Introduction to Conjunctions and Disjunctions ("And" and "Or")
The simplest deductions are those involving two or more things connected by "and" (a conjunction) or "or" (a disjunction), and are governed by the following rules:
"And" (and "but"): Affirm all, negate one.
- To affirm an "and" claim, all parts of the "and" must be affirmed as true.
- To negate an "and" claim, at least one part of the "and" must be negated as false.
"Or": Affirm one, negate all.
- To affirm an "or" claim, at least one part of the "or" must be affirmed as true.
- To negate an "or" claim, all parts of the "or" must be negated as false.
Example 1. Consider the claim, "Pat and Juan have arrived." If that claim is affirmed as true, we can conclude that both Pat and Juan have arrived, because all parts must be affirmed. If, however, the claim is false, then we can conclude that at least one of the terms must be negated: either Pat has not arrived, or Juan has not arrived, or neither has arrived.
Example 2. Consider the claim, "Farida or Marcia has won the race." If true, then (because at least one part of the "or" must be affirmed) one of the following must be true: Farida has won, Marcia has won, or they both have won. If false, then neither Farida nor Marcia have won.
These rules always apply, even when the deductions are complicated by more elements ("Farida, Marcia, Pat, and Juan"), the use of negatives ("Farida and not Marcia"), or some combination of these. So, to affirm the claim "Farida and Pat but not Marcia or Juan have finished," we would employ the following steps:
- To negate the "or" ("not Marcia or Juan"), we would negate both parts, concluding that Marcia has not finished and Juan has not finished.
- To affirm the "and" ("Farida and Pat"), we would affirm both parts, concluding that Farida has finished and Pat has finished.
- To affirm the "but" (which operates logically as an "and"), we would affirm all parts, concluding that Farida has finished, Pat has finished, Marcia has not finished, and Juan has not finished. All these must be true in order for the claim "Farida and Pat but not Marcia or Juan have finished" to be true.
Inclusive and Exclusive "Or"
Sometimes, "or" is used in an exclusive sense. For example, you might read on a menu, "Soup or salad comes with the dinner." This means, "soup or salad, but not both," because the menu is describing what is included with the price of the meal. However, if there is no contextual reason to think otherwise, assume every "or" is inclusive--that is, "A or B or both." The difference between the inclusive and exclusive "or," then, has to do with cases in which "both" are true. Since "Soup or salad comes with the dinner" is exclusive, it places "soup and salad" outside the range of things that are included in the price of the meal.
Now suppose an advisor tells you that you can take English 7 or History 60 to satisfy a critical thinking requirement. Though, in this case, it's clear that you don't need to take both, that "or" is still inclusive, because if you did take both, you would still be satisfying the requirement: English 7 or History 60 or both satisfy the requirement.
As a result, the use of "and/or" is unnecessarily confusing and should be avoided, since "or" by itself, in the absence of any exclusionary language or context, means the same thing. Using "A or B or both" makes the possibilities clearer but, in most cases, a simple "or" will suffice.