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ln2-page2 - The!O means that we arrived at this line by...

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The “ ! O” means that we arrived at this line by means of modus ponens . We’ll learn abbreviations for a number of different derivation steps as we go along. We still have not gotten to our conclusion. Now that we have line 4, and keeping in mind what we had in line 2, we can perform another modus ponens . 1. 2. 3. 4. 5. A ! ( A ! B ) A SHOW: B A ! B B Pr Pr 1,2 ! O 2,4 ! O Again we write “ ! O” on the right because this was another modus ponens , but notice now we made use of lines 2 and 4 in our reasoning instead of 1 and 2. In doing a derivation, every new line we introduce should be “justified” by some of the earlier lines. However, we can never make use of SHOW: lines in justifying a new line. In this case, it was OK for us to appeal to 1 and 2, and to appeal to 4 once we had shown it, but we couldn’t have appealed to line 3, since that is what we were trying to prove! What we have at 5, B , is what we were trying to show at line 3. (QED, or “ Quod erat demonstrandum ” as they say in Latin.) Therefore, we have done what we were trying to do. In
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