Maybe This Ios Okay

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  • 7/24/2019 Maybe This Ios Okay

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    As we know from the truth table for conjunction, a conjunction P & Q is true just in case each of its conjuncts is true. If we know that two formulae P and Q are true, then we can validly infer from this that their conjunction P & Q is alsotrue. This is exactly the pattern of inference that the conjunction introduction rule, or &I, captures.We represent it as follows:

    a. Pb. Q.. ...d. P & Q &I: a, bBefore going on to some examples, we should point out a few conventions that we're using in the representation of the rules of inference. First, you'll note that we've used letters rather than numbers to distinguish the different lines in the representation of the rule. This is because it doesn't matter what lines therule is applied to in an actual derivation, though it does matter that the linesto which the rule is applied are properly cited on the line of the inference made. Second, you'll notice that the order in which we cite the lines matches theorder in which the conjuncts appear in the inference line. If we wanted to derive Q & P, we'd have to cite lines a and b in the opposite order, i.e., b and a. Finally, you should note that the formulae appearing on the lines are not actually sentential letters, which is why we have formatted them differently. They areactually variables ranging over all well-formed formulae of sentential logic. The reason we use variables in the representation of rules rather than sententialletters is simply because the rules can be applied to any lines in a derivation,

    not just those with a sentential letter as its formula.

    All that being said, we should go through a few examples.

    Consider the following impossible conversation:

    Godel Godel: A bird in the hand is worth two in the bush.Kant Kant: A stitch in time saves nine.Socrates Socrates: Well then my friends, it must be that a bird in the hand is worth two in the bush and that a stitch in time saves nine.Symbolising this as an argument in standard form, we get the following:

    B

    S? B & SSo, we now proceed to perform a derivation in order to show that this argument is valid. We start out with our premises:

    1. B Prem2. S PremSince the argument itself is really just an instance of the pattern of inferencecaptured by &I itself, we only need to add one line to the derivation to reachthe conclusion:

    1. B Prem2. S Prem

    3. B & S &I: 1, 2So much for that trivial example. How about something a bit more interesting? Here we'll just skip ahead to the point where we have our premises as lines in ourderivation, so we'll specify the conclusion we want to derive.