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Logic

Hi, I'm trying to work through Hodges' book on Logic atm - I would really appreciate some help on a question I can't do.

The question is: (p103 ex 23C Q2)

Use a truth-table which of the following semantic sequents are correct; indicate a conterexample to each incorrect one.

..
2. P I= [Q>P]


(Sorry, I can't work out how to type logic symbols on the computer. I= is meant to be what Hodges calls the 'semantic turnstile' and > an arrow :o: )


This was my answer:

P Q P I= Q>P
T T T .......T
T F T....... F
F T F ......T
F F F........T

And so it's incorrect, because there is a structure - when P is true and Q is false - where P and Q>P are defined but P is true and Q>P is false. (And that's how he tells us to approach questions like this...)

But in the back he says the sequent is correct and he has a truth table like this:

P Q P I= Q>P
T T T .......T
T F T....... T
F T F ......F
F F F........T

Help!

Thank you anyone who managed to understand my muddled thinking..

Anna
I don't have anything to add other than I don't know and I hate logic :frown: Maybe if you posted in the uni philosophy forum you'd get more of a response?
Reply 2
Mods, could this be moved to the univeristy philosophy forum? Thanks
Yeah, your truth table is wrong. If you look in the back of Hodges you will find the tableaux rule for the arrow. It states that QPQ\rightarrow P is true whenever the antecedent (QQ) is false or the consequent (PP) is true (or both). The only case in which it can be false is when the antecedent (Q) is true, and the consequent (P) is false. In your truth table you have marked the sentence false when the consequent is true, and marked the sentence true when the consequent is false and the antecedent true.

This makes sense if you think about what the arrow means. QPQ\rightarrow P means "If Q, then P", or "Q entails P".
Reply 4
Rustlessbowl
Yeah, your truth table is wrong. If you look in the back of Hodges you will find the tableaux rule for the arrow. It states that QPQ\rightarrow P is true whenever the antecedent (QQ) is false or the consequent (PP) is true (or both). The only case in which it can be false is when the antecedent (Q) is true, and the consequent (P) is false. In your truth table you have marked the sentence false when the consequent is true, and marked the sentence true when the consequent is false and the antecedent true.

This makes sense if you think about what the arrow means. QPQ\rightarrow P means "If Q, then P", or "Q entails P".


Oh yes, thanks! :smile:

I kept reading it as P>Q, not the other way round :o: oops :rolleyes: