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Pure Objectivity

  1. Mar 13, 2005 #1
    Pure Objectivity... possible or impossible?

    Sorry, but this post will be unmercifully long, so if you have a short attention span, you may want to skip this one. :wink:
    Last Thurs., I asked my Logic professor about a certain fascinating property of PL (it either stands for Propositional Logic or Provisional Logic, I can't seem to remember :redface:) that I cannot seem to get my mind around: for any deductively valid argument phrased in PL, there is an infinite number of correct, valid conclusions. This can be verified by constructing logical proofs, which allow you to make any number of disjunctions (using the vi rule, called "wedge in") for the conclusion of any valid argument. Here is where my "big picture" question lies. If there are infinitely many correct solutions to any logical argument, how can one be objective in arriving at conclusions, or is it even possible to be objective? The way I see it, no matter which conclusion you arrive at for an argument, that conclusion was one out of an infinite number of other valid conclusions. In addition, no matter how many valid conclusions you find for some argument, you will be no closer to objectivity than you were upon reaching the first conclusion, because there ahead of you stand an infinite number of equally correct conclusions that you did not choose to use, meaning that in choosing one out of that infinite number, you are willingly or unwillingly being subjective.
    Well, maybe this is an unimportant and/or insignificant observation of logical form, which I would be more than willing to accept. But I figured that it is possible for some weight lie within this idea, so I posted :smile: . To be honest with you though, I'm not certain.
    My professor's response was basically: I'll get back to you on that one.

    He did mention that parameters can be set for arguments which filter out most of the alternate conclusions, similar in my view to bounds set for polar equations when they have an unyielding number of solutions or possible steps. Also, when contextually speaking or using NL, there will almost always be subjectivity in the conclusion of an argument since it deals directly with the relevant premises of that argument.
    It's just that within PL, which is used to symbolically represent all possible NL statements through atomic and molecular formulas, (ex. "C" in the formula (-W&C) can represent "Andrew hiked through the Grand Canyon" or "Jessica is a member of the Rebulican Party", it doesn't matter) it seems impossible to arrive at a conclusion for a specified argument without personally wanting to find that conclusion. If this is so, how can we logically arrive at unbiased conclusions? Sure, I have heard about logical efficiency and choosing valid conclusions based upon aesthetic need or simplicity, but still... why say that one conclusion is "better" than another if they are both equally valid?
     
    Last edited: Mar 13, 2005
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  3. Mar 13, 2005 #2

    honestrosewater

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    Propositional Logic makes sense. I've never heard of Provisional Logic, and google didn't turn up much. Anywho...
    I think I know what you mean, but an argument has only one conclusion. I think you mean that if P is in your set of premises, you can conclude (P v Q). Of course, if P is in you set of premises, you can also add (P v Q) to your set of premises (this rule is also called "Addition").
    BTW, the converse isn't a tautology. You cannot infer P from only (P v Q) since you don't know which proposition- P or Q (or both)- is true.
    I don't understand how you're tieing objectivity into this.? You don't really choose conclusions; Conclusions are entailed by a set of premises. You may have some proposition in mind that you want to prove, but it must still be entailed by the set of premises.
    When you infer (P v Q) from P, you aren't saying anything about the truth-value of Q. If P is true, you can infer (P v Q) precisely because it doesn't matter whether Q is true or false; Q can be any proposition [Edit: Actually, I should say "propositional form"], even an arbitrarily large compound, even a contradiction- it doesn't matter- the disjunction will still be true if P is true. Also, just because you can infer an infinite number of propositions by addition doesn't mean you can infer any proposition whatsoever; The proposition must be a disjunction with at least one true disjunct.
    The only way to rule out (P v Q) is by being given ~(P v Q) in some form or another.
    I don't understand what you're getting at. The validity of an argument doesn't depend on what you personally want to conclude. Could you give an example of an argument you have this problem with?
    As far as logic is concerned, good, bad, ugly, simple, and such don't apply to conclusions. Bias doesn't enter the picture unless the rules allow it to. (The rules I know don't allow it to.)
     
    Last edited: Mar 13, 2005
  4. Mar 13, 2005 #3
    Okay, an example could be: (P&W) THUS ((P&W)vZ)
    This same argument could have the conclusion: (((P&W)vZ)vR)
    While each of these conclusions are equally valid, and the truth value of (P&W) does not depend on Z and R, the point is this: to arrive at the conclusions I arrived at, I chose to include Z and R, they were chosen. I could have added another few hundred disjunctions or more to the conclusion of this argument. So by including or not including v in my conclusion, I am making the choice of whether or not to include the possibility of "or". I know that it exists independantly of whether or not I choose to include it (as there will always be another recursive "or" possible), but I just don't like the way it leaves such an indefinite number of ways to express the same thing. I makes logical sense, I just don't think I'm comfortable with the idea yet. Although honestrosewater, you're reply did help.
    -(Y&Q) THUS ([-(Y&Q)v-T]vU)
    -(Y&Q) THUS (((-(Y&Q)vK)v-T)vS) are both equally valid, right?

