Wednesday, October 21, 2009

Can beliefs be inconsistent?

Last night I listened to the podcast of last week's show with Matt and Don. I am looking forward to being back in the "other" studio again, but not next weekend as scheduled, since I have plans to fly to Pennsylvania.

One of the responses to the callers caught my attention. Matt and Don, beginning at around 42:30 in the audio, were speaking to Gregory in Eugene, Oregon. Gregory first wanted to propose his own uninteresting (IMHO) redefinition of God. Then, later, he claimed that he "considered himself just as much an atheist as I do a theist."

Matt asserted that this was ridiculous -- which it is. To be a theist means that you believe in a god, while to be an atheist means that you do not believe in a god. Obviously these positions are mutually contradictory, and so it makes no sense to hold both of them.

But then Matt went one step further, claiming that Eugene could not hold both of these positions simultaneously, effectively accusing him of either lying or being deluded about his own beliefs. It is this point that I wish to respond to, because -- as an enthusiast of formal logic -- I think one can't make such a blanket statement about other people's beliefs.

Of course, it's perfectly reasonable to assert that people should not hold contradictory beliefs, and to point it out vigorously when they try to slip that sort of thing past. However, it does not follow that one cannot hold contradictory beliefs, and in fact, I think that they do all the time.

Seen in abstract terms, you could say that a person's state of mind is a set of propositions that they assert to be true. "The sky is blue" and "the sky is not green" are two such propositions; "There is a God" is another. Not all propositions need to be definite; they could be probabilistic, as in "There is probably no God" or "There may be intelligent life elsewhere in the universe."

Now, given a particular proposition P, P can be either true or not true; and according to propositional logic it must be one or the other but not both. However, just because a proposition is false in reality does not mean that it is not a part of someone's belief system. Indeed, we know that some people have false beliefs -- just consider that many people are theists and many others are atheists. Either there is a God or there isn't, and therefore one of these groups clearly holds a false proposition to be true, and most likely a host of related propositions as well.

Still, believing a false proposition P does not make your belief system inconsistent; you can easily believe something that is false but does not contradict any other proposition in your universe of beliefs. However, my point is that there is no reason in principle why somebody cannot comfortably believe the assertion P1: "X is true" and P2: "X is false", at the same time.

As a computer geek, I happen to believe (though not entirely backed by affirmative evidence) that the human mind can be represented as a formal system, not entirely unlike a computer could behave in principle if it was outfitted with the right software. I am a believer in the possibility of artificial intelligence, though I definitely do not believe it has been achieved yet, and it may not be achieved within my lifetime, or even the entire span of the human race.

There is a famous theorem formulated by Austrian mathematician Kurt Gödel, which states that all formal systems are either incomplete or inconsistent. In other words, either there is some proposition P for which the system holds P to be both true and not true (inconsistent); or else there is some proposition P which really is true, but the system cannot prove it (incomplete, as P is true but missing).

Some opponents of AI see this as a fundamental limitation of computers, a proof that humans are somehow better than computers because they have intuition which is capable of somehow "jumping outside the system" and directly perceiving truths that cannot be proven formally. I see that as a fallacy. Sure, humans are capable of making logical leaps of intuition, but that doesn't mean the leaps lead exclusively to true beliefs. We know for a fact that brains are often misled into believing things which are not true, and may even be contradictory, which we sometimes call "hypocrisy."

Imagine a person who is completely insane, in the sense that he believes everything that is false and nothing that is true. Such a person must necessarily be inconsistent as well. Why? Well, consider the following untrue statements. P1: 2+2=3. P2: 2+2=5. These statements are contradictory -- they cannot both be true at once. Yet the insane person must believe both, because they are both false.

But you don't have to go so far as complete insanity in order to hold contradictory beliefs. In fact, I would speculate that everyone in the world probably has some beliefs that contradict one another. I do. Matt does. I'm not saying that this is desirable, or that you can't minimize the number of contradictions you believe, but the mind is full of shortcuts and logical leaps and rules of thumb that let us analyze reality without becoming immediately paralyzed by an in-depth comparison of new information against every other single proposition you already believe.

In fact, I once read a beautiful proof by logician Raymond Smullyan that every person must necessarily be either inconsistent or conceited. It goes like this:

The human brain is finite, therefore there are only finitely many propositions which you believe. Let us label these propositions p1, p2, ..., pn, where n is the number of propositions you believe. So you believe each of the propositions p1, p2, ..., pn. Yet, unless you are conceited, you know that you sometimes make mistakes, hence not everything you believe is true. Therefore, if you are not conceited, you know that at least one of the propositions, p1, p2, ..., pn is false. Yet you believe each of the propositions p1, p2, ..., pn. So you believe at least one of these statements to be both true and false; hence you must be inconsistent.

Believing a contradiction does not make you crazy or a liar. Continuing to believe both "there is a god" and "there is no god," even after the contradiction is explicitly pointed out to you, might make you a bit thick. But there's no impossibility there. We know thick people exist, and most of us encounter them on a daily basis.


  1. That is a beautiful little proof, I might just have to borrow it!

    On the topic of holding contrary believes. Could it be that we add limits to our beliefs? One could think of it as a way to keeping the contrary beliefs apart, thereby maintaining a ‘sort of’ consistent world view. One could picture it in this way:

    X is true in a given interval [X1,X2] and X is not true in a given interval [X3,X4]

    While the two statements might be mutually contradictory (They don’t actually ‘have’ to be contradictory, but let’s say so for the sake of the argument) to, they will never come in conflicts, unless the intervals are changed so that they overlap.

    As for the caller being thick? He didn’t sound thick really, he just sounded as if he didn’t want to label himself, or provoke any party. But that tactic tends not to work very well on Matt ;)

  2. The statement might make more sense if you look at the semantic containers being used, the terms, imo aren't very precise.

    Saying one is a theist, especially when trying to redefine god, can mean deist or syncretist, or what I would call a small t theist.

    So stating you are as much an atheist as a theist can make sense if you allow that the person speaking does not believe in a monotheistic-type god but does believe in some type of god that is ill-defined or yet to be defined.

    Sure there is an element of ignorance here, after all, theist means something specific. However, I think a lot of the confusion lay with the christian apologists who have, in an attempt to appear rational, erased the edges of the capital G god and replaced them with abstract logical principles no one actually believes, but can defend in a syllogism.

    And you are correct in saying people can believe in contradictory things, you provided a good proof, and I think Godel alludes to this as well.

  3. Moreover, there's a related issue: Belief in a contradiction from a purely matter of formal logic implies that every statement is both true and false.

    Quick sketch: We agree that "P or Q" and "not P" allows us to conclude "Q". Also we agree that "P" allows us to conclude "P or Q" for any Q.

    Ok, so now assume P and assume not P. Let Q be any fixed statement. Then by the second of these two rules we have "P or Q" and then by the first rule we conclude "Q". So having any contradiction in first order logic leads to everything being true and false.

    Here's the key though: Humans don't use formal logic in their reasoning. At least, we don't naturally. If someone has a contradiction in their belief they don't throw their hands up in decide that the moon must be made out of green cheese. Instead, you generally try to find the error in your reasoning. If you can't, you eventually shrug your shoulders and move on.

  4. i felt a little bad for the caller because i could understand his position having been there myself.

    i think, given time, he will recognize the absolute proposition matt was talking about, but until i started listening to podcasts like AE and NP, i didn't really have anyone to discuss this stuff with, so i was in the more abstract camp that the caller was coming from. that is, it seemed reasonable to say "i don't necessarily believe in a god, but i also don't discount the possibility, therefor i don't not believe in a god."

    i understand the error in that statement pertaining to the logical question of atheist vs theist, but many people are used to thinking more abstractly and emotionally overreact to being forced into committing to one side or the other because they put some much weight onto it. as if to say because you don't currently believe in something, you are immediately closing yourself off to changing your mind.

    for me to understand, it was the detail that "i don't believe P is true" doesn't mean "P cannot be true." it just means "i haven't got sufficient reason to believe P is true."

  5. Bertrand Russell famously stated in a lecture that if you accept a false proposition then you can derive any proposition. He was challenged: "2+2=5. Prove that you are the pope." He allegedly came up with this proof on the spot:

    1. 2+2=5.
    2. Simplifying, 4=5.
    3. Reducing both sides by 3: 1=2.
    4. The pope and I are two.
    5. But 2=1, therefore the pope and I are one.
    6. I am the pope.
    7. QED.

  6. Gregory was becoming a bit annoying when Matt finally jumped on his illogic. I got the idea that Gregory was new to some of the ideas he was trying to understand and was having trouble expressing himself. He may have been trying to express the idea that he is an atheist when thinking about the religious idea of god but might be a theist when it comes to some redefined version he just came up with.

    Most of us have not taken the time to put our ideas under the scrutiny of formal logic. We can than seem dense when we make our first attempts to sort out ideas that were just packed away. When one does this in front of people who deal with these ideas all the time it is hard for them to continue to have patience, particularly when one does not seem to be listening.

    I personally decided that the supernatural had no place in my worldview a long time ago. God, ghosts and the tooth fairy have long ago been relegated to my mental outbox - no further processing required. Certainly we have many ideas both true and false that have been likewise filed away. Many are likely to be contradictory but if they are not brought out to consider at the same time we don't notice this.

    Logic is nice for sorting out our thoughts, but it does not dictate reality. If we were to try to logically defend one of the ideas that we have not thought about for some time, it would be fairly easy for someone like Ray Comfort to put us in a logical bind with his clickomatic "are you a good person". Winning the argument logically does not create god or any other reality. For that you also need evidence.

  7. I'm not really convinced that the proof is actually equivalent to the theist/atheist situation. I think the proof works well to demonstrate that we can believe both some number of propositions, and the proposition that one of them is probably or even certainly false. Specifically, that you can simultaneously believe a proposition that negates one unspecified proposition of all the other propositions you believe.

    That's not the situation with the claim to believe simultaneously that 'there is a god' and 'there is no god'. These are directly contradictory propositions and you can't believe both at once.

    The equivalent in the world of your proof is actually that you believe propositions, and also that "p1 is false". I think this is clearly impossible. Either you believe that p1 is true, or you believe that it is false, not both. In the proof, you don't know which proposition you believe is false, only that you believe one of them will turn out to be false. In the theist/atheist example, you know exactly which proposition your negating belief negates.

    As such, we can only really assume that the guy was making a statement in the heat of the moment as a sort of rhetorical move, or honestly doesn't know what the terms mean.

    Correct me if I'm wrong on this, but I don't really think the proof works, at least logically.

  8. The "proof" was mostly a demonstration of what should be the trivially obvious fact that people can and do hold mutually contradictory beliefs. Do you agree or disagree with this statement?

    If you agree, can you tell me how you justify this?

    "The equivalent in the world of your proof is actually that you believe propositions, and also that "p1 is false". I think this is clearly impossible."

    Why do you think it is impossible? Do you seriously think that nobody you know is inconsistent in their beliefs? What specific mechanism is in the brain that would prevent this, in your opinion?

  9. I personally think Matt is right on this issue. While people hold contradictory beliefs all the time, they don't look at a dog and believe it to be both a dog and not a dog at the same time. Two abstract ideas can but up against each other in ways that cause contradictions in our beliefs, but two directly contradictory straight-forward propositions can't both be believed at once. The fact is that if someone says they are both an atheist and a theist they are not using those words right.

  10. I believe Pippin is "spot on" here. Pippin, I'll give you the opportunity to answer Russell before I decide to jump in.

  11. Someone can actually accept two contradictory positions, but I'm convinced that this can only occur when they are unaware of the contradiction. The dilemma must be obfuscated in some way (including insanity).

    To flatly claim that you are both X and notX - as this caller did - is most likely resolved by:

    1. The caller is lying to us and/or themselves.
    2. The caller is crazy.

    To acknowledge the contradiction and still assert the claim is absurd. How many times have we pointed this out when someone says "I neither believe nor disbelieve..." BZZZT! Wrong. Once you've said "I neither believe" you have affirmed your disbelief.

