Beyond Symbolic Logic


 Cuthbert Berry
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1 Beyond Symbolic Logic 1. The Problem of Incompleteness: Many believe that mathematics can explain *everything*. Gottlob Frege proposed that ALL truths can be captured in terms of mathematical entities; and these entities are reducible to sets. For instance, 2 is really just the thing that the collections, e.g., {dog, dog}, {mug, mug}, {apple, apple}, and {table, table}, etc. all have in common. Unfortunately, Bertrand Russell cast suspicion on set theory with a paradox: The Barber Paradox: Consider the following claim: The barber shaves all and only those who don t shave themselves. So Who shaves the barber? If he DOES shave himself, then he doesn t (because he doesn t shave anyone who shaves himself). If he DOESN T shave himself, then he does (because he shaves everyone who doesn t shave himself). Seemingly, we have here a contradiction. This is a problem for set theory, because there is a similar contradiction for sets. Conceivably, some sets might contain themselves. (For instance, the set of all mathematical objects is itself a mathematical object.) And certainly, many sets do not contain themselves (e.g., the set of all prime numbers is not itself a prime number). But, now consider the set that matches the following description: The set of all and only those sets which don t contain themselves. Does this set contain itself? If it DOES contain itself, then it doesn t (because it doesn t contain any of the sets that contain themselves). If he DOESN T contain itself, then it does (because it contains all of the sets that don t contain themselves). So, here we have a contradictory set. Why is this a problem? Well, remember the Rabbits Out of Hats (Hat) sequent? This sequent stated that anything follows from a contradiction. Well, in mathematics, if there are contradictory sets, it turns out that we can prove anything whatsoever! 1
2 This is very bad. It means that set theory (and therefore mathematics) is inconsistent (that is, contains contradictions). In light of this, mathematicians scrambled to repair mathematics so that it did not allow for such contradictions. Gödel s Incompleteness Proof: Unfortunately, such selfreferential statements could be further used to show that no mathematical system could EVER be shown to be complete. To see why, begin by considering: This statement is a lie. (or, if you prefer: This statement is false.) This is known as The Liar s Paradox. The paradox is: If it s true, then it s false (because it s a lie). If it s false, then it s true (because it s not a lie). Now, note that mathematicians were searching for a mathematical system that was: (1) Complete: A complete mathematics is one in which all true statements can be proven. (2) Consistent: A consistent mathematics is one which contains no contradictions. Kurt Gödel demonstrated that any consistent mathematics is NECESSARILY incomplete by pointing out a statement similar in form to the Liar s Paradox. This statement is unprovable. (call this G for Gödel statement) The paradox: If it false, then the sentence IS provable. Unfortunately, then your system is inconsistent (because you can prove something that is false). If it s true, then it is unprovable. Unfortunately, now your system is incomplete (because you have a true statement that cannot be proven). Conclusion: Unless we want a mathematics that is inconsistent, then we must accept that G is TRUE. But, then, any consistent mathematics is necessarily incomplete. 1 1 Interesting side note: If G is unprovable for any consistent mathematical system and yet it is so obviously true to us perhaps the human mind is not a mathematical system? But, all computers are mathematical systems. So, is the obviousness of G a proof that the human mind is NOT itself (as many claim) just a sophisticated computer? Whoa For instance, see: Lucas, 1959 (here). An argument for the above might look like this: 1. No computing machine (i.e., mathematical system) can see or show that any Gstatement is true. 2. Humans can easily show and see that the above Gstatement is true. 3. Therefore, human minds are not computing machines. 2
