Assignment J

Usually, if the axioms in a logistic system are all axiom schemas, we will have Substitution Rule. However, when we introduce a new axiom, like $p\supset\sim p$, to the system, Substitution Rule does not hold in general. Many students pointed out the flaw in line (2) of Argument B. But notice that we can say more about it. Thanks to the idea in Argument A, it would be interesting to have that actually $\mathcal{P^*}\nvdash p\supset\sim p\supset \sim .p\supset\sim p$.

Common Mistakes:

• Many students wrote that Argument A is incorrect since it uses the soundness of $\mathcal{P^*}$ as a hidden hypothesis. But actually, the proof only uses the definition of valuation on wffs.
• Some students correctly pointed out the problem of Substitution in Argument B, and they added that because $p$ appears in the formula that we want to replace with, namely $p\supset\sim p$, it is illegitimate to replace $p$ by $p\supset\sim p$. But basically, the occurrence of $p$ has nothing to do with substitution.