Is it the case that proposition "x" must either be "p" or "-P"?
Do you believe this for only SOME propositions? If so, how can this assertion be a basic law of logic?
Do you believe this for ALL propositions? If so, wouldn't just one example of a proposition that is both true and false prove the case is only true for SOME propositions? Thus, if the example is shown, how can the assertion be a basic law of logic?
I'm hoping for your opinion and some serious, light-hearted discussion on the matter.
Well. To begin. This wasn't exactly the serious, light-hearted kind of response I was looking for. It borders on the Appeal to Authority Fallacy. But... whatever.
Actually, I have taken many logic and math courses up to and including Symbolic Logic, Analytical Geometry and Calc III. I passed them all with ringing bells and whistles. What I've found is that they are all beautifully complex and quite utilitarian.
However, since I consider myself to be a Pragmatic Utilitarian Soli Fallibalist, I have found that progressive thought often stalls on utility alone. All of my studies seem to send up red flags begging me to return to the basic principles from which such complex ideas form.
So, if you'd like to continue the discussion, I would be willing. But if you're just another one of those absolutist that pop up out here, spouting off simply to attempt a power postition, then I'd suggest we quit.
I do very much appreciate the definitions of words. They help us all better focus on communicating our symbologies. But the definition of a proposition (by your link) is fundamentally simply a declarative statement. Yes, it also includes the properties of being either true or false.
I will be direct. I am seriously questioning the added properties "of being either true or false" being applied to the definition. In fact, I'm at the point where the definition is wrong because of these added properties.
Yes, Classic (Formal) logic codifies true or false upon propositions and that is why I have posted out here for your awesome comments. I like that you call Informal Logic "fuzzy". That is hilarious... :)
I mentioned that if one proposition could be shown that is both true and false, then the concept of Non-Contradiction could not be a basic tenent of logic because it would only be true for some propositions. I am sure I have at least one proposition that fits this criteria.
Can you think of just one proposition that is both true and false?
We have to distinguish between formal logic, which is mainly a special branch of mathematics, and rhetorical logic, before we can meaningfully go further with this. If we are talking about formal logic in purely mathematical symbols, then the statement that x is either p or -p, i.e. either p or not p, is fundamental, because no proof could ever get started without it. Notice that this is not the same as either "true" or "false." The correct thinking is either "p" or "not p." It either is itself or is not itself. There may be some Kurt Gödel shenanigans that could give us a contradictory statement in formal logic for this proposition. I am not logician enough or versed enough in principia logic to know for sure.
In rhetoric, the idea that x can be both p and -p would be called a paradox. Inductive logic would lead me to say that probably all such paradoxes rest upon faults in language, overt or covert equivocations that allow one to say that the same statement is contradictorily true. Paradoxes can reveal interesting problems in language, but are not often really problems in logic, since the paradox can usually be resolved by changing the terms.
This is actually a very interesting question. Yes, if you take a BASIC logic course you will learn that every proposition is true or false. This is the classic logic. If you would take an ADVANCED logic course, you might learn that there are other logical systems. In "intuitionistic logic" statements are no longer true or false. This kind of logic was advocated by the dutch mathematician LEJ Brouwer.
To Brouwer, the statement "P or not P" is only valid if you have a proof of P, or you have a proof of "not P". Consequently, he also had to reject proofs by contradiction, because otherwise
you could deduce "P or not P". Without proofs of contradiction, you can proof much less, but the things
you can prove are, in a way, more meaningful. Intuitionistic Logic never gained popularity and is still fairly obscure.
Using a proof by contradiction, this is how you deduce "P or not P":
Suppose that "P or not P" is false
Suppose that P is true,
Then "P or not P" is true.
But we assumed it was false, so we have a contradiction.
Therefore, "not P" is true.
But then "P or not P" is true.
This again is a contradiction, because we assumed that "P or not P" is false.
So our assumption that "P or not P" is false, is wrong.
Therefore, "P or not P" is true (this is were we use the proof by contradiction).
BTW, in logic, whether intuitionistic or not, statements cannot both be true and false.
Or rather, if a statement is true and false at the same time than you have a contradiction
and ANYTHING follows.
Wow! I like your attention to detail and very much appreciate the link on Intuitionistic Logic. The preservation of justificaton is exactly the utilitarian concepts for which I want to observe.
However, Intuitionistic Logic seems to continue to assume the law of excluded middle. (Even though it claims it is not by stating with different terminology) "...a statement is 'only true' if there is a constructive proof that it is true, and 'only false' if there is a constructive proof that it is false." Intuitionistic Logic rejects statements which cannot be proven true, and rejects statements which cannot be proven false. But it does not include statements which can be proven to be both true AND false. This is why I posted and we're having this lovely discussion.
As you said, there are a lot of other logical systems out there. What I'm finding is that none of them actually reject the law of excluded middle. I'm certain that I can produce at least one proposition which is proven to be both true and false. Can you think of one?
BTW, thanks for posting the deductive proof of "P or not P". It's a wonderful example of why I posted. But, isn't it simply a tautology, assuming the law of excluded middle, BECAUSE it does not take into account "P and not P"?
If you prove that a proposition is true and false at the same time, then you have
a contradiction and anything follows. In particular, every proposition is true and false
at the same time. If you prove that a proposition is true and false using certain
assumptions, then one of those assumptions must be false. Which proposition is true AND false?
Within a certain framework of logic with certain fixed rules of deduction, "P or not P"
is not assumed to be true, but can be easily proven. That was the proof I gave.
So it is not entirely tautological.
