Why Obey the Laws of Logic?

December 4, 2007    By: Jacob J @ 12:29 am   Category: Uncategorized

There is a wonderful little book that I would blog if I were not so lazy called Walking the Tightrope of Reason. Robert Fogelin explores the paradoxes and problems associated with reason and logic. On the one hand, we are absolutely and unavoidably commited to the integrity of reason. On the other hand, we find that reason has a tendency to turn against itself and create incoherence when pursued unrestrained. In an early chapter entitled “Why Obey the Laws of Logic?” he makes the point that if we reject the law of non-contradiction, everything goes south fast:

The standard proof that everything follows from a contradiction depends on three seemingly unassailable principles. The first concerns conjunction. From the conjunction p & q we may infer p and we may also infer q. The second concerns disjunction: from p, we may infer p or q. The third principle also concerns disjunction and is a bit more complicated: from p or q, together with ~p, we may infer q. Given these three rules of inference, the proof of the spread principle—the principle that everything follows from a contradiction—is short and sweet:

p & ~p
p The conjunction principle
pq The first disjunction principle
~p The conjunction principle again
q The second disjunction principle

Because q can be any proposition whatsoever, this proof shows that if we start with a contradiction, we can derive any proposition we please. In order to block this line of reasoning, at least one of the three inference rules that generate it must be rejected. This makes tough intuitive going, because each of these three inference rules seems wholly plausible on its face.†

This proof packs quite a punch. It shows that if you allow a contradiction to be “true,” then from that you can prove absolutely anything else. For example, from a single contradiction, every other contradiction can be proven. Thus, if you accept a contradiction, you are, in effect, granting the following permission:

In interpreting what I say, you may add the phrase “It is not the case that” to the front of any sentence I utter. Do this as you please, for it will in no way alter the significance of my discourse. ‡

So, the next time someone comes through here arguing that our logic doesn’t apply to God, I will agree with them that, indeed, our logic does apply to God, and direct them to this post to explain why I conclude from their statement the opposite of what they stated. If it is true that “our logic” does not apply to God, then we can stop wasting our time arguing at all. For that matter, we can stop believing things, or thinking that what we believe is important. Accepting a contradiction to save your theology strips your theology of rational meaning, so I recommend against it.

I think this proof provides quite a good reason to obey the laws of logic. What say you?


† Robert Fogelin, Walking the Tightrope of Reason, pg 174, footnote 21.
‡ Ibid. pg 36.

66 Comments »

  1. My initial reaction to this post is that this sort of goes against what Blake said at Fair and Sunstone this year, regarding the spirit trumping reason. However, I think our reason does allow us to decide whether the spirit is the spirit or if it is some other influence, so I think you are correct ultimately.

    I hope Blake responds to this though, as I’d love the clarification.

    Comment by Matt W. — December 4, 2007 @ 7:13 am

  2. Matt. I don’t see reason and the spirit as in competition or at odds. My point about spiritual experience is that it doesn’t stand in need of an argument or logical proof to be a legitimate basis of commitment. That doesn’t mean that it’s ok to adopt a demonstrably incoherent view.

    So for example I don’t need an argument to show that chocolate tastes good to me. Nor can I prove that chocolate tastes good to me. In fact it does and my experience of tasting chocolate is sufficient to show that fact. It is self-evident and given in my experience of tasting chocolate (well at least a well-seasoned dark chocolate from Switzerland) that it tastes good to me. I can’t prove it by reason. However, I would not be justified in such experience if there were no such thing as chocolate, or no such sense as taste and so forth. Given such facts (if they were facts) my belief that it was chocolate causing me to taste something good would be incoherent. Further, imagine how absurd it is to taste chocolate and then try to reason to the fact it tastes good or try to deduce from reason whether it tastes good or bad to me. Such things are known only by experience and the experiencing of it is sufficient to demonstrate it. That is what I claim the experience of the spirit is like. However, the fact that I like chocolate doesn’t mean everyone will have the same experience. All that I can do is to invite you to give it a shot to see if you like it. If you have an open heart you’ll feel the spirit and know; just as if you taste chocolate you’ll find out whether you like it. It isn’t about reason.

    Comment by Blake — December 4, 2007 @ 7:31 am

  3. Blake: Here’s my hang up with what you are saying. In my practical non-philosopher’s world, my experience is an intrical part of my ability to reason. It is only by experience (ours or anothers) that we know the good from the bad and the logical from the illogical. After all, we can take our experience with Chocolate and use logic with that experience to make Chocolate taste better (well-seasoned dark chocolate, rather than the kind you get in a hershey’s kiss, for example)

    (I could go on and on, but this is Jacob’s thread, and I don’t want to take it in a direction he is not interested in)

    Comment by Matt W. — December 4, 2007 @ 7:48 am

  4. Matt,

    I think you are misunderstanding Blake’s point. We could not use reason to determine if light or dark chocolate universally tastes “better”. That is a completely subjective opinion. But we could use reason and our experience to determine whether or not there is such a thing as chocolate. The actual existence of something we call chocolate in the universe is not a subjective thing — it either really exists or it doesn’t. The same principle applies to the Holy Spirit — we can use our experience to reasonably decide if it/he exists or not.