3 2. Undecidability: Fortunately, both propositional logic and predicate logic are complete. That is, we can construct proofs for all theorems, and derivations for all valid sequents. Unforunately, predicate logic is undecidable. To understand this term, first consider this claim: Goldbach s Conjecture: Every even number is the sum of two primes. Here are a few examples of what Goldbach had in mind: 10 = ; 12 = ; 28 = ; 44 = Surely, Goldbach s Conjecture is either true or false. Right? But, which is it? We ve determined that this claim is true up into the hundreds of trillions but is it true for EVERY even number, to infinity??? We will never know. We are incapable of calculating out that far. If Goldbach s Conjecture turns out to be false, then maybe one day we will know that (for instance, if a computer finally finds an even number that is NOT the sum of two primes). However, if it is true, then we will never know it. So, on some definitions of the term, Goldbach s Conjecture is undecidable. But, here is a more rigorous definition of undecidability, for logical systems: Undecidability: A formal system of logic is undecidable if there is no mechanical method for determining whether a statement is a theorem or not, or if a sequent is valid or invalid. Now, Propositional Logic WAS decidable. Remember, we DID have a mechanical method for determining whether or not a statement was a tautology, and whether or not an argument was valid. We simply used truth tables. However, Predicate Logic is NOT decidable. (This is known as the ChurchTuring Thesis) Sure, if we CAN construct a derivation for some sequent, then we know that it is a valid one. However, what if we CANNOT construct a derivation? What then? Can we know for sure that the sequent is invalid? The best we can do is use the finiteuniverse method to determine its invalidity. But, failure to find a finiteuniverse interpretation on which an argument is invalid does not guarantee that it is valid! (It s always possible that you just haven t found the correct derivation or finiteuniverse counterexample yet.) 3
4 In short, there is no mechanical method for Predicate Logic (like the truth table method for Propositional Logic) that tells you with certainty whether a sequent is valid or invalid. Worse, if we can t figure out how to construct a derivation for some sequent (to show that it is valid), nor construct a finiteuniverse interpretation under which the sequent is shown to be invalid, then that sequent s validity or invalidity will remain a mystery. 3. Vagueness: Here are two common assumptions: (1) The Law of Excluded Middle: P P (2) The Transitivity of Identity: a=b, b=c Ⱶ a=c But, consider the following: The average human being has 100,000 hairs on their head. Surely, someone with 100,000 hairs is not bald. Now, pluck one hair. 99,999 hairs: Did you go from being notbald to bald? No, definitely not. Now, pluck another. 99,998 hairs: Did you go from being notbald to bald? No, definitely not. Now, pluck another. 99,997 hairs: Did you go from being notbald to bald? No, definitely not. Instead of continuing, let us suppose the following principle, which seems true: For any person who is not bald, plucking one hair from their head does not make them become bald. However, keep plucking hairs and apply this assumption every time. After plucking 100,000 times there will be no hairs left. Now, if no single pluck made you become bald, and transitivity is true, then even after the very last pluck you still weren t bald! But, that s clearly false. This paradox is known as The Sorites Paradox. It is a problem of vagueness. It seems that vagueness infects many of our terms (we could have gotten a similar paradox from the claim that adding a single grain of sand to something that is not a pile of sand does not make it become a pile; or that adding a single calorie to a candy bar that you re eating does not make you go from not fat to fat; or adding a penny to your bank account does not make you go from poor to rich; and so on). 4
5 What should we do about vagueness? Well, we could: (1) Deny vagueness. Perhaps there ARE sharp cutoff lines. For instance, maybe when you had 50,000 hairs you were not bald but when you plucked just ONE more to get 49,999 hairs, you suddenly became bald. (But, that seems crazy) (2) Accept vagueness. Maybe somewhere around the 40,000 to 60,000 hair range it is unclear whether or not someone is bald. They are neither bald nor not bald. But, then, B B would be FALSE for such a person. This is a denial of the law of excluded middle (which spells big trouble for logic). 4. Conclusion: Russell s Paradox, Godel s Incompleteness Proofs for mathematics, completeness proofs for Propositional and Predicate Logic, Goldbach s Conjecture, the ChurchTuring Thesis, vagueness, and transitivity (in addition to modal notions such as possibility and necessity, mentioned in the previous lecture) are just a few of the things you might learn about in an advanced logic course. Now, go forth and be logical! 5
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