• Nicholas Ferreira
    58
    Hello guys. I've been trying to prove a sentence for over a week and I really don't know what else to do. I have tried to do direct proof and reductio ad absurdum several times, but I always reach a dead end. So, since I am already exhausted from this, I would like to propose the challenge of proving that the sentence is true. There it is:
    ∴ "If exactly one being is F and exactly one being is G and nothing is F-and-G, then exactly two beings are F-or-G."
    ∴ (((∃x)(Fx∧¬(∃y)(y≠x∧Fy))∧(∃x)(Gx∧¬(∃y)(y≠x∧Gy)))∧¬(∃x)(Fx∧Gx)) ⊃ (∃x)(∃y)(((Fx∨Gx)∧(Fy∨Gy))∧(x≠y)∧¬(∃z)((z≠x∧z≠y)∧(Fz∨Gz))) 
    or
    ∴ (((∃x)(Fx • ∼(∃y)(∼y=x • Fy)) • (∃x)(Gx • ∼(∃y)(∼y=x • Gy))) • ∼(∃x)(Fx • Gx)) ⊃ (∃x)(∃y)(((Fx ∨ Gx) • (Fy ∨ Gy)) • (∼x=y • ∼(∃z)((∼z=x • ∼z=y) • (Fz ∨ Gz))))
    


    The first sentence is the english paraphrase of "1+1=2", and the second and third are the corresponding formula. Both the second and the third are exactly the same formula, they differ only in notation (the former uses "∧" for conjunction and "≠" to inequality, while the latter uses "•" and "∼=", and it's more spaced.).
    The challenge is to prove by natural deduction that this sentence is true. Good luck.

    (Oh, and it's not my homework or something, i'm just learning logic by myself. I took this example in the book "Introduction to Logic", by Harry Gensler (2ª edition: pg 207; 3ª edition: pg 294). I also sent him an e-mail asking for the solution, but he said he never tried to prove this, but that it's possible.)

  • Noah Te Stroete
    2.1k


    “1”, “+”, “=“, and “2” have specific meanings by convention. So, “1+1=2” is a tautology. It has to be true given the meanings of the terms used. There is nothing to prove.
  • wax
    301
    by true, it depends what you mean by 'true'.. :P

    We are complex beings in a complex mysterious universe, the workings of which we only have a vague clue.
    So we use these complex mysterious minds/brains to perceive and understand the question 'is 1+1=2 true'........people often try to prove it is true by showing a person gather a single apple and the place it next to another apple....and I think 'are you serious?'..have you any idea how complex an apple is?
    What exactly is an apple? It is the perception by a complex mind, in a complex universe of various components...we really can't be sure of the 'truth' of what is going on..are we supposed to build a model of reality, actually inside that reality, and compare the concept of adding two apples together and then counting the result, and the consider we have proved that 1 apple placed next to 1 apple leads to the conclusion that this process proves that 1+1=2 is a statement of truth somehow.

    And as Noah Te Stoete said, it is just a statement that doesn't mean anything outside of maths....it is no different than saying that 1=1...or 683=683.. :)
  • Nicholas Ferreira
    58
    Lol, what about interpretation? I'm not talking about mathematics foundings here neither about some philosophical trip. Actually, it doesn't matter if that sentence can really be equivalent 1+1=2 or not. The only thing that I wanted to propose is to prove the validity of that claim. If you want, you can forget what is written about 1 + 1 being equal to 2, whatever.
    Oh, and where did you get that tautology should not be proven? One thing is a self-evident tautology like 1=1 or p->p; another thing is something like "(((∃x)(Fx∧¬(∃y)(y≠x∧Fy))∧(∃x)(Gx∧¬(∃y)(y≠x∧Gy)))∧¬(∃x)(Fx∧Gx)) ⊃ (∃x)(∃y)(((Fx∨Gx)∧(Fy∨Gy))∧(x≠y)∧¬(∃z)((z≠x∧z≠y)∧(Fz∨Gz)))", which you can't trivially say whether it's a valid inference or not only by reading it's terms and connectives.
  • Terrapin Station
    11.7k
    I think it's worth (1) asking why there would be any need to prove this, and (2) analyzing just what it is that we're doing when we're constructing proofs in the first place.
  • Nicholas Ferreira
    58
    Do you ask your teacher the reason to prove something on a test? Man, it's a challenge, I'm assuming that people who frequent this section of logic and math philosophy know what it means to prove something, to show that a statement entails another, using inference rules, as I said. Sorry, but I really don't know whats the difficulty to understand that.
  • Terrapin Station
    11.7k


    You don't think either is worthwhile. Okay. So what am I supposed to do with that information now?
  • Nicholas Ferreira
    58
    I don't even know what are you talking about. I'm leaving the post for those who want to test, and possibly improve, their logical capacity. Thanks for answering.
  • Noah Te Stroete
    2.1k
    Oh, and where did you get that tautology should not be proven?Nicholas Ferreira

    Would you seek to prove this tautology: “A bachelor is an unmarried man”?

    I suppose you could try, but why?

