[quote author="StephenLawrence"]Ok so absolute truth exists inside self- referential systems.
In the case of big foot, does an absolute truth of the matter exist but we can’t know what it is, or is there no absolute truth of the matter?
“Truth” is a property of linguistic items. “Absolute truth” is a confusing thing, since people mean different things by it. Some people mean to make metaphysical claims by the word “absolute”, and others mean to make epistemological claims.
A metaphysical claim: “Absolute truths” are truths that could not have been false. E.g., 2+2 = 4 is an “absolute truth” since 2+2 could not have equalled anything other than 4. “Bigfoot does not exist” is not an “absolute truth” since bigfoot might have existed, even if in fact he does not. In this sense, all logical and mathematical truths are absolute.
An epistemological claim: “Absolute truths” are truths that we can know are true with 100% probability. But then we get into difficulties, and it is not clear that there are any really “absolute” truths. Sure, maybe 1+1 = 2 is an “absolute truth”, but what about this:
44267445 / 965 = 45873
If this is true, it’s just as true as 1+1 = 2. But are you sure this is true with 100% accuracy? Do you believe your calculator CAN’T make an error? The root of this sort of thinking comes from Descartes’s notions about “clear and distinct ideas”. It isn’t obvious to me at all that I have a clear and distinct idea about this mathematical operation, or that I couldn’t make an error in calculating it. Hence many complex mathematical and logical operations are not “absolute truths” in an epistemological sense. We can be wrong about them!
So ... re. bigfoot, there is definitely a truth to the matter whether he exists or not. (Though this is not an “absolute” truth in the standard metaphysical sense I’ve given it). And we can never be absolutely-100%-certain what that truth is, although we can be very, very probably right. So it’s not an “absolute” truth epistemologically either.
[quote author="StephenLawrence"]So if we consider two statements.
Big foot does not exist.
The possibility of big foot existing is very very small.
These both mean the same thing, is that right?
Strictly speaking, no. Stated baldly, your first claim means that there is a ZERO chance that bigfoot exists. However, in conversation we make this sort of over-strong claim because it is understood that we are talking roughly. There is an unstated conversational implication here.
It’s like the suppressed “I believe” before everything we say. If in conversation I say “God does not exist”, someone who disagrees with me might say, “You mean you believe that god does not exist.” Well, yes, everything I say can be taken to be implicitly preceded by an “I believe”. But I’m not going to go about repeating those two words every time I say something.
The same is true with claims like “Bigfoot does not exist”, or “Complex life arose on earth by Darwinian evolution”. Strictly speaking these are claims that appear to be made with 100% certainty, and so have the form of epistemically absolute claims. But we know that in fact they are not, due to the fact that it’s understood we are speaking about very high degrees of probability.
I would say that an absolute truth does exist, in what Doug calls the epistemological sense, because Bigfoot either exists or doesn’t. I don’t know w/ 100% certainty which is true, but I’m willing to say he doesn’t becasue my degree of doubt is very small.
Doug,
Very clearly and succintly put. The point about implied conversational convention is what I was trying, much less successfully, to say.
A metaphysical claim: “Absolute truths” are truths that could not have been false. E.g., 2+2 = 4 is an “absolute truth” since 2+2 could not have equalled anything other than 4. “Bigfoot does not exist” is not an “absolute truth” since bigfoot might have existed, even if in fact he does not. In this sense, all logical and mathematical truths are absolute.
The first thing to say is thank you for your input.
It is fantastic to wonder about these things and to be able to get good feedback.
Questions keep occuring to me and I don’t know which to pick but I’ll go for this one for now.
You say 2+2 could not have equalled anything but 4.
But am I right in thinking 2+2 does not necessarily equal 4?
[quote author="StephenLawrence"]You say 2+2 could not have equalled anything but 4.
But am I right in thinking 2+2 does not necessarily equal 4?
No, 2+2 necessarily equals 4.
That means, 2+2 equals 4 in every possible world.
There is no possible world in which 2+2 does not equal 4. Why? Because 2+2 not equalling 4 would lead to a contradiction. And contradictions are logically—hence absolutely—impossible.
[quote author="George Benedik"][quote author="StephenLawrence"]I’d like to explore necessity.
I’ll start by saying I know I don’t understand it.
Any thoughts?
Stephen
Now when you pick a pawpaw
Or a prickly pear
And you prick a raw paw
Next time beware
Don’t pick the prickly pear by the paw
When you pick a pear
Try to use the claw
But you don’t need to use the claw
When you pick a pear of the big pawpaw
Have I given you a clue?
There is no possible world in which 2+2 does not equal 4. Why? Because 2+2 not equalling 4 would lead to a contradiction. And contradictions are logically—hence absolutely—impossible.
Ok but I don’t quite understand.
Couldn’t what is logical in this world be a contradiction in another world and vice versa?
[quote author="StephenLawrence"]Couldn’t what is logical in this world be a contradiction in another world and vice versa?
No. A contradiction implies anything. Hence any supposed world that contained a contradiction would contain every contradiction, or every claim and its opposite.
So, for one thing, we can see that if it were possible to have a world with a contradiction, there could only be one, because any proposed SECOND world with a contradiction would contain all the same true/false claims. It would be impossible to distinguish the two. So given that we distinguish worlds by the truths they contain, there is at most one possible world with a contradiction.
But that isn’t even possible, because it is not possible to coherently describe a world that contains contradictions. There is no description possible for it, since any description you give would be falsified by the other part of the contradiction ...
[quote author="dougsmith"][quote author="StephenLawrence"]Couldn’t what is logical in this world be a contradiction in another world and vice versa?
No. A contradiction implies anything. Hence any supposed world that contained a contradiction would contain every contradiction, or every claim and its opposite.
So, for one thing, we can see that if it were possible to have a world with a contradiction, there could only be one, because any proposed SECOND world with a contradiction would contain all the same true/false claims. It would be impossible to distinguish the two. So given that we distinguish worlds by the truths they contain, there is at most one possible world with a contradiction.
But that isn’t even possible, because it is not possible to coherently describe a world that contains contradictions. There is no description possible for it, since any description you give would be falsified by the other part of the contradiction ...
[quote author="StephenLawrence"]There are things that are necessarily true.
Things that are true but not necessarily true.
Right.
[quote author="StephenLawrence"]There are things that don’t happen necessarily but nothing that does happen necessarily?
Is that right, I feel uncertain about the last statement, if wrong is there an example of somthing that happens necessarily?
Well, there are different kinds or degrees of necessity. The strongest kind is “logical necessity”, that we’ve already discussed. I am not sure that there is anything particularly interesting that happens by logical necessity, although one might say (for instance) that if two people get married, it happens necessarily that they became husbands (if men) or wives (if women), etc.
However there is another form of necessity that is weaker than logical necessity. That is, physical necessity: acting in accord with the physical laws of the universe. This is weaker than logical necessity, in that there are non-contradictory worlds that have different physical laws than ours. And then one can say that every causal thing that occurs in our universe occurs due to physical necessity.
No. A contradiction implies anything. Hence any supposed world that contained a contradiction would contain every contradiction, or every claim and its opposite.
So, for one thing, we can see that if it were possible to have a world with a contradiction, there could only be one, because any proposed SECOND world with a contradiction would contain all the same true/false claims. It would be impossible to distinguish the two. So given that we distinguish worlds by the truths they contain, there is at most one possible world with a contradiction.
Ok, now you lost me. A contradiction is something whose properties violates the rules that govern them. E.G, in mathematics, ordinary integers can only have one value, so 2+2=4 and 2+2=6 is a contradiction because for both to be true the value of 2 (or 4 or 6) would have to fluctuate.
In systems not invented by humans, i.e. in the physical world, there are rules (natural laws, whatever we want to call them) that govern the properties of physical events. So, a contradiction would be something that violates those laws. I cannot physically be in two places at once, so my doing so would be a violation of natural laws, and saying “I am in France” and “I am in Mexico” would be a contradiction (barring tricks like “black and white and read all over). But it is possible to conceive of a universe in which this is not impossible according to the prevailing physical laws, so the statements would not be contradictory there. Now, what I think Stephan was saying was, couldn’t something be a contradiction in our universe and not a contradiction in another? This possibility doesn’t seem to lead to what you’re response indicates.
