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Math Help - Logical Argument, valid or not.

  1. #1
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    Logical Argument, valid or not.

    Explicitly identify the component statements that are part of the argument and show the symbolic form of the argument.

    Decide whether the argument is valid or invalid. Here you may use truth tables, standard forms, Euler diagrams, and/or logical manipulation. If the argument is invalid, point to its logical fallacy.

    -------------------

    "It's impossible for pigs to understand logic, and it is also impossible for pigs to breathe underwater; so it must be the case that pigs neither breathe underwater nor understand logic."

    I have, P - Pigs to understand logic
    Q - Pigs to breathe underwater

    I am becoming familiar with LaTex so please bare with me until I become proficient at it. This is what I have for trying to represent it symbolically so far.

    (~P ^ ~Q) --> (~Q V P)

    I would assume it is valid, but My truth table skills are lacking. This is where I need help.

    My textbook does a very confusing job at explaining them to me in terms of logical arguments. The textbook I have is Mathematical Excursions the 2nd edition by Aufmann, Lockwood, etc.

    I am also assuming the word "impossible" means negating P and Q in the first statement, yes?
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  2. #2
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    One thing you must get accustomed to is the English can be a little ambiguous. There may be multiple solutions.


    In my view, "Neither A, nor B" means ~A ^ ~B = ~(A or B), using DeMorgan's Law
    Last edited by TKHunny; May 13th 2011 at 02:55 PM.
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  3. #3
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    Are you saying the first statement should read, ~(P ^ Q)?

    Sorry, having a hard time trying to grasp your post. What do you mean by the English can be a little ambiguous?
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  4. #4
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    Quote Originally Posted by stueycal View Post
    Are you saying the first statement should read, ~(P ^ Q)?

    Sorry, having a hard time trying to grasp your post. What do you mean by the English can be a little ambiguous?
    I have to agree with TKHunny that the 'consequent' of the English sentence is of the form \lnot(P \lor Q).

    When TKHunny says the English can be a little ambiguous, it means that English is not a formal language like Logic, so it can be interpreted differently by different people. In this case, it seems you interpreted "nor" more like how I would interpret the word "or."

    pigs neither breathe underwater nor understand logic
    So you interpreted the word "neither" as a negator on "pigs breath underwater" but not as a negator on "pigs understand logic." It seems that "neither" when paired with "nor" should imply a negation on each predicate (phrase) it is connected with. So:

    neither P nor Q should always mean:

    \lnot(P \lor Q)

    In other words, both P and Q are false. Or again in logic:

    \lnot{P} \land \lnot{Q}

    And that is (I think) the whole point of that exercise, to show you that these two ways of speaking are logically equivalent.

    So I think that solves your problem. There is another more subtle problem with this exercise that hinders its pedagogical objective. That being:

    While

    x\mbox{ is impossible}\Rightarrow x\mbox{ doesn't happen}

    is clearly true, the converse:

    x\mbox{ doesn't happen}\Rightarrow x\mbox{ is impossible}

    is not necessarily true.
    Last edited by Ontolog; May 13th 2011 at 12:00 PM.
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  5. #5
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    Sorry, having a hard time trying to grasp your post. What do you mean by the English can be a little ambiguous?
    Ha-ha-ha! You are confirming TKHunny's claim about English!

    Quote Originally Posted by TKHunny
    In my view, "Neither A, nor B" means ~A ^ ~B
    Quote Originally Posted by stueycal
    Are you saying the first statement should read, ~(P ^ Q)?
    There is no word "neither" in the first statement: "It's impossible for pigs to understand logic, and it is also impossible for pigs to breathe underwater." So, TKHunny's remark is about the second statement: "it must be the case that pigs neither breathe underwater nor understand logic." This statement can be reformulated as follows: "neither pigs breathe underwater, nor pigs understand logic." According to TKHunny's remark, this is the same as "pigs don't breathe underwater, and pigs don't understand logic," i.e., ~Q /\ ~P. So, the whole claim is ~P /\ ~Q -> ~Q /\ ~P.

    Actually, this exercise somewhat abuses modal constructions "impossible" and "must." If this is introductory logic, then "it is impossible that P" probably just means ~P. However, there are more sophisticated logics which make a distinction between "P is impossible" and "P is false," but this is beyond the scope of this question. I would just rather that introductory logic texts dealt with what is true/false rather than what is necessary/possible.
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  6. #6
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    ~P /\ ~Q -> ~Q /\ ~P

    The symbol "and" on the last statement, we are replacing the word "nor" with the word and, pretty much re stating the second statement? Is the above symbolic representation correct, or should I be using a V,

    And also, do I have to prove this with a truth table, or are the symbolic representations proof enough that it IS valid....right?
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  7. #7
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    "Neither Q, nor P" means ~Q /\ ~P, so yes, your representation is correct.

    And also, do I have to prove this with a truth table, or are the symbolic representations proof enough that it IS valid....right?
    The easiest way is to prove it with truth tables or Euler (Venn) diagrams. It is also possible to give a syntactic derivation, but you need to fix logical equivalences that will allow you to rewrite formulas in an equivalent way (such as ~P /\ ~Q <=> ~(P \/ Q)) or specific axioms and rules of inference.
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