1.) A > B
2.) B > C
3.) ~C.D
//:. ~A
To determine the validity of the argument ~A based on the given premises, we can use the method of deductive reasoning known as the syllogism. A syllogism is a logical argument that consists of a major premise, a minor premise, and a conclusion.
1.) A > B (Premise 1)
This premise states that A implies B, meaning whenever A is true, B must also be true. This can also be read as "If A, then B."
2.) B > C (Premise 2)
This premise states that B implies C, meaning whenever B is true, C must also be true. This can also be read as "If B, then C."
3.) ~C.D (Premise 3)
This premise states that not C (the negation of C) is true, and D is also true. The dot "." represents the logical AND operator, meaning both ~C and D must simultaneously be true.
//:. ~A (Conclusion to be reached)
To determine if ~A (the negation of A) can be logically deduced from the given premises, we can combine premises 1 and 2 using the transitive property of implication.
Transitive property of implication:
If A > B and B > C, then A > C.
Using the transitive property, we can deduce:
4.) A > C
This shows that if A is true, then C must also be true.
Now, let's combine premises 3 and 4 using the modus ponens inference rule.
Modus ponens:
If A > C and ~C.D, then ~A.
Applying modus ponens, we can conclude:
5.) ~A (Conclusion)
Therefore, based on the given premises, we can logically deduce that ~A is true.