Question

Consider the following (true) statement:

“All birds have wings but some birds cannot fly.”

Part 1

Write this statement symbolically as a conjunction of two sub-statements, one of which is a conditional and the other is the negation of a conditional.

Use three components (p, q, and r) and explicitly state what these components correspond to in the original statement.

Hint: Any statement in the form "some X cannot Y" can be rewritten equivalently as “not all X can Y,” or "it is not the case that X can Y." Here X denotes a class of objects and Y denotes a specific property pertaining to these objects.

Part 2

Use your result from part a) to write the negation of this statement symbolically as a disjunction. Explain your reasoning and identify the logical rules used in your answer.

Hint: The negation of the conditional "p-->q" is the conjunction "p ^ (~q)".

Part 3

Write the negation of this statement verbally using your result form part b). Keep in mind that while there are multiple logically valid ways to write this negation, some are clearly at odds with the way we use sentences in everyday life. Try to write a sentence that is not awkwardly phrased.   

Answer 1

1.

First, define three statements (about the item/animal being considered)

p = it is a bird
q = it has wings
r = it can fly

Statement #1 (S 1)

p ===>> q "If it's a bird, then it has wings"

(i.e. birds have wings)

Statement #2 (S2)

p =//=>> r " something being a bird DOES NOT imply that it can fly"

(i.e. birds are not necessarily capable of flying)

the CONJUNCTION of these statements ( S1 AND S2) is as follows:

[ p ==> q ] ^ [ p =/= > r ]

( i.e. birds have wings but birds are not necessarily able to fly )