    Comment by Geoff J — December 4, 2007 @ 9:13 am

  5. BTW Jacob — I agree with your point in the post. If we can’t count on logic to discuss God then we should not ever waste our time discussing God at all since it would all be end up being meaningless anyway.

    Comment by Geoff J — December 4, 2007 @ 9:22 am

  6. Geoff:
    wow, that really helped. Thanks. You were right, I was misunderstanding.

    Comment by Matt W. — December 4, 2007 @ 9:32 am

  7. Matt,

    I am happy for the thread to take its own course, don’t worry about a threadjack (especially when your question arises from your reading of the post).

    The interplay between reason and experience is actually quite interesting and complex. I agree with you that it can be hard to separate what we know by reason and what has been influenced by our experience. There was a long proud tradition in philosophy of trying to jettison everything we infer from our experience and come up with a system of thought based entirely on reason. Of course it is well known now that such efforts never legitimately got off the ground. In fact, at the end of the book when Fogelin offers his thoughts on how to deal with the suicidal tendencies of reason, he talks a lot about restraining reason by experience. He makes a good case for the idea that reason uncovers a lot of insoluable problems when left untethered (unfortunately it uncovers a few even when we try to tether it down).

    Comment by Jacob J — December 4, 2007 @ 10:08 am

  8. And someday, my impaired logic will catch up to God’s perfect logic. But that is another tangent . . . Job and sinful depravity.

    Comment by Todd Wood — December 4, 2007 @ 10:09 am

  9. Reason loosely used isn’t just logic. Reasoning includes gathering premises whereas in logic the premises are taken for granted. As they say, garbage in garbage out.

    Comment by Clark — December 4, 2007 @ 10:40 am

  10. Geoff: The same principle applies to the Holy Spirit — we can use our experience to reasonably decide if it/he exists or not.

    Actually, I don’t believe that we can use our reason to decide whether the Holy Ghost exists or not. If I experience the Holy Ghost, then I have a good and sufficient reason to believe that the Holy Ghost exists. I know if it tastes good to me, as Alma suggests. I don’t need a reason to justify my experience nor is the existence of the Holy Ghost subject to logical proofs. However, arguments can be given that my experience ought not be trusted. However, these arguments will take the form of suggesting that my experience is mistaken. The problem is that like tasting chocolate, I know that what I have experienced is good and I like it; however, I could be convinced that it wasn’t really chocolate but an electrode in my brain that caused the experience of tasting the chocolate. Only once I have some reason to doubt and a defeater to call my experience into question does the interplay between reason and experience begin.

    However, I agree with Jacob. We are all convinced that logic is compelling because otherwise we could just babble anything and it would have as much meaning as any other assertion we care to make. Not even God can make the statement “X created a perfectly round square” meaningful simply because we replace “X” with “God.” Nonsense is nonsense even if we assert that God is somehow behind the nonsense.

    Comment by Blake — December 4, 2007 @ 11:01 am

  11. Blake:

    Would it be fair to say that reason, loosely speaking, would be more than logical proofs, as your experience, in your example, gives you reason to believe?

    Jacob J: I think any philosophy that fails to incorporate experience is doomed. However, I guess there are so few universal experiences that any philosophy that relies too heavily on experience is also limiting itself.

    Comment by Matt W. — December 4, 2007 @ 11:17 am

  12. Jacob:

    I am quite stupid with this stuff. Would it be possible for you (or someone) to walk through a practical example of this proof. It sort of tastes like unsweetened chocolate to me.

    Comment by Eric Nielson — December 4, 2007 @ 3:24 pm

  13. Eric,

    Here is a simple one showing the consequences to arithmetic:

    (1) 1 = -1
    (2) 1 + 1 = -1 + 1
    (3) 1 = 0
    (4) 0 = -1
    (5) 10 + 10 = 20
    (6) 10 * 1 + 10 * 1 = 20
    (7) 10 * 1 + 10 * 0 = 20
    (8) 10 = 20
    (9) 20 = 0

    And so on…

    Comment by Mark D. — December 4, 2007 @ 4:45 pm

  14. We are all convinced that logic is compelling because otherwise we could just babble anything and it would have as much meaning as any other assertion we care to make.

    I disagree. ;) First, we are not all convinced that logic is compelling. (Just watch a presidential debate sometime. Or try to convince a devout follower that WWF wrestling is fake.) Second, for those who are convinced that logic is compelling, not all of them believe that without it every statement would be meaningless. Third, which logic do you speak of? Standard propositional logic? Logic without ~~p=>p? Logic with predicates? Many valued logic?

    My point of view is that we use logic because it seems to accurately represent the universe. If we ever did find an actual contradiction in the universe, say “p&~p”, we would probably modify our logic accordingly. I imagine that the second disjunction would be the natural axiom to remove/modify in such a circumstance.

    Comment by P. Nielsen — December 4, 2007 @ 7:37 pm

  15. P. Nielsen. The fact that someone makes logical fallacies doesn’t mean we aren’t persuaded by sound logic. In fact, your very example about WWF has absolutely nothing to do with logic. And yes, we are talking about various kinds of logic, but usually S5 and standard axioms that depart from the law of non-contradiction. Further, it is still the case that virtually anything follows from contradictory premises.

