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2. <math>(\varphi\rightarrow(\psi\rightarrow\chi))\rightarrow((\varphi\rightarrow\psi)\rightarrow(\varphi\rightarrow\chi))</math> | 2. <math>(\varphi\rightarrow(\psi\rightarrow\chi))\rightarrow((\varphi\rightarrow\psi)\rightarrow(\varphi\rightarrow\chi))</math> | ||
| − | 3. <math>(\lnot\varphi\rightarrow\lnot\psi)\rightarrow(\psi\rightarrow\varphi | + | 3. <math>(\lnot\varphi\rightarrow\lnot\psi)\rightarrow(\psi\rightarrow\varphi )</math> |
Revision as of 05:43, 5 September 2013
Independence of Axioms of Propositional Logic
Chenkai Wang
It is an interesting question that whether some mathematical statement(s) can or cannot prove some other statement. To show a statement is provable, simply present a proof. But is there a way to show a statement is unprovable? For example, the Euclid's fifth postulate turned out to be unprovable from other postulates. Because one can find a geometry where all Euclid's postulates are ture (including the fifth) and another where all but the fifth are true. Since proof preserves truth one can say the fifth is unprovable. I am going to demonstrate a simple technique of showing the three axioms for propositional logic is independent from each other.
The axioms for proposition logic are
1. <math\varphi\rightarrow(\psi\rightarrow\varphi)</math>
2. $ (\varphi\rightarrow(\psi\rightarrow\chi))\rightarrow((\varphi\rightarrow\psi)\rightarrow(\varphi\rightarrow\chi)) $
3. $ (\lnot\varphi\rightarrow\lnot\psi)\rightarrow(\psi\rightarrow\varphi ) $