However, perhaps you took him to be asking (and perhaps he truly was asking) could a world contain a contradiction by the rules of that world. You seem to be saying this is impossible, because to contain a contradiction a world would have to have no stable rule structure at all, since one contradiction implies that even simple logical truth is not present in that world. I suppose this makes sense, but I’m drawn to the same argument I made when we talked about agnosticism vs atheism; can the fact that we cannot make logical sense of the possibility of a world that contains a contradiction yet does not fall into incoherence mean such is not possible, or simply that our minds are structured to view things in certain terms (including cause comes before effect, logical truths must actually be true, etc) and that we cannot extend our imagination effectively beyond these hardwired operating principles? In other words, even so self-evident a truth that 2+2 cannot = 4 and 6 may be a failure of conception or imagination on our part, stemming from the materials and natural history of our cognitive apparatus, rather than an underlying principle of reality. Purely thought experiment territory, since I’m an empiricist at heart and such a claim doesn’t lead to much useful in the real world. But I think humans have a unfortunate tendancy to confuse our understanding or concept of reality with reality, and I’m wary of asserting, at least in abstract philosophical discourses like these, that what we cannot conceive of cannot be. What do you think?
[quote author="mckenzievmd"]In systems not invented by humans, i.e. in the physical world, there are rules (natural laws, whatever we want to call them) that govern the properties of physical events. So, a contradiction would be something that violates those laws. I cannot physically be in two places at once, so my doing so would be a violation of natural laws, and saying “I am in France” and “I am in Mexico” would be a contradiction (barring tricks like “black and white and read all over). But it is possible to conceive of a universe in which this is not impossible according to the prevailing physical laws, so the statements would not be contradictory there. Now, what I think Stephan was saying was, couldn’t something be a contradiction in our universe and not a contradiction in another? This possibility doesn’t seem to lead to what you’re response indicates.
A contradiction is quite simple, logically. It is:
A & not-A
We don’t need to talk about violating physical laws to talk about something that, as you say, would be “a contradiction in our universe and not a contradiction in another”, if I understand your point. For, if bigfoot existed in our universe (and keeping all else the same, including the fact that bigfoot does not exist), we would have:
bigfoot does not exist & bigfoot exists
and that is a contradiction.
But in another universe, bigfoot might exist by himself without contradiction.
To be clear, “I am in France” and “I am in Mexico” is not a contradiction, even in our universe. First of all, I have been in both places, so we will at least need a time indicator. But even so, is it really logically necessary that I can’t be in France and Mexico at the same time? I don’t think so. We can think of someone large enough (or spaghettified enough) that he could be in both simultaneously. So that’s not a good example.
You may think I’m splitting hairs here, but the point is that contradiction is a fact of logic, and so the precise way we formulate the hypothesis can make the difference between an actual contradiction and something that seems contradictory on first glance but is in fact not.
[quote author="mckenzievmd"]However, perhaps you took him to be asking (and perhaps he truly was asking) could a world contain a contradiction by the rules of that world. You seem to be saying this is impossible, because to contain a contradiction a world would have to have no stable rule structure at all, since one contradiction implies that even simple logical truth is not present in that world. I suppose this makes sense, but I’m drawn to the same argument I made when we talked about agnosticism vs atheism; can the fact that we cannot make logical sense of the possibility of a world that contains a contradiction yet does not fall into incoherence mean such is not possible, or simply that our minds are structured to view things in certain terms (including cause comes before effect, logical truths must actually be true, etc) and that we cannot extend our imagination effectively beyond these hardwired operating principles? In other words, even so self-evident a truth that 2+2 cannot = 4 and 6 may be a failure of conception or imagination on our part, stemming from the materials and natural history of our cognitive apparatus, rather than an underlying principle of reality.
No, it’s a matter of logic. ” One may prove any proposition from a set of axioms which contain a contradiction .” It’s a simple proof, and I have it written down somewhere in my notes. This has nothing to do with epistemology or opinion. The point is that a world in which a single contradiction is allowed is a world in which every proposition is true.
It’s simple obscurantism to allow the possibility of contradiction. This is a standard ploy of woollier-headed post-modernists, and I counsel us to avoid it like the proverbial plague.
I hope I am making myself clear. If not, I will be happy to hammer at it again.
The first issue is that I may be missing your point because I am trying to use “contradiction” in a more general way, rather than according to a formal definition from logic.