    Further, because I am Kantian I don’t believe that logic or what is a priori depends on experience but the other way around.

    Comment by Blake — December 4, 2007 @ 8:09 pm

  16. Blake,

    The post was made half jokingly (although I do stand by, in a strict sense, everything I said).

    You are welcome to be Kantian. I was just pointing out, jokingly, that not everyone in the world finds logic (say, modus ponens with the usual axioms) compelling (in many of the different senses of the word), which is how your initial sentence read. I was just picking on your use of the absolute “we”.

    But just to clarify: (1) I don’t disagree with the idea that *some* find logic persuasive, even in the face of people often employing logical fallacies. In fact, my post had nothing whatsoever to do with logical fallacies. I was rather (in humor) taking issue with the idea that *all* people are convinced that logic is compelling, for the reasons you gave. (2) My example about the WWF has quite a bit to do with logic, broadly construed. But it does depend on the context, and how the WWF-believer tries to defend said belief. (3) I understood that the thread was implicitly talking about a very narrow set of logics. My questions were more to the point of *universal* acceptance of *a* logic. (4) “Virtually anything follows from contradictory premises” only in some logic systems. (5) I never insinuated that “logic” or “what is a priori” depends on experience. Rather, I said that experience helps us decide which axiomatic system to utilize to try and accurately express the universe. On the other hand, in my opinion, how the universe truly behaves does affect how logic works.

    You may assert that whether there are any square circles in Euclidean geometry does not depend on how reality really is. But I would counter that this view takes a leap of faith. A leap of faith which is fundamentally alogical. A leap of faith which is based upon a value judgement of the logic and intuition which you employ to judge your understanding of Euclidean geometry (and circles and squares) to be correct. A leap of faith ultimately tied to your experience with the universe.

    Comment by P. Nielsen — December 4, 2007 @ 8:45 pm

  17. Todd (#8),

    And someday, my impaired logic will catch up to God’s perfect logic.

    Just to be clear, are you referring to your imperfect use of logic, or are you claiming that God’s logic is fundamentally unrelated to the sort of logical inferences found in the proof?

    Eric (#12),

    Which are you looking for, some concrete examples of the logical inferences, or some concrete things substituted for p and q?

    Comment by Jacob J — December 4, 2007 @ 10:47 pm

  18. P. Nielsen,

    If someone determines (in ordinary logic) that for some p, ‘p & ~p’ is true the problem is most definitely not the logic, but rather the formulation of p. We do not guide probes to the outer reaches of the solar system with a mathematical infrastructure having a foundation as faulty as asserting that 1 = 0.

    In addition, it doesn’t really matter what logic a person uses as long as the rules are consistent with the application and the user takes appropriate care when translating to and from natural language.

    It is not like someone can discover that ‘p & ~p’ in absolute terms. The best that can be done is show that it is an implication of some choice of axioms demonstrably incompatible with ordinary usage.

    Comment by Mark D. — December 4, 2007 @ 11:58 pm

  19. If someone determines (in ordinary logic) that for some p, ‘p & ~p’ is true the problem is most definitely not the logic, but rather the formulation of p. We do not guide probes to the outer reaches of the solar system with a mathematical infrastructure having a foundation as faulty as asserting that 1 = 0.

    I don’t view `p& ~p’ as either a problem in the logic nor as a problem in the formulation of p. If it is true that `p& ~p’ in whatever framework you are using (and if some other things are true, which I will leave unsaid), then everything is both true and false (as asserted by the initial post). Mathematicians work with such systems all of the time [because if you can prove that a system is inconsistent, that tells you something meaningful about the system]. It doesn’t mean the logic is faulty. Nor that p was improperly formulated.

    Second, believe it or not, but in some very useful contexts, 0=1. Of course, in those contexts, 0 and 1 mean different things to mathematicians than to lay people. You think of the counting numbers. We think of elements of what are called “rings” satisfying certain properties. My point being that this is a perfect example of where a different system than what is usually used might reflect reality better than the current one (in some situations).

    Just because we emply Euclidean geometry to guide probes that doesn’t mean we believe Euclidean geometry accurately reflects the universe (in fact, it doesn’t). Only that it safely “gets the job done”, according to our experiences.

    In addition, it doesn’t really matter what logic a person uses as long as the rules are consistent with the application and the user takes appropriate care when translating to and from natural language.

    Just to be clear, I believe you are correct. But, as I said earlier, your statement is a faith based claim. One built upon experiencing the universe via using one’s brain to think.

    It is not like someone can discover that ‘p & ~p’ in absolute terms. The best that can be done is show that it is an implication of some choice of axioms demonstrably incompatible with ordinary usage.

    By “like someone can discover” I’ll take you to mean “discover in the real world” (since mathematicians discover p&~p in mathematical systems quite often). Under that reading, these are faith based claims that I don’t necessarily believe. I am open to the possibility that the logic we employ does not accurately reflect the real world. (I am not open to a universe in which every statement is true. But that is a personal preference.) In other words, I am open to the possibility of a meaningful statement p, about the universe, with both p and ~p true. Or to a universe in which the second disjunction doesn’t hold for statements about it. (I’m even open to a universe in which I am simply wrong that ‘logic’ works correctly). I don’t think these will happen, which is why I personally continue to use logic to talk about the universe, but I’m open to the possibilities.