I think it might be fair to say that anything will follow if a proposition is proven to be both true and false. But follow as what, is my question. With the lack of properties available to properly conclude a proposition, it's often concluded as (-P) which I think can be an improper perspective. My thinking goes something like this:
I proposition can be (P), (-P), (P and -P) or (-P and --P)
I threw in the last one because I am convinced I can argue its validity without reducing it to (P and -P) It's where propositions about gods would fall due to the lack of any recognizable properties for determination of T or F. I say that (--P) does not reduce to (P) because not not something does not guarantee its opposite. [It would only be so if we conform to the traditional excluded middle concept]
I guess a better way for me to explain where I'm coming from is: Contradictions exist. And as an existing contradiction they should be recognized as non-contradiction. Yes, this is a contradiction. But recognizing that contradictions do exist means we can properly categorize them rather than putting them by default in the (-P) conclusion. [Thus continuing this age-old argument over whether or not one can prove a negative]
I don't think I really have a problem with being able to prove everything. In fact, it might actually help. Any proposition would be concluded as either (T), (F), (T and F), or (-T and -F).
I do appreciate your awesome response to this. It's been bugging me for years. LOL
REVISION - Shucks... I knew I should not have included the (-P and --P) portion yet. I should have stuck to my guns and waited until I could thoroughly cover the topic. My deepest apologies that this post causes confusion. I see several mistakes.
I wrote "I proposition can be (P), (-P), (P and -P) or (-P and --P)." I should have written "Any proposition's conclusion should include the possibility of (P), (-P), (P and -P), or (-P and --P)
I wrote "...because not not something does not guarantee its opposite." I should have written "...because not not something does not guarantee the something."
I now realize this also creates more confusion if I continue without discussing how the dash (-) and the "not" are used. Equating the (P) to (T), the (-P) to (F) and so on directly without clarifying the association better was an oversight on my part. Unfortunately, I don't have time to continue right now. I will proceed with it tomorrow.
I have received some great feedback on this topic from another link. In order to not have to update both, I invited them to come here to continue. So, I'll update where we're at.
I claim there is at least one proposition which is both true and false at the same time. Here is the one I presented: "The cup on my desk is at rest."
Some are trying to convince me that the conclusion of this proposition is not inherently a contradiction. I will paraphrase since it was reiterated in several ways. They say the cup is at rest relative to the wall (or whatever "at rest" object around it); and separately, the cup is in motion relative to the sun. They say this alleviates my contention because the proposition implies a relative distinction. Furthermore, they claim: in order for a proposition to be truly contradictory, the entities involved must contain specific properties that display provable opposable characteristics.
Again, I am paraphrasing. So, I welcome any response where I may have erred. Also, I wanted to make sure, if the others do come to this link, they know I was not avoiding their position.
I contend that it is the same cup and it is displaying properties in contradiction of both "at rest" and "in motion" at the same time. All other properties contained within that proposition remain the same. My argument is that the very mental effort of ONLY attributing (P or -P) to propositions creates a tautology such that we are forced to have to ONLY conclude either (P) or (-P).
This is similar to when someone argues: The Bible says God exists. Therefore God exists because the Bible says so. In our example: A declarative, provable statement must only conclude as either (P) or (-P). Therefore, the conclusion must be either (P) or (-P) for declarative, provable statements.
Is anyone prepared to make the argument that the presented proposition is not a declarative, provable statement? [I hope not because I've met some extreme skeptics who'll argue over everything and that can be exhausting.]
So, what appears to be happening when someone objects to a valid contradicting proposition is: They must force it into either a (P) or (-P) position. To do that they demolish the proposition into its perceived contradictory parts and rationalize those parts as separate non-contradictory concepts so as to conclude either (P) or (-P). In fact, they rewrite the proposition in order to fit it into the 'excluded middle' box. The more correct conclusion is to make allowances for (P and -P).
Marc, you're throwing the baby out with the bathwater here. You are missing the possibility that the declarative statement in question is not sufficiently complete to allow a true or false determination at all. Surely any system of logic requires that we are completely clear on definitions. Otherwise we sink into a semantic morass.
The only way that "The cup on my desk is at rest" is both true and false at the same time is if the interpretation where the statement is true refers to a different scenario than the interpretation where the statement is false. You're just squeezing one end of a balloon so your problem bulges out on the other end. The more correct conclusion is to go back at your declarative statement to figure out what's ambiguous about it and resolve that before proceeding to a true/false determination.
More to the point, I fail to see what you hope to accomplish by weakening the distinction between true and false. Don't we already have enough trouble with that? My passing understanding of even Fuzzy Logic is that it attempts to assign a truth value somewhere between true and false, but still never true and false simultaneously. A fuzzy logic truth value of 0.5 would imply something like "partially true and partially false", rather than "simultaneously true and false". Partially true and partially false can certainly apply to very complex or multifaceted truth claims, as in a court case when trying to assign liability amongst several parties, but I strongly suspect that any such claims can be broken down into their component parts and analyzed using strict boolean logic (or perhaps Bayesian probability), time and effort permitting.
In any case, your particular example is clearly a case of imprecise definitions; properly defined, there is no contradiction inherent in the statement. Different frames of reference produce different truth values, but not simultaneously. Don't confuse the fact that both truth claims use the same English words in the same order with the idea that both meanings are the same. They are not. This is like saying "Chris is a man" is both true and false when there are two different people named Chris in the room. You don't have a simultaneously true and false declarative statement. You have (at least) two separate declarative statements masquerading as one. Suggesting that this calls for a radical new approach to logic (that would seem to throw out a key underpinning of logic itself) is missing the point at best, and a dangerous idea to an easily confused species at worst. The proper response is not to imprison George Boole in Schrodinger's catbox, but rather to ask, "What do you mean? An African or European swallow?"