    It is, though, possible to CLAIM to be a theist and not a theist - as this caller did.

    I remain convinced that this caller is not simultaneously a theist and not a theist...despite his claims.

  12. i think the point i and some others on here were trying to make is that the limitation to "lying" or "crazy" is oversimplified. it assumes the person fully understands the proposition, which i don't think the caller did.

    i certainly don't defend that one can truly be both theist and atheist at the same time in term of logical absolutes, but i think the subject is sufficiently heavy in baggage and intricacies to lead someone to a third position along the lines of "confused."

    once i am confident someone really understands th question, i am comfortable with labeling them as either a lier or crazy, but until then, i was willing to cut the person some slack.

    all that said, i was glad when the call ended because i don't think his forward progress was at an end.

  13. Someone can actually accept two contradictory positions, but I'm convinced that this can only occur when they are unaware of the contradiction. The dilemma must be obfuscated in some way (including insanity).

    I think our callers disprove that assertion every week. Consider this hypothetical dialogue:

    Caller: "I believe that every word of the Bible is true, and a perfect moral guide."
    Host: "What about the part where God orders them to slaughter everyone except the virgins?"
    Caller: "The Bible does not say that."
    Host: "[Chapter and verse]"
    Caller: "...I'll have to get back to you on that."

    Very, very rarely this kind of dialogue can lead to a conversion. But that's an unusual case, so I'll assume the caller doesn't change his mind.

    At this point the caller simultaneously holds these propositions in his head:

    1. Everything the Bible says is moral.
    2. Genocide is immoral.
    3. The Bible is endorsing genocide.

    Probably in short order, he will somehow manage to crowd one of those propositions out. Perhaps they are just not bright enough to process the implications of the contradiction. But I think there are definitely periods of time when someone holds beliefs in his head that are in obvious contradiction to one another, and they are neither insane nor lying.

    Unless, of course, you believe that everyone is entirely rational.

  14. I'm with Matt on this one. Of course we accept the premise that people hold contradictory beliefs, but not about the exact same thing.

    For example, you believe that the boiling point of water is 200 degrees. You believe that a kettle heats to 100 degrees. You believe that you can boil water in a kettle.

    The person is not associating the beliefs.

    It's different to say you believe that the boiling point of water is both 100 degrees AND 200 degrees.

    The first has to do with logical deduction, the second has to do with your actual belief on a single topic.

  15. I think if we add a third option to 'crazy' and 'lying', we're probably all saying the same thing: Ignorant. Especially when talking about something like the term 'atheist', which people use in many different ways (including, "I'm an atheist with respect to Thor, but not FSM"). I can see how someone could say they're both at the same time by not understanding the terms, or even by failing to communicate their position clearly. That said, my gut feeling when listening to the podcast was that the caller wasn't being honest (even with himself).

  16. It's really hard to add to much to what Matt and Pippen say, I think they nail it.

    My two cents:

    I feel it depends on definitions and how specific you are defining what you are having the belief about.

    For example, the following "seemingly" contradictory statements could be true:

    "David hates all ethnic groups other than his own"

    "David loves all ethnic groups other than his own"

    The problem is the definitions of "love and hate" and the context based on the statements. David could love the way that they look different than him, but hate the way they all talk differently.

    I think someone looking at the above could say, "yeah, i see how the two statements can be conflicting and true at the same time".

    But, then you take these two statements:

    "David ALWAYS gets violently ill whenever he sees a white albino male human with green eyes, wearing red pants, red shoes, and a red shirt. No other feature matters for this to happen."

    and then,

    "David NEVER gets violently ill whenever he sees a white albino male human with green eyes, wearing red pants, red shoes, and a red shirt. No other feature matters for this to happen."

    Those are definitely contradicting statements, and no honest, rational person could say that someone could believe that both could be true.

    Another example could be:

    David says, "I'm an atheist"
    David says, "I'm a theist"

    Expounding on that you could call him out...but then he could say something like, "well, I'm a theist in regards to Thor, but an atheist in regards to Yahweh."

