exercise 5.8
write an informal proof of
Premises: LeftOf(a,b) | RightOf(a,b)
BackOf(a,b) | ~Leftof(a,b)
FrontOf(b,a) | ~RightOf(a,b)
SameCol(c,a) & SameRow(c,b)
conclusion BackOf(a,b)
State if you use proof by cases.
To write an informal proof of the given argument, we will use logical reasoning based on the given premises. Let's break it down step by step.
Premises:
1. LeftOf(a, b) OR RightOf(a, b)
2. BackOf(a, b) OR NOT LeftOf(a, b)
3. FrontOf(b, a) OR NOT RightOf(a, b)
4. SameCol(c, a) AND SameRow(c, b)
Conclusion:
BackOf(a, b)
To prove the conclusion, we need to consider the possibilities based on the premises. We will first examine the premises and make logical deductions using common methods of reasoning rather than using formal proof methods like proof by cases.
Let's analyze the premises one by one:
1. LeftOf(a, b) OR RightOf(a, b)
This premise suggests that either a is left of b or a is right of b.
2. BackOf(a, b) OR NOT LeftOf(a, b)
This premise suggests that either a is back of b or a is not left of b.
3. FrontOf(b, a) OR NOT RightOf(a, b)
This premise suggests that either b is front of a or a is not right of b.
4. SameCol(c, a) AND SameRow(c, b)
This premise states that c is in the same column as a and in the same row as b.
Now, let's consider the possible cases:
Case 1: a is left of b
If a is left of b, then it means a is not right of b. Based on premise 3, NOT RightOf(a, b). This allows for the possibility of NOT BackOf(a, b) according to premise 2. Thus, we cannot conclude BackOf(a, b) in this case.
Case 2: a is right of b
If a is right of b, it gives us no information about whether a is back of b or not. We cannot draw a conclusion about BackOf(a, b) in this case.
Since neither case allows us to draw a definitive conclusion about BackOf(a, b), we cannot prove the conclusion using proof by cases.
In summary, based on the given premises, we cannot prove that BackOf(a, b) is true. The argument neither confirms nor denies BackOf(a, b) with certainty.