# Should I perform a disjunctive syllogism directly on three expressions simultaneously?

Consider

\begin{align*} &1.\quad \lnot R \lor \lnot T \lor U\\ &2.\quad R\\ &3.\quad T \end{align*}

It seems clear that you can end up with this: $$4.\quad U$$

Now then, number $4$ was made by using disjunctive syllogism on three statements at the same time. Is that alright (for, say, a formal proof in a test), or am I supposed to do it "slower" and only do inferences with two statements at a time?

Another title for this question could have been: Is it alright to do inferences with more than two statements simultaneously? But, apparently, the destructive dilemma inference rule actually uses three statements anyway, so I guess I should keep this to just disjunctive syllogisms.

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It would be perfectly clear for me, but I know some who pay strict attention to such details and want students to labor through all the rules one by one (especially freshmen). The fact is that it is easier to grade, e.g. with more verbose solution, if you make the mistake in the middle, there would be the next derivation to fix it. – dtldarek Dec 2 '12 at 23:24

## 2 Answers

$$\quad\quad \quad \;\;\;\;\;\;1. \quad \lnot R \lor \lnot T \lor U$$ $$2.\quad R$$ $$3. \quad T$$ $$4. \quad U$$

Given the three premises $(1), (2), (3)$, yes indeed, $(4)$ is true. But assuming your task is to prove that $U$ follows from the first three premises: why wouldn't you add the intermediate step $(3.5)$, citing the premises and/or derivations used?:

$$(3.5)\; \lnot T \lor U\tag{ (1) (2) Disjunctive Syllogism}$$ $$(4)\; U \tag{ (3), (3.5) Disjunctive Syllogism}$$

You may want to speak to your instructor about how much detail to show. The disjunctive syllogism is typically taught as:

$$p \lor q$$ $$\lnot p$$ $$\therefore q$$

so you may also need to add $\lnot\lnot R$ from $(2)$, and $\lnot\lnot T$ from $(3)$, and then move to what I've numbered as $(3.5), (4)$, citing the respective premises to which double negation applies.

Again, it depends on your instructor, how explicit you need to be in your proofs.

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The numbering can give away to ambiguity since the number '4' is used in two steps it may be confusing (even though there is parenthesis) – Sniper Clown Dec 3 '12 at 0:55
I was just editing! Thanks, that's what I just corrected! – amWhy Dec 3 '12 at 1:04

Given:

$$1. ¬R∨¬T∨U$$ $$2. R$$ $$3. T$$ $$4. U$$

Using parentheses:

$$1. ¬R∨(¬T∨U)$$

Disjunctive syllogism is: $$p∨q$$ $$¬p$$ $$∴q$$

Next step:

$$1. ¬R∨(¬T∨U)$$ $$2. R$$ $$3. T$$ $$4. U$$ $$5. (¬T∨U) //justification: 1,2 DS$$ $$6. U //justification: 3,5 DS$$

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