I'm currently reading 'A Modern Formal Logic Primer' by Paul Teller which features the following derivation as a solution to exercise 7-1 l:
I can mechanically follow this derivation but feel uncomfortable with steps 12-20 so I want to make sure that I understand it correctly.
The strategy here appears to be the deliberate introduction of a contradiction by assuming $(¬S\land-J) \land S$. As far as I can tell, one could use this strategy to derive $S \to Q$ where $Q$ stands for any statement.
If my interpretation is correct then this strategy appears to be subtly different from the inference rule called negation introduction (denoted $¬I$) which was previously introduced in the book. With negation introduction, one makes an assumption and derives a contradiction with the premises and concludes that the assumption must, therefore, be false: $((P \to Q) \land (P \to ¬Q)) \leftrightarrow ¬P$. But in the derivation above the contradiction seems to be there by design. Is this true and there is something else going on here or is my unease unjustified?
Best Answer
Yes. That is a typical use of the Rule of Explosion.
$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{\neg S}{\fitch{S}{\neg S\qquad\text{Reiteration}\\\bot\qquad~~\text{Negation Elimination}\\Q\qquad~~\text{Explosion}}\\S\to Q\quad~~~\text{Conditional Introduction}}$$
It is reasonably common in a Proof by Cases when one case leads to a contradiction, then whatever is derived in the other case is true.
It is always there by design - assumptions should be made with a goal in mind.