Given the following symbolic meanings:

- $A@B$ means "All A are B"
- $A&B$ means "Only a few A are B"
- $A$\$B$ means "No A is B"
- $A\%B$ means "Some A is B"

Analyze the statement:

$$P \& R \% T \$ Q @ M$$

and determine which of the following conclusions logically follow:

I. Some R being not Q is a possibility

II. All M can be T

III. Some T being not P is a possibility

IV. All P can never be R

Question

Given the following symbolic meanings:

- $A@B$ means "All A are B"
- $A&B$ means "Only a few A are B"
- $A$\$B$ means "No A is B"
- $A\%B$ means "Some A is B"

Analyze the statement:

$$P \& R \% T \$ Q @ M$$

and determine which of the following conclusions logically follow:

I. Some R being not Q is a possibility

II. All M can be T

III. Some T being not P is a possibility

IV. All P can never be R

Accepted Answer

**Solution:**

1. Break down the statement:
   - $P \& R$ means "Only a few P are R".
   - $R \% T$ means "Some R is T".
   - $T \$ Q$ means "No T is Q".
   - $Q @ M$ means "All Q are M".

2. Analyze each conclusion:

- **Conclusion I:** "Some R being not Q is a possibility."
     Since no T is Q and some R is T, those R that are T cannot be Q. Hence, some R may not be Q. So, conclusion I is possible.

- **Conclusion II:** "All M can be T."
     Since all Q are M, but no T is Q, it is impossible for all M to be T because Q is a subset of M and Q and T do not overlap. So, conclusion II does not follow.

- **Conclusion III:** "Some T being not P is a possibility."
     Since only a few P are R and some R is T, it is possible that some T are not P. So, conclusion III is possible.

- **Conclusion IV:** "All P can never be R."
     Since only a few P are R, it means not all P are R. Hence, conclusion IV follows.

3. From the above, conclusions III and IV logically follow.

**Answer:** Both III and IV

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