The principle of explosion as a rule of inference?

Question

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?

Answer 1

The strategy here appears to be the deliberate introduction of a contradiction by assuming $(¬S∧−J)∧S$. As far as I can tell, one could use this strategy to derive $S→Q$ where $Q$ stands for any statement.

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.

But in the derivation above the contradiction seems to be there by design.

It is always there by design - assumptions should be made with a goal in mind.

Answer 2

I am quite familiar with Natural Deduction, but this version of it has some different notation that I'm unfamiliar with, so I am not 100% sure of what I'm writing is correct.

It looks like the strategy is to eliminate the disjunction. This by introducing two implication with each has the same consequent. You can see that in lines 11 and 29. In my tradition of writing Natural deduction you would write after line 11 -

11.5 $(s \wedge j) \rightarrow ( s \leftrightarrow j )$

and also after line 29

29.5 $(\neg s \wedge \neg j) \rightarrow (s \leftrightarrow j )$

and so line 30 would read

30. $ s \leftrightarrow j $ E $\vee 1, 11.5, 29.5$

Answer 3

As shown, the -I rule is {$\alpha$ ... $\beta$, $\lnot$$\beta$} $\vdash$ $\lnot$$\alpha$, where $\lnot$$\alpha$ has one less scope than $\alpha$. The notation given by wikipedia isn't correct, since it uses a conjunction, and you won't find an actual conjunction symbol when wikipedia's negation introduction gets used. It also goes one way (though the equivalence does hold). It should get notated something like {($\alpha$$\rightarrow$$\beta$), ($\alpha$$\rightarrow$$\lnot$$\beta)$} $\vdash$ $\lnot$$\alpha$. That rule of inference does NOT involve any change in scope, and requires two implications, while the above -I rule does not require any change in scope.

There's nothing invalid about either rule.

Yes, the contradiciton gets there by design. But, I'm not so sure if you should be unease or not. It depends on how the rule got stated before the (purported) proof. If it's consistent with the description of the rule, it's fine. If it didn't get described that way, then you should be uneasy, as what gets written doesn't follow the rules of inference.

Really, you probably should feel uncomfortable with the whole thing, because the definition of a formal proof needs to say that every step consists of a well-formed formula (or an equivalent term). But J>S is not well-formed and other strings are also not well-formed, and consequently there is no formal proof above.