    Comment by P. Nielsen — December 5, 2007 @ 5:57 am

  20. Jacob:

    I guess I was looking for something to put into p and q that might be discussed here. It seems like one would start from a contradiction – is that correct?

    So maybe a starting point that the wild thangs would believe to be a contradiction. Would absolute foreknowledge and free will being a ‘contradiction’ be a good place to start with a p and q.

    Using math like Mark D. just makes things look silly (although it might help illustrate).

    Comment by Eric Nielson — December 5, 2007 @ 6:44 am

  21. P. Nielsen,

    ‘p & ~p’ cannot be true in an ordinary logic. Either p is ill-formed, or the logic isn’t ordinary. Otherwise ordinary logic would be useless.

    Eric N.,

    The dispute is not one over logic, but rather over semantics. Once upon a time we had a “will” – some insisted that the will was free (hence the term “free will”) and others that the will was determined.

    The former won the argument so thoroughly that now a new faction (the compatibilists) assert that “free will” is determined, which is basically a contradiction in terms and an abuse of the language.

    However, once you reduce the semantics of “free will” to a formal definition (albeit one that may or may not be the case), the dispute goes away. No one argues that foreknowledge is logically incompatible with determinism, for example.

    Comment by Mark D. — December 5, 2007 @ 9:35 am

  22. Mark D.,

    That depends on what you mean by “logic” and “ordinary”. I use logic in the sense of “rules of inference” or “first order logic”. In that sense of the word, *nothing* is true because there is no context in which to apply the rules of logic.

    In some of the common contexts in which I use logic, such as Zermelo-Franekel set theory, it is *hoped* (but not known) that the system does not contain any true statements of the form “p&~p”. When a mathematical set of axioms does not contain a true statement of the form “p&~p” the system is said to be consistent.

    Mathematicians (or I should say, many mathematicians) do not limit themselves to working in consistent systems because, more often than not, one cannot prove that the system one is working in is consistent. Of course, if one proves that a certain system is inconsistent most mathematicians will abandon said system.

    Comment by P. Nielsen — December 5, 2007 @ 12:32 pm

  23. Eric,

    Here is an example. Let q=”all trees are orange”. Let’s take p=”some cats are alive”, so ~p=”all cats are dead”.

    We are trying to prove that all trees are orange from an arbitrary contradiction, like “Some cats are alive and all cats are dead.”

    So, here we go. Assume that some cats are alive and all cats are dead. In particular, some cats are alive. Hence, it is true that some cats are alive or all trees are orange. On the other hand, we are assuming that all cats are dead. Using this fact along with the bolded statement, we conclude that all trees are orange.

    Comment by P. Nielsen — December 5, 2007 @ 12:46 pm

  24. #17 – your first phrase

    Now, Jacob this has been puzzling thought for me.

    Would some LDS say the Bible is both inspired and impaired (or logically incoherent)?

    And Blake, I have some beginning questions for you on my blog if you have a moment to spare some time.

    Appreciate it.

    Comment by Todd Wood — December 5, 2007 @ 2:05 pm

  25. Correction:

    Would some say the present Bible . . .

    Comment by Todd Wood — December 5, 2007 @ 2:07 pm

  26. Eric,

    In addition to the help from Mark (#21) and P. Nielsen (#23) (thanks guys), let me try to phrase my point in a way that will seem less like unsweetened chocolate.

    When I am talking to someone about, let’s say, the nature of God, there are often statements which are potentially contradictions. It is important to make a distiction between people who are explaining why their view does not contain a contradiction as opposed to those who are saying that the existence of a contradiction doesn’t matter.

    Let’s take a concrete example, someone says that God is both nowhere and everywhere and I say that sounds like poppycock. They may respond that there is a sense in which God is everywhere and in a different sense, God is nowhere. This response is an attempt to show that there is no contradiction and their theology is logically sound. However, a different person might respond by telling me that God transcends logic and I am foolish and vain to try to apply human reasoning to the nature of God. It is God’s greatness that makes him appear as a contradiction!† You see the difference I hope. Although both hold the same view (God is both everywhere and nowhere), one person cares about obeying the laws of logic and the other does not. It is the person who cavalierly disregards logic that I think is being foolish. In the past, I have accused Todd of being an anti-logic kind of guy, and to my knowledge he has not proven me wrong on that. Maybe he will. P. Nielsen on this thread seems to be quite comfortable with the possibility of disregarding logic.

    My argument is not so much about a specific doctrine which I am claiming is contradictory so much as it is about our attitude toward logic. When someone waives off a contradiction as though it doesn’t matter, I think they are making a catastrophic mistake, and the proof in the post demonstrates how catastrophic it is. Is it starting to taste like delicious Clark chocolate yet?

    † The exclamation point is here to mimic the the kind of person who glories in contradiction. They just seem to love exclamation points.

    Comment by Jacob J — December 5, 2007 @ 3:53 pm

  27. They just seem to love exclamation points.

    Har!
    They generally enjoy all caps too I’ve noticed.