    On this:

    Caller: "I believe that every word of the Bible is true, and a perfect moral guide."
    Host: "What about the part where God orders them to slaughter everyone except the virgins?"
    Caller: "The Bible does not say that."
    Host: "[Chapter and verse]"
    Caller: "...I'll have to get back to you on that."

    Russell, I believe you're comparing apples to oranges here.

    It's not like the guy says:

    Caller: "I believe that every word of the Bible is true, and a perfect moral guide."

    AND THEN in the next breath says:

    Caller: "I also do NOT believe that every word of the Bible is true, and a perfect moral guide."

    I truly believe that as Matt says, if the statements are precisely and clearly defined, two clearly contradictory statements can not be believed at the same time.

    Now, I will give you - as you wrote your title, "Can beliefs be inconsistent", that is true, up to the point where the beliefs and statements are SPECIFICALLY defined, but at the point they can not.

    "Can specific beliefs be inconsistent"? I'd have to say no unless you're mentally ill, insane/whatever.

    But, maybe that's the point you're trying to make :)

  17. People seem to zero in on the ”liar or insain” options, but I’d say there are at least two other options.

    The caller might use other definitions of the words atheist and theist. The caller might even be unaware of the proper definitions of the words and as such confused.

    The 2nd option I don’t think apply in this case, but is worth noting. What do we call a person who has never thought about the god issue? That person wouldn’t have taken a stance on the issue, and as such neither affirmed nor rejected the idea of a god or gods. I would consider that person neither an atheist or a theist.

    So to sum up, he could have been a lying, insane, confused or unaware. Or possibly Russel is right, and we can hold mutually excluding ideas in our minds without a mental meltdown 

  18. While I don't claim this to be the case, my thought process centered around an assessment of a perfectly 50% chance that a claim is true. Under this situation, one cannot use evidence to claim that something is true or false, and it seems reasonable to assume that it could be either. You must accept that it is true just as much as you reject it if you are perfectly logical.

    Now, for theistic claims, I don't think anyone has put forth one that is remotely close to 50%, and I don't think the caller was struggling with a "it's just as likely" thought. In the end I think I agree with Matt's choice of words on the show, and with his follow up here in the comments, I have to agree even further.

  19. @ Dines
    A person who has never thought about the god issue and holds no explicit position on it is an atheist by definition. Just because the type of people you usually call atheists are people like us who talk about this stuff all the time, doesn't mean you ignore the definition of the word. The people on here are all, for the most part, going to be activist atheists but anyone who doesn't hold a positive position of belief in a god is an atheist whether they think about the issue or not. This has been addressed many times on both of the ACA's shows as well as many other places.

  20. I agree with Kazim on this issue - it is definitely conceivable that one might hold two contradictory propositions simultaneously.

    And yet... I agree with Matt - it is impossible to believe X is true and believe X is not true simultaneously.

    Wait... does that mean that I believe two contradictory propositions?!

  21. It is possible to hold two contradictory beliefs simultaneously. George Orwell called it doublethink.

  22. "At this point the caller simultaneously holds these propositions in his head:

    1. Everything the Bible says is moral.
    2. Genocide is immoral.
    3. The Bible is endorsing genocide.

    Not true. In your example, the caller has just had a dilemma clearly exposed to them and has said they'd get back to you - because they're trying to reconcile it.

    This is where rationalization and compartmentalization serve to obfuscate the dilemma.

    In fact, your particular example tends to result in apologists rationalizations like "God knew which ones could be saved". Why? Because they recognize the conflict and need to resolve it.

    If anything, your example supports what I've said. I don't believe you could find a (sane and honest) believer who openly states that they believe slavery is both moral and immoral...they'll claim it used to be moral, or any number of other twists that disguise the contradiction.

  23. I think you are treating this as a qualitative difference when it fact it is a quantitative one. The key is in your final phrasing, which is poorly defined: "or any number of other twists that disguise the contradiction."

    I think every person in this thread has agreed with me that anybody can hold contradictory positions in their mind, as long as the nature of the contradiction is sufficiently obfuscated that they don't see the problem.

