7.6 In the following question, the domain of discourse is a set of male patients

Question

7.6
In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates:

    P(x): x was given the placebo
    D(x): x was given the medication
    A(x): x had fainting spells
    M(x): x had migraines

Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate:
   P(x)    D(x)    A(x)    M(x)
Frodo    T    F    F    T
Gandalf F    T    F    F
Gimli    F    T    T    F
Aragorn T    F    T    T
Bilbo    T    T    F    F

For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English.
(e)
x (M(x) A(x))

(f)
x ((M(x) A(x)) ¬D(x))

(g)
x (D(x) ¬A(x) ¬M(x))

(h)
x (D(x) (A(x) M(x)))

Explanation / Answer

x (M(x) A(x))

this statement is not a proposition as all the values are not true according to the value given in the table

The english translation is as follows:
For all x ,x had migraines if and only if x had fainting spells

(f)
x ((M(x) A(x)) ¬D(x))
this statement is not a proposition as all the values are not true according to the value given in the table

The english translation is as follows:
For all x ,x had migraines and x had fainting spells if x was not given the medication

(g)
x (D(x) ¬A(x) ¬M(x))

This statement is a proposition as for some value of x it is true.
The english translation is as follows:
For some x ,x was given the medication and x had no fainting spells and x had no migraines

(h)
x (D(x) (A(x) M(x)))
this statement is not a proposition as all the values are not true according to the value given in the table

The english translation is as follows:
For all x ,x was given the medication if x had fainting spells or x had migraines

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