    Comment by Geoff J — December 5, 2007 @ 4:31 pm

  28. Jacob J,

    There is a way out of this conundrum, that has some practical application beyond dialetheism: Paraconsistent logic.

    The usefulness of paraconsistent logic is not so much allowing one to maintain a belief in logical contradictions (dialetheism), but rather to avoid believing falsehoods when a contradiction sneaks into one’s belief system by accident. This is a big deal in machine learning systems in particular.

    Comment by Mark D. — December 5, 2007 @ 7:19 pm

  29. Thanks Jacob. The mathematical example and the orange tree example were perhaps good illustrations, but they seemed so silly to me that it really did not seem to help. But perhaps that was the point – to show how silly things can get when we just ignore what seems to be a clear contradiction.

    You bring up an excellent point when it comes to explaining why something is not a contradiction. There are often a few possibilities available if you are willing to stretch a little bit. These are often speculative stretches, but that is preferrable to me over just letting the contradiction just sit.

    Comment by Eric Nielson — December 6, 2007 @ 6:35 am

  30. Eric,

    If you want to make it serious do the following. Let q=”God does not exist”, and let p=”sentence A is true” where sentence A=”This sentence is false.”

    Is sentence A true or false? If it is true, then the sentence itself asserts its own falsity. On the other hand, if it is false, then that implies it is true. So if you accept the premise that all well-formed statements have a truth value then either (1) sentence A is not well-formed [which doesn’t seem to be the case], (2) sentence A is both true and false.

    So, assuming the law of excluded middle, you have the true contradiction p&~p. I’ll leave it to you to work out why q is then true if one assumes the conjunction and disjunction principles.

    Comment by P. Nielsen — December 6, 2007 @ 8:08 am

  31. Pace: I’m sure you meant by “sentence A” actually to state: “proposition p” didn’t you? Now maybe you could deal with the rules of S5 dealing with self-referential sentence where P and Q are not truly two different propositions that can contradict themselves. It so happens that in this case, P and Q have the relation of either P or Q, but not P and Q. Further, p above is not a proposition but a statement as to truth value whereas q is not a statement about truth value. So these statements are not “well-formed.” Further, I would reject the view that all well-formed propositions (why are you speaking of sentences?) have truth value. “I will freely have eggs an hour from now” doesn’t have a determinate truth value on my multivalued logic, but it is a well-formed sentence.

    Comment by Blake — December 6, 2007 @ 8:19 am

  32. Blake,

    Yes, I did equivocate a little bit. I wasn’t trying to actually make a valid argument, but rather try to help Eric get the jist of the original post using serious examples, even if I had to bend things a little (since I personally don’t know of any true dialetheisms in the universe, do you?). Maybe to have avoided this criticism, I could have simply written p=”p is false”, but the problem with that is it doesn’t satisfy Eric’s criteria. Such a statement is confusing, and it isn’t clear it has any meaning, or is well-formed, etc… Maybe I could have written p=”There exists a barber which cuts everyone’s hairs who don’t cut their own” but that suffers from the ridiculousness of the existences of said barber. (Again, I don’t know of any true dialetheisms!) So I was trying to make due with what Eric allowed. Make lemonade out of lemons, and all that.

    Further, I would reject the view that all well-formed propositions (why are you speaking of sentences?) have truth value.

    Why did I use “sentence” instead of “proposition”? Because it is a sentence. Because I doubt Eric knows what “proposition” means in the technical sense. Because in some contexts, sentence A is not a well-formed proposition. etc…

    And you are more than free to reject the view that all well-formed propositions have truth value. I explicitly brought up that point in my last post because many people do reject the law of excluded middle. Further, if you even object to the idea of accepting the law of excluded middle just for my previous post, we can do away with that assumption, and simply assume p&~p, for sake of argument. (But again, this complicates matters, in my opinion, and I was trying to keep things simple.)

    Now maybe you could deal with the rules of S5 dealing with self-referential sentence where P and Q are not truly two different propositions that can contradict themselves.

    Unfortunately, I am a mathematician, who specializes in algebra, and am not a logician. Hence, I cannot acquiesce to your request due to my lack of experience with said system specifically. I know that in Peano arithmetic one can create well-formed, meaningful sentences (i.e. propositions) which are self-referential such as “This system is inconsistent” but I don’t know all the ins and outs. I would imagine that S5, by itself, has no self-references. Maybe you could tell me the answer to your questsion.

    Best,
    P. Nielsen

    Comment by P. Nielsen — December 6, 2007 @ 2:29 pm

  33. Sorry Pace, I was just being way too technical. That happens after teaching logic. However, math majors are the worst because they can’t deal with the formal/semantic divide easily.

    Comment by Blake — December 6, 2007 @ 3:05 pm

  34. You may enjoy “Godel, Escher, Bach: An Eternal Golden Braid” by Douglas Hofsteader. Godel’s theorem says a lot about the limits of logic.

    Comment by DavidF — December 6, 2007 @ 8:12 pm

  35. Blake,

    What is the formal/semantic divide? (One of the biggest weaknesses I’ve noticed about mathematicians is that we *look* for mistakes. Makes it hard to appreciate the good things!)

