# Is this argument valid without premise 1.1?

Source: 11 mins 49 s juncture, Lecture 3-7 (transcription), ... How to Reason and Argue,
by Prof W Sinnott-Armstrong. Caution: My enumeration differs from the Prof's. For brevity, I abbreviate 'anything else that rattles when shaken' with definition 2 of the noun 'rattler'.

1. This box rattles when I shake it.
1.1. A box doesn't rattle when shaken if it contains only a sweater and not any other rattler.
Original: A box doesn't rattle when shaken if it contains only a sweater and not anything else that makes a rattling noise when shaken.
1.2. If this box contains a sweater, then it contains only a sweater, and doesn't contain any other rattler.
Original: If this box contains a sweater, then it contains only a sweater, and doesn't contain anything else that rattles when shaken.

2. So this box does not contain a sweater.

At the 9 mins 32 s juncture, the Prof did not introduce 1.3 yet (I inserted it above to save space, but apologise if it deceives). He explains that without 1.3, the argument is INVALID:

Well, no, for the same reason we saw before, because my wife might be a trickster who puts rocks around my sweater in the birthday present box, in order to fool me.
Then, the premises can be true, and the conclusion, false.

My question: Is the argument valid if we delete 1.1, and keep only and 1.2?

We have :

1) Is_a_box(a) and Rattles_when_shaken(a)

1.1) For_all(x) [ if Is_a_box(x), then (if Contains_only_a_sweater_and_nothing_else(x), then not Rattles_when_shaken(x)) ]

We apply Universal Instantiation with the term a to 1.1 getting :

2) if Is_a_box(a), then (if Contains_only_a_sweater_and_nothing_else(a), then not Rattles_when_shaken(a)

from 1) by Simplification (or conjunction elimination) we derive :

3.1) Is_a_box(a)

and

3.2) Rattles_when_shaken(a)

with 3.1 and 2, by modus ponens, we derive :

4) if Contains_only_a_sweater_and_nothing_else(a), then not Rattles_when_shaken(a)

from 4 by Contraposition (with double negation) we derive :

5) if Rattles_when_shaken(a), then not Contains_only_a_sweater_and_nothing_else(a)

from 3.2 and 5, again by modus ponens, we conclude with :

not Contains_only_a_sweater_and_nothing_else(a).

But the denial of : "the box does not contain only a sweater and nothing else" (using De Morgan) is :

"either the box a does not contain a swater, or it contain a sweater and something else".

Thus, in order to conclude with :

the box a does not contain a sweater

we need an additional premise, in order to apply Disjunctive syllogism to the disjunction :

either the box a does not contain a swater, or it contain a sweater and something else.

This additional premise must be the denial of :

the box a does contain a sweater and something else;

the negation of (p∧q) is (¬p ∨ ¬q) as well as (p → ¬q), and thus the additional premise must be :

if the box a does contain a sweater, then it does not contain something else

that is 1.2 above.