The “!O” means that we arrived at this line by means ofmodus ponens.We’ll learn abbreviations for a number of different derivationsteps as we go along.We still have not gotten to our conclusion.Now that we have line 4, and keeping in mind what we hadin line 2, we can perform anothermodus ponens.184.108.40.206.5.A!(A!B)ASHOW:BA!BBPrPr1,2!O2,4!OAgain we write “!O” on the right because this was anothermodus ponens, but notice now we made use of lines 2 and 4in 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 cannever 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 toappeal to 4 once we had shown it, but we couldn’t haveappealed 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 atline 3. (QED, or “Quod erat demonstrandum” as they say inLatin.)Therefore, we have done what we were trying to do. In
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