    Comment by P. Nielsen — December 6, 2007 @ 9:12 pm

  36. Jacob, no doubt, friend, you would be better trained in logic than me.

    Is there anything recorded in the Bible that you don’t logically understand?

    Comment by Todd Wood — December 6, 2007 @ 9:29 pm

  37. Pace: One can do mathematics without semantics; but one cannot detect the truth value of a proposition and whether it equivocates without knowing the semantic meaning of the terms. For instance:

    (1) No one is perfect;
    (2) I am no one;
    (3) Therefore, I am perfect.

    It works out this way formally:

    (1*) x = A.
    (2*) B = x.
    (3*) Therefore, A = B.

    However, it is clear that though formally valid, not just any semantic substitution will work because of the meaning of the terms.

    David, the limits of logic arise from self-referring or self-including sets. There are a number of logical theorums that have been proposed to resolve the problem — though it isn’t clear which, if any, are successful.

    Comment by Blake — December 6, 2007 @ 10:01 pm

  38. Todd,

    My comment is not about who is better trained in logic. After all, my training is in engineering, not logic. Once again, I am talking about our attitudes toward logic. You seem to be comfortable with the idea that your theology may contain contradictions when it comes to describing God. I view contradiction as evidence of an error. It is the different attitude toward logic that I am pointing out. Of course, if I have mischaracterized your view, please correct me. I am going off my remembrance of things you’ve said here in the past, but I could be misunderstanding you.

    As to the Bible, I don’t know of anything in the Bible which defies logic.

    Comment by Jacob J — December 6, 2007 @ 10:49 pm

  39. Jacob, how about Donkeys talking, women turning to salt, two of every kind of beast on a boat sufficient to repopulate the earth, embodied persons walking through walls and plucking your eye out if you don’t like what you see?

    Comment by Blake — December 7, 2007 @ 8:26 am

  40. Blake, that doesn’t defy logic, it just defies experience.

    Comment by Kent — December 7, 2007 @ 10:18 am

  41. Actually, it does defy logic if we include the semantics and soundness of premises in the equation (which we must because logic includes assessment of logical fallacies in argumentation).

    Comment by Blake — December 7, 2007 @ 10:38 am

  42. Blake,

    I am with Kent on this. There is no constraint of logic which would prevent God from making a donkey talk. Furthermore (and more importantly in my opinion), depending on our assumptions about how the Bible came to be, the existence of such stories as those you point can be interpreted in a wide variety of ways, many of which are not challenging in any way to logic. Does it really defy logic for Jesus to tell us to pluck out our eyes, if we understand him to be employing hyperbole? Of course I agree that certain readings of the Bible may be problematic, but belief in the Bible does not require me to abandon logic.

    Comment by Jacob J — December 7, 2007 @ 11:26 am

  43. Jacob: The point is that logic can be experientially based to the extent the validity or soundness of a premise is based on its probability. Remember, logic is not merely deductive but also inductive. There is no way, barring a miracle that defies natural laws, that two animals of each kind is enough to re-populate the earth without almost certain extinction. Is it a logical problem? Certainly, it is mathematically impossible or zero probability. Thus, it is inductively so improbable that unless one buys into a violation of the laws of nature as a means of performing miracles, what the Bible reports is just impossible — and that is a logical argument. Sabe?

    Comment by Blake — December 7, 2007 @ 2:50 pm

  44. Blake,

    Yes, I see your point about deduction vs. induction. My point is that I don’t have to believe that two animals of each kind literally re-populated the earth starting 6,000 years ago in order to believe in the Bible. If your point is that a certain kind of strict literalism when reading the Bible leads to logical difficulties, then I can accept that point. However, your comment reminds me that one of these days I need to post on miracles in relation to the laws of nature since the standard Mormon position is a bit bewildering to me.

    Comment by Jacob J — December 7, 2007 @ 3:06 pm

  45. I don’t think logic (in the narrow sense) has anything to with experience or with probability. It is an abstraction more like mathematics.

    Inductive logic is a bit of a misnomer. It can’t be done in the abstract. As soon as you introduce experience and probability, you are beyond logic into the realm of natural philosophy and science. Probability is among other things a metaphysical position about the nature of unknown quantities.

    Of course any real world application of deductive logic is going to involve extra-logical metaphysical, semantic, and interpretive issues as well.

    Comment by Mark D. — December 7, 2007 @ 3:07 pm

  46. Mark: You are of course entitled to have your own idiosyncratic view of what constitutes “logic,” but inductive logic has well-established rules (and some that are quite controversial about probability), and it is definitely a branch of logic traditionally. It is a bit like saying that as far as you are concerned “food” is a bit of a misnomer for bread, but most people use it that way.

    Comment by Blake — December 7, 2007 @ 4:35 pm

  47. Let’s talk about logic and plucking your eye out. In no possible world is it morally obligatory to pluck one’s eye out if it sees something unworthy. Thus, it is logically necessary that “you must pluck your eye out” is false. That makes it a logical truth in possible worlds logical systems. Now you could reinterpret it to mean: “don’t look at anything unworthy,” but that rather changes the radical meaning that is the hallmark of Jesus’s teachings.

