all kinds of stuff, things very different from each other. It seems to me to be the case that anything requiring deduction has to be deduced. It doesn’t matter that this deduction happens so swiftly it is beneath the threshold of our awareness.
Good point about “so swiftly it is beneath the threshold of awareness.” This gets into a theory of the threshold between conscious and unconsciout thought. I’m still working this part of my ideas out, but I think there is no hard cut-off; it’s just that some stuff happens so fast that you immediately forget it, hence have no conscious memory of having thought it - which is a good thing, because we only have so much short-term memory to work with.