# Why can we use multiple rules of inference?

In developing the argument where the premises $$p$$ and $$r$$ lead to conclusion $$t$$, we can use a rule of inference on $$p$$ such that $$p\implies q$$ and one on $$r$$ such that $$r\implies s$$, after which we can apply another rule of inference on $$q$$ and $$s$$ such that $$q \wedge s\implies t$$, thus showing that $$p\wedge r\implies t$$.

For example, I was reading the following example:

1. $$\exists x(A(x)\wedge \neg B(x))$$ [premise $$p$$]
2. $$A(c)\wedge \neg B(c)$$ [existential instantiation from $$p$$]
3. $$A(c)$$ [simplification from the above; this is statement $$q$$]
4. $$\forall x(A(x)\rightarrow C(x))$$ [premise $$r$$]
5. $$A(c)\rightarrow C(c)$$ [universal instantiation from $$r$$; this is $$s$$]
6. $$C(c)$$ [modus ponens on $$q$$ and $$s$$; this is $$t$$]

where steps 1 to 3 establish $$p\implies q$$, steps 4 to 5 establish $$r\implies s$$, and step 6 is $$q\wedge s \implies t$$. Intuitively, I can understand why the example works. This feels like a double case of hypothetical syllogism (two chains of implications converging at the modus ponens step), but anyhow I can't justify rigorously through the use of symbols. I've tried $$(p\rightarrow q)\wedge (r \rightarrow s) \wedge (q\wedge s \rightarrow t)\Leftrightarrow(p\wedge r \rightarrow t)$$

but I can't seem to get the manipulations to work. Any guidance will be much appreciated.

The argument needs quantification rules; with only propositional logic we cannot show its validity.

The fact is that there is a "link" between $$p \to q$$, where $$q$$ is $$A(c)$$ and $$r \to t$$ via $$s$$, that is $$A(c) \to C(c)$$.

Without using predicate logic to "see inside" the formulas, the argument :

$$(p \to q) \land (r \to s) \vDash (p \land r) \to t$$

is not valid.

The argument is :

1) $$∃x(A(x) ∧ ¬B(x)) \vdash A(c)$$ --- using Existential instantiation

2) $$∀x(A(x) → C(x)) \vdash A(c) → C(c)$$ --- using Universal instantiation

Thus :

3) $$∃x(A(x) ∧ ¬B(x)), ∀x(A(x) → C(x)) \vdash C(c)$$ --- by modus ponens.

In conclusion, the issue is not "why can we use multiple rules of inference ?" but "why we have to".