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I'm beginning to study logic and was reading Graham Priest's Logic: a Very Short Introduction, and he includes this inference:

$q, \lnot q / p$.

He then provides a truth table. But, for the life of me, I can't figure out what he's doing (he doesn't state it, which makes me feel even dumber) The final configuration of Ts and Fs for the final 'p' is (from top to bottom) TFTF, with the initial truth-value reference table for q and p being (naturally):

q: TTFF p: TFTF

Whence does he derive that particular configuration for the final 'p'? Is it that because the premise states 'q' and then negates it, it says nothing and therefore the truth values for p stay the same? I know I could have asked this more clearly, but I'm a little desperate

Mauro ALLEGRANZA
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FredFrege
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    I am a bit confused by your notation. Presumably ${\sim}q$ is the negation of $q$, but what does ${\sim}q / p$ mean? – Xander Henderson Apr 10 '18 at 18:17
  • It means that p is the conclusion of the premises 'q, ~q'. It means that 'q, ~q / p' is a syllogism in which 'p' is the conclusion. – FredFrege Apr 10 '18 at 18:28
  • See Principle-of-explosion](https://math.stackexchange.com/questions/1120251/principle-of-explosion-other-arguments). – Mauro ALLEGRANZA Apr 10 '18 at 18:32
  • Thank you, Mauro. I did know what the truth table looked like. I just couldn't find a way to reproduce it in the question, but thanks for finding that link, and for answering! – FredFrege Apr 10 '18 at 20:10
  • HINT: Try producing some truth-tables for some more 'normal arguments, just to get a sense for them, see how they work and what, most importantly, you are looking for when the table is filed out. In particular, note how a valid argument is one where, for each row where all the premises are true, the conclusion is also true. It's just that in this particular, there are no rows with all true premises, making it vacuously true that for all (zero) rows with all true premises, the conclusion is also true. So that's why it is valid. – Bram28 Apr 10 '18 at 21:10

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Please, check again; the truth table has three columns :

$q$ : TTFF | $¬q$ : FFTT | $p$ : TFTF,

because we have an argument with two premises and a conclusion.

We have two variables ($p$ and $q$) and thus we need $2^2=4$ rows.

We have to check that in every line of the truth table where all premises have T also the conclusion has T.

See Tautological consequence.

Mauro ALLEGRANZA
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