Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English.Sample question: Some patient was given the placebo and the medication.
∃x (P(x) ∧ D(x))
Negation: ¬∃x (P(x) ∧ D(x))
Applying De Morgan's law: ∀x (¬P(x) ∨ ¬D(x))
English: Every patient was either not given the placebo or not given the medication (or both).
(a) Every patient was given the medication.
(b) Every patient was given the medication or the placebo or both.
(c) There is a patient who took the medication and had migraines.
(d) Every patient who took the placebo had migraines. (Hint: you will need to apply the conditional identity, p → q ≡ ¬p ∨ q.)

Answer:

P(x): x was given the placebo

D(x): x was given the medication

M(x): x had migraines

Explanation:

(a) Every patient was given the medication

Solution:

∀x D(x)

∀ represents for all and here it represents Every patient. D(x) represents x was given the medication.

Negation:¬∀x D(x).

This is the negation of Every patient was given the medication.

The basic formula for De- Morgan's Law in predicate logic is:

¬(P∨Q)⇔(¬P∧¬Q)

¬(P∧Q)⇔(¬P∨¬Q)

Applying De Morgan's Law:

          ∃x ¬D(x)

∃ represents there exists some. As D(x) represents x was given the medication so negation of D(x) which is ¬D(x) shows x was not given medication. So there exists some patient who was not given the medication.

Logical expression back into English:

There was a patient who was not given the medication.

(b) Every patient was given the medication or the placebo or both.

Solution:

∀x (D(x) ∨ P(x))

∀ represents for all and here it represents Every patient. D(x) represents x was given the medication. P(x) represents  x was given the placebo. V represents Or which shows that every patient was given medication or placebo or both.

Negation: ¬∀x (D(x) ∨ P(x))

This is the negation or false statement of Every patient was given the medication or the placebo or both.

Applying De Morgan's Law:

∃x (¬D(x) ∧ ¬P(x))

∃ represents there exists some. As D(x) represents x was given the medication so negation of D(x) which is ¬D(x) shows x was not given medication. As P(x) represents x was given the placebo so negation of P(x) which is ¬P(x) shows x was not given placebo. So there exists some patient who was neither given medication nor placebo.

Logical expression back into English:

There was a patient who was neither given the medication nor the placebo.

(c) There is a patient who took the medication and had migraines.

Solution:

∃x (D(x) ∧ M(x))

∃ represents there exists some. D(x) represents x was given the medication. M(x) represents x had migraines.  ∧ represents and which means patient took medication AND had migraines. So the above logical expression means there exists a patient who took medication and had migraines..

Negation:

¬∃x (D(x) ∧ M(x))

This is the negation or false part of the above logical expression: There is a patient who took the medication and had migraines.

Applying De Morgan's Laws:

∀x (¬D(x) ∨ ¬M(x))

∀ represents for all. As D(x) represents x was given the medication so negation of D(x) which is ¬D(x) shows x was not given medication. As M(x) represents x had migraines so negation of ¬M(x) shows x did not have migraines. ∨ represents that patient was not given medication or had migraines or both.

Logical expression back into English:

Every patient was not given the medication or did not have migraines or both.

(d) Every patient who took the placebo had migraines.

Solution:

∀x (P(x) → M(x))

∀ means for all. P(x) represents  x was given the placebo. M(x) represents x had migraines. So the above logical expressions represents that every patient who took the placebo had migraines.

Here we are using conditional identity which is defined as follows:

Conditional identity, p → q ≡ ¬p ∨ q.

Negation:

¬∀x (P(x) → M(x))

¬∀ means not all. P(x) implies M(x). The above expression is the negation of Every patient who took the placebo had migraines. So this negation means that Not every patient who took placebo had migraines.

Applying De Morgan's Law:

∃x (P(x) ∧ ¬M(x))

∃ represents there exists some.  P(x) represents  x was given the placebo. ¬M(x) represents x did not have migraines. So there exists a patient who was given placebo and that patient did not have migraine.

Logical expression back into English:

There is a patient who was given the placebo and did not have migraines.