2 a : a proposition, statement, or phrase that asserts or implies both the truth and falsity of something b : a statement or phrase whose parts contradict each other <a >
3 a : logical incongruity b : a situation in which inherent factors, actions, or propositions are inconsistent or contrary to one another
Definition 2 seems to be what you’re using, and 3b is closer to what I’m using. I raised the issue of physical laws because I’m interested in exploring the meaning of a contradiction outside simple artifical systems like logic or mathematics. In this sense, I thought my example might work. FWIW, the present tense implies the time frame (simultaneously), and I tried to exclude any tricks such as ambiguity of words, two countries with adjacent borders, physically impossible human beings, and that sort of thing. But it was off-the-cuff so not very precise. What I was aiming for was something that was logfically impossible according to some set of rules larger and more realistic than those of mathematics.
I would be interested in the proof you mentioned that allowing a single contradiction allows all propositions to be true, since I can’t seem to follow the reasoning there.
Otherwise, saying “it’s a matter of logic” evades my argument, because “logic” is a system of rules for reasoning that humans invented, and so it is subject to whatever limitations we are (http://en.wikipedia.org/wiki/Logic). Of course, I don’t really think we have much choice except to assume that our understanding of reality represents reality in a meaningful way. But I don’t think it is “obscurantism” or “wooley-headed postmodernism” to argue that we may possibly be able to understand it only within limits set by our physiology and natural history, and that we cannot necessarily know what those limits are. In fact, it seems difficult to argue cogently that there are no limits to our capacity for understanding. Dogs don’t get basic arithmetic, and there’s probably stuff we don’t get either. Now, following this to the extreme of saying no truth is sufficiently reliable to base values or actions on and we should give up science and the effort to understand would be falling prey to the errors you identify, but the it’s a bit of a straw man technique to assume that admitting of the possibility that our understanding and imagination has limits and flaws (up to and including the possibility that what we consider mathematic or logical truths may, in some sense we don’t understand, not be) is the same as following this possibility to these extremes.
Anyway, I understand that certain arguments may be undefeatable and yet practially useless. You can’t prove to me anything outside my own head is real, but it would be pointless to live my life in a way dictated by the proposition that everything I see is an illusion. Similarly, I’m not sure you can prove that human-derived constructs of reasoning (e.g. formal logic, mathematics) are aboslute truths in the sense that they must be true in any universe that is coherent, but I don’t plan on living my life by the proposition that all truth is relative. Still, I can’t help but wonder how we can be so sure of even simple truths given the inherent cognitive limitations we must have as evolved organic life forms, just as all the other life forms we know about have such limitations. We’re the bright kids on our block, but assuming there are no cognitive “humans” to our cognitive “annelids” out there some where is a but presumptious.
Re. the proof that a contradiction implies anything ... I wish I could reconstruct it from memory (it’s quite simple), but right now I can’t. I’ll try to remember to post it here if I can find it.
As for the issue of the status of logic and mathematics, you are right to point out that there are two schools of thought on the matter. One school of thought believes that logic and math (I’ll just say “logic” from now on, since math reduces to logic plus set theory) are human inventions, and that they wouldn’t exist but for us, and that they have no other reality. This could be called the “subjectivist” school.
The other school, to which I belong, says that logic is a discovery of ours, not an invention. It is the discovery of the deepest laws of reality, that uncovers the basic machinery of everything. Thus, it is impossible to reason without logic. It is impossible to have a world that contains a contradiction. All physical laws have their form in mathematics. This could be called the “objectivist” school. (FWIW, most professional mathematicians are “objectivists").
The problem with the subjectivist school is that the claim appears to be that logic depends upon us for its existence. That would seem to imply that if we hadn’t existed, contradictions might have been possible, and that’s even worse than claiming that the universe depends on us for its existence. Similarly, many mathematical proofs are about infinities, or depend upon a notion of infinity. Do we really feel that we can do that sort of operation “by ourselves”? Or rather, if logic is subjective, are any proofs involving infinities simply invalid?
Can we really be sure that addition works “all the way up”, even to numbers bigger than we can conceive, if mathematics is subjective?
Where the obscurantism and wooly-headedness comes in is where post-modernists use these sorts of arguments to denigrate reason and logic. When one points out to them that they are being unclear, illogical or unreasonable in arguments, they then say that there is nothing wrong with unclarity, unreasonableness or illogic since these are human constructions and might be wrong. I don’t know how one can be more clearly obscurantist than that ... (And BTW, I have heard postmodernists making these sorts of claims).
I certainly don’t mean to say you are doing any such thing. There are many good, non-post-modernist subjectivists about logic. But I don’t find their arguments forceful, and at any rate, one has to be a very sophisticated subjectivist indeed to state the position clearly and get round the apparent difficulties of the position.