    Well, how much obfuscation is required? Obviously it varies from person to person. Logic puzzles that present a dozen or so propositions like "Bob is a painter," "The painter has a daughter," "The architect is single" etc, can easily slip a contradiction in that is hard to detect unless you put in a whole lot of work. Therefore, I can read a dozen statements that wind up being obscurely but provably contradictory, and believe them all without causing any mental anguish at all. Probably if the same contradiction was reduced to three or four statements, I would notice it and complain.

    But then again, I'm trained in that sort of thing through programming and gaming. A person who is not used to that sort of thinking might easily miss a contradiction in three or four statements, or even less if he's dumb, unobservant, or just plain apathetic.

    What I'm saying is that there is no clear amount of obscurity that is "the right amount" to allow people to believe a contradiction, just as there is no definition of the creationist weasel word "kind." And I'd say the specific level of tolerance a given person has for contradictions is less important than the universal fact that everyone can believe a contradiction no matter how smart. After that, we're just arguing over a matter of how low this threshold has to be before you are allowed to apply the (also poorly defined) "insane" label.

  24. Um, is it just me or is there a tiny bit of an "apples and oranges" thing going on here?. Mr. Dillahunty seems to be using "contradiction" to mean something like "no sane or honest person can believe god is perfectly moral AND god is NOT perfectly moral at the same time." Mr. Glasser(Kazim) on the other hand appears to mean something like "it's possible to believe a contradiction like: god is perfectly moral AND the Bible is inerrant."

    The latter sounds like a believer holding two contradicting beliefs but rationalising the contradiction(if the Bible is inerrant, how can you say the thing god did are moral?) by making excuses and ad hoc explanations, which is possible and in fact, happens all the time. The former sounds more like someone stating that A = A and A = -A at the same time, which is impossible('cause things are what they are; they're not what they're not; and it's not neither or both).

    I dunno... just throwing stuff out there. Don't pay too much attention to me. :P

  25. I don't think it's an apples and oranges thing. When I say "contradiction" I mean it in the literal, conventional sense of the word. As I say before, I feel like we can agree that people can and do believe contradictory information. I think the only difference being expressed here is in terms of how starkly drawn the contradiction has to be before a non-crazy person will sit up and say "Hey, wait a minute..."

    Unfortunately, this will have to be my last contribution to this discussion for the week, as I'm headed home, after which I'll be catching a flight in the morning and I have no laptop to bring with me. (Yet. In point of fact I just ordered one and it should be available shortly after I return. I'm quite excited about it.)


  26. it does not follow that one cannot hold contradictory beliefs, and in fact, I think that they do all the time.

    This seems (to me) to be an issue of semantics. The objection to contradictory beliefs is not that one can't have them. It's that in doing so, the person doesn't understand the nature of his/her belief.

    It's similar to what you said later in your post. But I will side with those who claim that the person can't actually believe what he/she is claiming.

    Personally, I moved away from religion once I found out that many believers uphold contradictory ideas. Few are honest enough to admit that they don't care about the contradictions, and even fewer understand the implications of what they claim.

  27. I'm still with Matt. You can have two inconsistent beliefs. But you can't have a belief and at the same time not have it. These are two different things.

  28. I think you can hold two opposing positions at once, but only if you're batshit insane. I agree with Matt here. If you know that you are wrong, and don't care, you are a moron!

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  30. I think you guys have really misrepresented Gregory’s position, and/or, perhaps it’s better put that you’re interpretation of him was rather unforgiving, and I would suggest, incorrect….. Gregg’s position is that he is both a Theist and an Atheist, at the same time.

    From his opening remarks it seems quite evident that what Greg rejects in God (that makes him call himself an atheist) is God in the sense of a Platonic conception (Realism or Materialism is you will. i.e. he rejects the same God you do). The notion that Greg accepts is a Pragmatic conception, perhaps God as metaphor (that requires significant unpacking of course – think, Nietzsche’s notion of truth as a mobile army of metaphors).

    You guys on the show seem to have definite Materialist/Realist leanings (and you’re skilled in that rhetoric) so of course you’re going to do a good job pulling Greg into your world, forcing him to argue from the foundations of your assumptions. I’d be willing to bet Greg would feel quite comfortable in the hands of say, a William James or Richard Rorty.