    Like I said, if this is insignificant, that's fine with me. I just want to make sure that I am understanding this correctly. By choosing to use either of these conclusions, am I making a subjective choice, despite the fact that they are essentially equivalent? Thx.
     
  5. Mar 14, 2005 #4

    honestrosewater

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    I think that if it's significant to you, it's significant enough. The quick "solution": Yes, if -(Y&Q) is true, then ([-(Y&Q)v-T]vU) and (((-(Y&Q)vK)v-T)vS) have the same truth-value, namely, truth. But I don't think this solves your problem; It's rather just treating the symptoms. However, I'm not sure exactly what's causing the problem so I'll just run through some things and hope for the best. I'll err on the side of caution and risk telling you things you already know.
    [P -> (P v Q)] is a tautology (it's always true- try a truth table if you aren't sure*see below). Here, "P" and "Q" are propositional forms, meaning you can substitute any proposition, atomic or compound, for them (as always, as long as you substitute them consistently). For instance, [(M & N) -> ((M & N) v (Z -> (~M v Y)))] is a possible substitution. You can use this to check your examples.
    Do you see when addition is useful and when it's useless? Say I want to prove the following:
    A. R, therefore S -> R.
    This is very easy, using addition.
    1. R [premise]
    2. R v ~S [1, Addition]
    3. ~S v R [2, Commutation]
    4. S -> R [3, Implication/QED]
    Of course, I could have added other steps and inferred (R v S) or (R v (Z -> (~K & P)) or (R v R) or (R v ~R) or so on, but none of these would have helped me reach my conclusion.
    Does it bother you that you can represent a number in an infinite number of ways? For instance, you can represent the number zero as "0" or "0 + 0" or "1 - 1" or "2 * 5 - 12 + 2" or so on. Similarly, I can assert that some proposition P is true by writing "P" or "~~P" or "~~~~P" or so on. I'm sure you can think of several other examples. By choosing to write "P" instead of "~~~~~~~~~~~~~~P", you are making a subjective choice, if you want to call it that, but that choice has nothing to do with logic; Those two propositions are logically equivalent.
    In the case of addition, you're making another choice: You're (possibly) introducing new atomic propositions (the "possibly" is there since, from P, you can infer (P v P) or (P v ~P) or the like). For instance, when I introduced "~S" by addition in the above proof, I chose "~S" for a reason, but I could have chosen any proposition whatsoever. If this is what bothers you, I'm not sure what more to say. Your conclusion only says what's already contained, in some way, in your premises; Your conclusion makes the implicit explicit. I can infer an infinite number of propositions from just P, but they all say the same thing- that P is true; They just say it in different ways. Has your book or teacher dealt with language, meaning, syntax, semantics, interpretations, structures, etc. yet?
    Eh, I don't really feel like I'm helping you. If there's still something bothering you, just say so. If you want to get comfortable with a rule, grab a reference containing all the rules (just in case), and just work through problems. If your book doesn't have enough problems, let me know, and I'll find some.

    *Edit: Actually, no, if you aren't sure, think it through, and try to understand it. You should be able to construct the truth table anyway. If P is false, [P -> (P v Q)] is true, since a conditional is true whenever its antecedent is false. If P is true, [P -> (P v Q)] is true, since the consequent is a disjunction with at least one true disjunct, namely, P. Notice I didn't even have to consider the truth-value of Q. :wink:
     
    Last edited: Mar 14, 2005
  6. Mar 14, 2005 #5
    Well, I do see what you're saying now. The use of propositons and searching for validity/ logical equivalence is what really turns the implicit into explicit form. So unless I have a specific conclusion in mind for a premise, (Y->(-T&B)) for example, the conclusion itself can be anything which obeys the rules of logical form since the point is to test specific conclusions for validity, not to exhaust all possible forms of the conclusion.
    This is more helpful... While I may subjectively choose which conclusion to use for an explicit argument with given premises, that doesn't negate the validity or logical equivalence of all its other forms. I had an abstact thought that just needed a little clarification, but I think that now it's as clear as can be, so thanks.
     
  7. Mar 15, 2005 #6

    honestrosewater

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    My pleasure. I would just add that you should be careful when you use "valid" and "consistent"; Read their definitions; They are not interchangeable.
     
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