    Why not try to prove something that isn’t self-evident instead?
  • Nicholas Ferreira
    58
    Well, I guess you are confunding logic tautologies with linguistic tautologies. In logic, a tautology is a formula, or a truth-function, that returns always "true" to whatever interpretation you give to the variables. That is, to any possible combination of the truth-value of the variables, the tautology is always true (opose to the contradiction, which is always false to any combination). Linguistic tautologies are just redundant statements that are true due the meaning of the terms and their internal relations. Your example isn't true if I define a bachelor as being an apple (I guess apples aren't unmarried man). On the other hand, the formula ((P⊃Q)∧¬Q)⊃¬Q), for instance, is always true, regardless what you attribute to "P" and "Q", and you easily can verify it's a tautology with a truth table.
  • Noah Te Stroete
    2.1k
    “1” and “2” are not variables, and “1+1=2” is basically a linguistic tautology.
  • Nicholas Ferreira
    58
    But you example was about bachelors. Anyway, it doesn't matter, the logical formula in question have variables, and I still not understanding the reason of the difficulty to understand what was proposed...
  • Noah Te Stroete
    2.1k


    My objection was that self-evident equations shouldn’t have to be proven (given the meanings of the terms used). Why not prove something interesting?
  • Nicholas Ferreira
    58
    Are you really saying that "(((∃x)(Fx • ∼(∃y)(∼y=x • Fy)) • (∃x)(Gx • ∼(∃y)(∼y=x • Gy))) • ∼(∃x)(Fx • Gx)) ⊃ (∃x)(∃y)(((Fx ∨ Gx) • (Fy ∨ Gy)) • (∼x=y • ∼(∃z)((∼z=x • ∼z=y) • (Fz ∨ Gz))))" is self-evident?
  • Noah Te Stroete
    2.1k


    I’m saying that “1” has a meaning. “+” has a meaning. “=“ has a meaning. “2” has a meaning. Given these meanings, “1+1=2” must be true all the time. You’re making something simple more complicated than it really is. What kind of nut goes about “proving” something that every child can show with apples as @wax suggested?
  • sime
    378
    The conclusion "exactly two beings are F-or-G" does not follow from

    "exactly one being is F and exactly one being is G and nothing is F-and-G"

    Because the conclusion permits (x or y) or (x & y) to be F & G.
  • Nicholas Ferreira
    58
    Are you reading anything i'm writing? I said at least twice that the "1+1=2" is absolutelly irrelevant to what i'm proposing, it's just a detail. The proposal is to prove that the sentence "(((∃x)(Fx • ∼(∃y)(∼y=x • Fy)) • (∃x)(Gx • ∼(∃y)(∼y=x • Gy))) • ∼(∃x)(Fx • Gx)) ⊃ (∃x)(∃y)(((Fx ∨ Gx) • (Fy ∨ Gy)) • (∼x=y • ∼(∃z)((∼z=x • ∼z=y) • (Fz ∨ Gz))))" is true, valid, call it what you want. I'll not repeat, and I'll not reply to this kind of comment again.
  • Noah Te Stroete
    2.1k


    How is it irrelevant when it is in the topic title?
  • Terrapin Station
    11.7k
    The first sentence is the english paraphrase of "1+1=2",Nicholas Ferreira

    I brushed over that the first time I read your post. That's an English paraphrase of 1=1=2 according to whom? It certainly bears no resemblance to anything at all that I think when I think about "1=1=2"
  • Nicholas Ferreira
    58
    If you had read what I said you would notice that the title isn't anything than a flashy title.
    Why does the conclusion permits "(x or y) or (x & y) to be F & G" if it is said that nothing is simultaneously F and G?
    The reference is in the text, just read. And it's not 1=1=2, it's 1+1=2.
  • sime
    378
    Why does the conclusion permits "(x or y) or (x & y) to be F & G" if it is said that nothing is simultaneously F and G?Nicholas Ferreira

    Your stated conclusion

    (∃x)(∃y)(((Fx∨Gx)∧(Fy∨Gy))∧(x≠y)∧¬(∃z)((z≠x∧z≠y)∧(Fz∨Gz)))

    does not say that X or Y cannot simultaneously be F and G.

    Therefore your conclusion is weaker than your premise.
  • Terrapin Station
    11.7k
    Typo--I'm trying to get used to a new keyboard. I guess I wasn't hitting the shift key right. :wink:

    Anyway, so I guess I'd need to ask Mr. Gensler what the heck he's talking about re that being an English translation.
  • Nicholas Ferreira
    58
    Wtf are you doing on this forum?
    This is said in the antecedent, not in the conclusion. Maybe this image can help the visualization.
    -bl8zTUxQveRz8kFID5gmQ.png
    Well, I guess it can be taken as a paraphrase that is an interpretation of the mathematical assertion. I really don't know, and I really don't care if it's really 1+1=2, I just wanted to see if someone could prove the validity. Anyway, he's e-mail is . He usually answer quickly, so please let us know if you get an answer.
  • sime
    378
    sime
    This is said in the antecedent, not in the conclusion
    Nicholas Ferreira

    Correct, hence your conclusion permits a possibility denied by your premise.
  • Nicholas Ferreira
    58
    But why would the conclusion need to explicit something that already has been said in the premise? I mean, in "P⊃Q", for instance, if you analyze only the consequent, you'll see that "Q" permits "¬P", which is denied by the antecedent. I'm not quite sure what kind of analysis are you doing but I think that analyzing only the conclusion without considering what was stated in the premise isn't the right way.
  • fdrake
    2.5k
    ITT people try to convince an OP explicitly asking for help with a natural deduction proof in a specified system that it isn't worth the bother.