    Comment by Blake — December 7, 2007 @ 4:39 pm

  48. Blake,

    I said “a bit” of a misnomer on purpose. Deductive logic follows from the nature of the language or formal system itself. Inductive logic does not. They belong in different categories. Here is the SEP definition of classical logic:

    Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language.

    And so while I recognize that “inductive logic” is standard usage, the nature of inductive logic is so radically different from deductive logic as to make the question of whether such and such an event violates the laws of logic ambiguous unless that distinction is made. There are no inductive laws of logic, because inductive logic is all systematic guesswork. The foundation of any inductive conclusion lies in experience, not in any sort of logical certainty.

    Comment by Mark D. — December 7, 2007 @ 6:29 pm

  49. I would like to see a series where Jacob or others work through the miracles of Scripture, logically.

    And not end up being a material rationalist.

    Comment by Todd Wood — December 7, 2007 @ 9:20 pm

  50. Mark: The laws of probability for given populations are not mere guesswork and inductive fallacies are easily detectable. There are issues of which rules and inferences to apply in both deductive and inductive logic.

    Todd: It so happens that many Mormons are quite comfortable with materialistic rationalism. That is, everything real is material in some sense (or supervenes on something material) and reality is ultimately rational (something which I believe no theist could dispute unless you maintain that God is arbitrary and irrational).

    Comment by Blake — December 7, 2007 @ 9:46 pm

  51. Blake,

    I said “systematic guesswork” – big difference. Probability is all systematic guesswork. In an environment of perfect information, probability is nigh unto useless. In the real world, it is a systematic means of accounting for what one does not know.

    Also, starting with a “given” population implies that one is at the end of a long road of inductive inquiry – an inquiry into the laws of nature, not the laws of logic. An inductive inquiry can have neither rhyme nor reason without a theory as to the nature of the objects or events being studied.

    Every purported law of inductive reasoning is merely an artifact of some metaphysical assumption like the constancy of the laws of nature. If you take away natural philosophy, induction is an empty void.

    Comment by Mark D. — December 7, 2007 @ 11:09 pm

  52. In an environment of perfect information, probability is nigh unto useless.

    Chaos and quantum theories suggest otherwise. Moreover, since we never have “perfect” information, inductive logic is the way that beings having access to the kinds of information we actually possess assess claims.

    Comment by Blake — December 8, 2007 @ 9:13 am

  53. Blake,

    Chaos theory is deterministic. If you have perfect information and sufficient computational capacity, there is no need for probability. The Bohr interpretation of quantum mechanics is as instrinsically probabilistic as it is surreal. I find it hard to take a theoretical interpretation seriously that cannot manage to separate perception from reality.

    Of course I agree that probability theory is a tool of enormous practical utility. My point though is that deductive logic is a kin to mathematics, where inductive logic is essentially indistinguishable from science. So the question of whether something is contrary to logic is ambiguous.

    An assertion contrary to deductive logic is a contradiction – something that can be distinguished from a disagreement with the premises. But there is no corresponding distinction in inductive logic. There is no way for an assertion to disagree with inductive logic in a manner independent of disagreement with some scientific theory.

    Without further information, it is ridiculous to claim that the virgin birth, the resurrection, and immortality are ‘illogical’. They are simply contrary to available scientific evidence. If we are to have a rational discussion about miracles, that is a critical distinction to make.

    Comment by Mark D. — December 8, 2007 @ 10:34 am

  54. Blake (#47),

    You are starting to baffle me on this eye plucking point. Are you claiming that Jesus really meant that we should literally pluck our our eyes if we saw something offensive?

    Todd,

    And not end up being a material rationalist.

    For some reason I always find myself guessing at what you are claiming. Could it be because you rarely commit yourself to a disputable claim, choosing rather to insinuate claims?

    At any rate, it seems that the implication of your comment is that miracles are fundamentally incompatible with logic (strangely, you and Blake may be on the same page on this one). If that is your claim, then I disagree. I don’t think miracles are illogical with respect to the kind of logic discussed in the post. If I have correctly identified your claim, I would love to see your argument for why they are incompatible, that might make an interesting discussion.

    Mark,

    Chaos theory is deterministic. If you have perfect information and sufficient computational capacity, there is no need for probability.

    That’s all fine and good until you realize that for many things it is impossible to get “perfect information,” even in principle. Infinite precision on an irrational number does not exist.

    Comment by Jacob J — December 8, 2007 @ 11:17 am

  55. Jacob: Did Jesus mean we should pluck our eye out? No, it was a colloquialism. But he did mean to love our neighbors and enemies literally so that literally we have no enemies. My point is that what the scriptures says isn’t logical, the semantic twist of the colloquialism is necessary to bring sense to it.

    Do I mean that miracles are a breach of logic? Yes, if by miracle you mean that what is naturally possible is suspended or breached, it is illogical and otherwise inductively impossible (having a probability of exactly 0). Do I mean that the LDS view of miracles is irrational? That depends on what you take the LDS view of miracles to be. On my view they aren’t. I doubt that there are any real miracles requiring any divine intervention or anything more than a deistic god on your view if I have understood it correctly as merely a natural event that we don’t yet understand.

