Detachment and Syllogism Examples

"Calicos are cats" and "All cats are cute." What can be inferred from these statements?

Reducing this into p → q form, we have "calicos → cats" and "cats → cute." That means we can make "calicos → cats → cute" or, more simply, "calicos → cute." That means our final inference is, "All calicos are cute," which is obvious.

Given the statements, "There is no school during the weekend," and "Bob does his homework during the weekend," what can be inferred?

Our shorthand will look like this: "weekend → no school" and "Bob's homework → weekend." We can condense these down into "Bob's homework → no school." Don't get too excited because it doesn't mean what you think it means. Remembering what we started out with, it translates to "Bob does his homework when there is no school."

"All cats are cute" and "All dogs are cute." What can be inferred from these statements?

Uh. Nothing. If we write this in shorthand, we have "cats → cute" and "dogs → cute." As cute as they are, we can't write "dogs → cute → cats" or "cats → cute → dogs" because the arrows can't be reversed. So nothing new can be inferred from these statements.

"Parties are fun" and "Birthdays should have parties." What can be inferred from the inverse of these statements?

First, let's write them all in p → q form. The first one becomes, "parties → fun" and the second is "birthdays → parties." Inversing these statements just means putting a "not" in front of each atom. So they become, "not parties → not fun" and "not birthdays → not parties." Squishing them together, we get "not birthdays → not fun." In normal English, "If it's not a birthday, it's not fun." Even if that were true, buck up, because every day is someone's birthday!