    ∴ (((∃x)(Fx • ∼(∃y)(∼y=x • Fy)) • (∃x)(Gx • ∼(∃y)(∼y=x • Gy))) • ∼(∃x)(Fx • Gx)) ⊃ (∃x)(∃y)(((Fx ∨ Gx) • (Fy ∨ Gy)) • (∼x=y • ∼(∃z)((∼z=x • ∼z=y) • (Fz ∨ Gz))))


    Only thing I can offer is that if the author didn't attempt it but knows that it is true, maybe the important thing to cultivate in the exercise is an intuition for why it must follow somehow. I imagine you already have this intuition - F is an exclusive property of x, G is an exclusive property of y, for some x Fx, for some y Gy; ie only x if F and only y is G. The only way for there to be a z such that Fz or Gz is if z=x or z=y, and the only way to be both is z=y=x. By eliminating the x=y case, you force the implication that (such a z exists implies z=x or z=y) which by the exclusivity of F and G we know can't be the case. Since we've exhausted the only way such a z can exist and it lead to a contradiction, no such z exists.
  • sime
    378
    But why would the conclusion need to explicit something that already has been said in the premise? I mean, in "P⊃Q", for instance, if you analyze only the consequent, you'll see that "Q" permits "¬P", which is denied by the antecedent. I'm not quite sure what kind of analysis are you doing but I think that analyzing only the conclusion without considering what was stated in the premise isn't the right way.Nicholas Ferreira

    P⊃Q doesn't permit Q⊃¬P in a consistent logic. That case is different to the set-theoretic case, where Fx ∨ Gx permits Fx ∧ Gx and is therefore a weaker statement than the latter.

    Of course, in a sense your antecedent might be said to contain your "conclusion" as a weaker premise, but i think it is a mistake to think of your right-hand side as a conclusion because it must forever remain tied to the antecedent if it isn't to be misinterpreted as allowing F and G to be overlapping sets containing multiple members... assuming of course, that you want to represent the number 2 as a union of pairwise disjoint singleton sets.
  • Michael Ossipoff
    1.7k


    Do you think you can prove that 1+1=2?

    No. That's the definition of 2.

    ...at least as I'd say it.

    The positive integers can be defined by repeated addition of the multiplicative identity (1).

    Such things as 2 + 2 = 4 can be proved by the additive associative axiom.

    Michael Ossipoff

    10 F
  • Nicholas Ferreira
    58
    P⊃Q doesn't permit Q⊃¬P in a consistent logicsime
    Of course, but you were saying about the consequent only, not about the entire implication. That is why I said that in "P⊃Q", "Q", alone, permits "¬Q".

    That case is different to the set-theoretic case, where Fx ∨ Gx permits Fx ∧ Gx and is therefore a weaker statement than the latter.sime
    Hm, I don't know if I understood. For instance, in the sentence "(¬(∃x)(Fx∧Gx) ∧ Fx) ⊃ (Fx∨Gx)", you would say that the consequent "Fx∨Gx" permits "Fx∧Gx"? (It's just an example for me to understand, I'm not saying this is the case)

    Of course, in a sense your antecedent might be said to contain your "conclusion" as a weaker premise, but i think it is a mistake to think of your right-hand side as a conclusion because it must forever remain tied to the antecedent if it isn't to be misinterpreted as allowing F and G to be overlapping sets containing multiple members... assuming of course, that you want to represent the number 2 as a union of pairwise disjoint singleton sets.sime
    Well, why couldn't we treat it like so? I mean, I could say that the antecedent is the premise and the consequent is the conclusion, and since the conclusion follows from the premise, I could represent they in a conditional statement. I didn't understand part of your latter paragraph... English isn't my native language and i'm not familliar with a lot of terms you used.
  • tim wood
    2.9k
    The - one - way to prove 1+1=2 is to create a system in which 1+1=2. That is, in your system 1+1=2 follows from the rules and the axioms of your system. Then you show that your system is the same as arithmetic; I think the word is isomorphic. Then it follows that your proof in your system carries over into arithmetic, thus proving that 1+1=2.
  • andrewk
    2.1k
    “1”, “+”, “=“, and “2” have specific meanings by convention. So, “1+1=2” is a tautology. It has to be true given the meanings of the terms used. There is nothing to prove.Noah Te Stroete
    The statement of "1+1=2" in Peano arithmetic is:

    S0 + S0 = SS0

    Because "1" means S0 and "2" means SS0. S is the 'successor' function.

    That is not a tautology. It has to be proved. The proof, as I recall, is not long.
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