    Comment by Blake — December 8, 2007 @ 11:22 am

  56. Blake,

    My point is that what the scriptures says isn’t logical, the semantic twist of the colloquialism is necessary to bring sense to it.

    This seems to imply that the text has some a-contextual meaning. Its like you are saying we are all irrational for using figures of speech or something. You say that “what the scriptures say isn’t logical” but for that to be true, you have to hold to the view that “what the scriptures say” must refer to the words on the page take out of context and interpreted with a strict literalism like that an alien robot my employ. I would take a different view, which is that “what the scriptures say” refers to an interpretation in which the colloqualism is understood.

    On miracles: Just to be clear, the view that miracles are “merely a natural event that we don’t yet understand” is the view that baffles me. It is not my view.

    Comment by Jacob J — December 8, 2007 @ 11:59 am

  57. Jacob: The scriptures aren’t logical — but not being logical doesn’t mean illogical.

    I misunderstood your view of miracles. It baffles me too. But then, miracles are supposed to be baffling, aren’t they?

    Comment by Blake — December 8, 2007 @ 12:39 pm

  58. Jacob,

    That’s all fine and good until you realize that for many things it is impossible to get “perfect information,” even in principle. Infinite precision on an irrational number does not exist.

    I agree as a matter of practice of course. However, there is no logical problem with perfect information. If their were, no rational person could maintain divine omniscience without contradiction.

    And certainly no one is going to represent arbitrary irrational numbers perfectly in any physically realizable computer. However, there is no logical problem with such information. An assertion to the contrary is equivalent to the claim that irrational numbers themselves do not exist.

    Comment by Mark D. — December 8, 2007 @ 2:00 pm

  59. Blake,

    The proposition that the laws of nature can be suspended is equivalent to the claim that there are no laws of nature.

    Since our knowledge of the laws of nature is imperfect and incomplete, the proposition that miracles involve a suspension of natural laws tends to boil down to the claim that no conceivable set of natural laws could accomodate the miraculous, and further that miracles are not only contrary to law, that they are contrary to rationality itself.

    I disagree on both counts.

    Comment by Mark D. — December 8, 2007 @ 2:15 pm

  60. Mark: The claim that miracles are just what happens by natural law means there are no miracles since what occurs is what just naturally occurs. What I claim is that miracles aren’t just events that would occur in a completely a-theistic universe without a god; rather, miracles entail divine intervention. If something occurs that is a miracle, it occurs because God intervened and not because it is the result of the natural order alone like deists hold.

    Of course there are natural laws even if there are miracles in a classical sense. There is the natural order of how things occur defined by natural laws and there is the order events that occur in contravention of natural laws. But since you assume a Humean view of natural laws, I doubt that there are any natural laws in that sense.

    Comment by Blake — December 8, 2007 @ 2:51 pm

  61. Blake,

    I understand Hume’s view to be the following:

    (1) Miracles are violations of the laws of nature
    (2) The laws of nature cannot be violated

    Thus Hume defines miracles out of existence. However, Talmage’s view (which I agree with) is rather:

    (1) The laws of nature cannot be violated
    (2) However, miracles are not violations of the laws of nature

    Since the latter view is in question here, I ask: where is the counterargument? Does anyone have a good reason to believe that miracles require the suspension of the laws of nature?

    Comment by Mark D. — December 8, 2007 @ 4:07 pm

  62. What I think is funny is the idea that talking donkeys are “illogical”, when I’ve seen talking animals on TV for years now.

    Comment by P. Nielsen — December 8, 2007 @ 4:15 pm

  63. Yeah Mark, dead men don’t come back to life, donkeys don’t talk and bodies don’t pass through walls given the natural laws that govern our universe.

    Pace, you’re right, if it is on TV it must be logical!

    Comment by Blake — December 8, 2007 @ 6:19 pm

  64. Blake,

    It sounds like that you are being sarcastic. If not, you have not provided any evidence for that assertion. Surely you can do better than that.

    Comment by Mark D. — December 8, 2007 @ 7:42 pm

  65. Blake,

    Yep. TV is a universal oracle. If it says something is true, it must be true.

    By the way, are you asserting that there is no amount of technology which would allow us to make dead men come back to life, donkeys talk, or bodies pass through walls? From my own personal experience with science, I’d say these things are not illogical (nor even improbable) for a being who is super-intelligent and super-powerful.

    Comment by P. Nielsen — December 8, 2007 @ 8:20 pm

  66. P. Nielsen,

    That’s the crux of the debate. Arthur C. Clarke expressed a comparable opinion when he said “Any sufficiently advanced technology is indistinguishable from magic”.

    It seems to me that if we are committed to the eternality of the elements (D&C 93:33), and the corresponding impossibility of ex nihilo creation, that commitment directly corresponds to a commitment to the principle that energy is conserved. What else other than the conservation of energy prevents God from creating matter out of nothing?

    And so if God wants to multiply loaves and fishes, it seems quite necessary for him to transfer energy from one place to another (or at least from one form to another) to do so.

    Comment by Mark D. — December 8, 2007 @ 11:36 pm

Leave a comment

RSS feed for comments on this post.