Identify what you can properly conclude from the given premises: P: not A → B P: A → not Z P: not Z → F C: ?
BenMingov May 31, 2020
Hi Brianna, thanks for reaching out.What we are trying to do is combine all of the sufficient/necessary premises and create a transitive chain.
We are provided with the following:
Not A -> B
A -> Not Z
Not Z -> F
The second and third statements easily combine to the following
Not A -> B
A -> Not Z -> F
Now we have to think about whether the first premise is somehow able to chain up. Since we need the common term between the two statements to be sufficient in one premise and necessary in the other. It would serve us well to take the contrapositive of the first premise.
Not B -> A
A -> Not Z -> F
Now A is both necessary and sufficient in the two chains.
We can combine to the following
Not B -> A -> Not Z -> F
The final deduction from this is our conclusion: Not B -> F (as well as the contrapositive... Not F -> B)
I hope this helps. Please let me know if you have any other questions.
Zachary August 24, 2020
BenMingov,I follow your reasoning about using the transitive property. However, I do not think it is fair to assume a priori that the contra-positive of the first premise is necessarily true, as there is no indication from the other premises that this is the case.
In these types of questions, must we assume that the contra-positive of all premises are necessarily true?
ZG
Mariyam June 20, 2024
I don't know if anyone else will find this useful but I write everything out in longhand FIRST and then try to make my connections. I also label everything:P1: Not A --> B CP: Not B --> A
P2: A --> Not Z CP: Z --> Not A
P3: Not Z --> F CP: Not F --> Z
First Example:
P1CP P2 P3
Not B --> A A --> Not Z Not Z --> F
If you remove the parts that repeat then you can see what's left
P1CP P2 P3
Not B --> {A A} --> {Not Z Not Z} --> F
P1CP P2 P3
Not B --> --> --> F
leaves Not B --> F
Mariyam June 20, 2024
Sorry, the system removed my spaces, the P1CP, P2 & P3 are supposed to be directly above the place where they join the chain.
Emil-Kunkin June 21, 2024
This is great!bsmoyj November 17, 2024
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