Using only negation and disjunction, we can analyze the form of one common pattern of reasoning, which is called process of elimination or, more technically, disjunctive syllogism. As an example, consider this argument:
Her phone line is busy, so she must either be talking on the phone or using her modem. She is not using her modem, since I just tried to e-mail her and she did not respond. So she must be talking on the phone.
After trimming off assurances and subarguments that support the premises, the core of this argument can be put in standard form:
(1) She is either using her modem or talking on the phone. (2) She is not using her modem.
(3) She is talking on the phone. (from 1–2)
This core argument is then an instance of this argument form:
1. p q ~p
It does not matter if we change the order of the disjuncts so that the first premise is “She must be either talking on the phone or using her modem.” Then the argument takes this form:
2. p q ~q
Both of these argument forms are valid, so the core of the original argument is also valid.
in more than one way, symbolize each interpretation and describe a context in which it would be natural to interpret it in each way.
1. It won’t rain tomorrow.
2. It might not rain tomorrow.
3. There is no chance that it will rain tomorrow.
4. I believe that it won’t rain tomorrow.
5. Joe is not too smart or else he’s very clever.
6. Kristin is not smart or rich.
7. Sometimes you feel like a nut; sometimes you don’t. (from an advertisement for Mounds and Almond Joy candies, which are made by the same company and are exactly alike except that only one of them has a nut)