    Anyway, bottom line is I think Greg was being quite honest, but could simply not articulate himself properly and find his way out of Platonist assumptions. In this way, I don’t think he was being in the least bit inconsistent, merely inarticulate.

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  32. I also have to agree that it is only possible to have contradictory beliefs when the person is unaware of the contradiction.

    The way I see it is that any time two beliefs can be expressed as the same proposition P1 being both true and false then there is no room to hide the contradiction.

    For example:

    P1 "Thor exists". I claim that it's impossible to believe that this statement is both true and false.

    However if we look at two other statements:

    P1 "Thor exists" P2 and "No gods exist". Now it's possible to believe that both P1 is true and P2 is true because since we are talking about two diffrent statements, it's possible that the contradiction between the two statements is not clearly seen. (For example if one doesn't know that Thor is a god).

  33. Ingemar,
    Your example simply won’t do – you are essentially mistaking the finger for the moon.

    If your fictitious person thinks Thor exists, but doesn’t think Gods exist, and by your admission doesn’t think Thor is a God, then he hasn’t contradicted himself. You interpreting him the way you are (as being wrong about Thor) drains your case that you are interpreting him correctly. For all you know Thor is his neighbor, of some new video game, etc. etc. The bottom line is (and you’ve demonstrated it) the person doesn’t think Thor is a God, and as such there is no contradiction.

  34. Andrew Louis:

    My example with Thor is exactly the same thing as Russels example with someone believing that the bible is moral but thinking that genocide is immoral. The person saying that the bible is moral simply doesn't know or understand what the bible says about genocide.

    The point is that any contradiction that someone believes can only be pointed out by someone else who identifies the beliefs as a contradiction. I claim that noone believes two contradicting things if they know that it's contradicting.

  35. Kazim:

    Great, thought-provoking post!
    I just got one question - aren't you mixing properties of formal systems and human belief systems too readily?

    Inconsistent formal systems contain paradoxes - like the famous 'This statement is false'. You can say that in the formal system of aristotelean logic the statement can be both proved true and false at the same time. But can a human mind grasp that concept of both true and not-true? Or is it rather the case that the human mind merely cannot decide whether the statement is true or not?

    In my case, no matter what I do, I cannot wrap my mind around such paradoxes and honestly believe they are true and false at the same time. And I suspect no human mind can do that.

    In a similar fashion if you look at Escher's Convex and Concave - you know that the objects in the pictures can be interpreted in two different ways and you can make your mind see them convex or you can make your mind see them concave, but you cannot make your mind see both at the same time.

  36. Stenlis,

    Inconsistent formal systems contain paradoxes - like the famous 'This statement is false'. You can say that in the formal system of aristotelean logic the statement can be both proved true and false at the same time.

    Actually, it's the opposite. A paradoxical statement like the one above is considered by logicians to be neither true nor false.

    But can a human mind grasp that concept of both true and not-true? Or is it rather the case that the human mind merely cannot decide whether the statement is true or not?

    I think you are mixing up the default position, which is simply complete apathy with regard to the truth or falsehood of a given statement. It takes work to ferret out a contradiction in your beliefs. Many people, on encountering the claim "This statement is false," might just say "Whatever" and not think through the implications. That's the sort of person that I think can easily accept contradictory claims without a problem.

    Many who have contributed to this thread would probably say that this doesn't count, because those people haven't really grasped the contradiction and so do not really "believe" it. But I think that's the point -- it's a common occurrence to accept contradictory claims precisely because it's uncommon to think hard about the logic involved. The only way to weed out contradictions is to really work hard at the logical process that isolates the contradiction, or else make the contradiction so simple and obvious that it takes no work. But I hardly think this means that one person can't accept a contradiction which is readily apparent to another person. It happens all the time.

  37. Kazim:

    OK, the liar paradox was perhaps not the best example, but *inconsistency* of formal systems in general means that there is a statement P such that both P and 'not P' are provable by the axioms of the system. Perhaps the Berry paradox is a better example of something that is both true and not true at the same time.

    In any case, I see what you are saying with the misunderstanding of logic that people profess.


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