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?

1

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.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.