LOGIC proving/disproving general laws

In summary, the correct statement is (b) and a counterexample to (a) is when X is false and Y is true.
  • #1
dburton
1
0
One of the following general laws is true, the other false. Prove the true one. Find a counter example to the other.(We must do this in writing out each statement, no truth tables or anything.

(a) If X implies Y, then the disjunction of X or Y is equivalent to X.
(b) If X implies Y, then the disjunction of X or Y is equivalent to Y.

I don't know, could you help me write out each step to prove/disprove?
 
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  • #2
dburton said:
One of the following general laws is true, the other false. Prove the true one. Find a counter example to the other.(We must do this in writing out each statement, no truth tables or anything.

(a) If X implies Y, then the disjunction of X or Y is equivalent to X.
(b) If X implies Y, then the disjunction of X or Y is equivalent to Y.

I don't know, could you help me write out each step to prove/disprove?

Okey. We have as an assumption "X implies Y". That means that either (1) X is false and Y is false, (2) X is false and Y is true, or (3) X is true and Y is true (because "X implies Y" is false if and only if X is true and Y is false). "[T]he disjunction of X or Y" (this is phrased a little funny) is false for (1) and true for (2) and (3). So we need to figure out if X or Y is similarly false for (1) and true for (2) and (3). That will be your answer. We see that Y satisfies these criteria, so (b) is the correct answer. A counterexample to (a) is the case where X is false and Y is true. Then "X implies Y" is true, but "the disjunction of X or Y" (true) is not equivalent to X (false).
 

Related to LOGIC proving/disproving general laws

1. What is the process of proving or disproving a general law using logic?

The process of proving or disproving a general law using logic involves analyzing the given statement and breaking it down into smaller logical statements. Then, using deductive reasoning and various logical techniques, the truth or falsity of the general law can be determined.

2. How do you determine the validity of a general law through logic?

The validity of a general law can be determined through a logical proof. This involves using established rules of logic, such as modus ponens or modus tollens, to show that the general law follows logically from the given premises. If a valid logical proof can be constructed, the general law is considered to be true.

3. Can a general law be proven or disproven using only deductive reasoning?

Yes, a general law can be proven or disproven using only deductive reasoning. Deductive reasoning is a logical process in which conclusions are drawn from premises that are assumed to be true. By carefully analyzing the premises and using valid logical arguments, the truth or falsity of a general law can be determined.

4. Are there any limitations to using logic to prove or disprove general laws?

Yes, there are limitations to using logic to prove or disprove general laws. One limitation is that the validity of a logical proof depends on the accuracy of the given premises. If the premises are false, the conclusion drawn using deductive reasoning will also be false. Additionally, there may be statements or concepts that cannot be proven or disproven using logic alone.

5. How can we ensure that the logical proof of a general law is sound?

To ensure that the logical proof of a general law is sound, it is important to carefully analyze the premises and make sure they are accurate. Additionally, it is important to use valid logical arguments and avoid fallacies or errors in reasoning. Peer review and further testing can also help to verify the soundness of a logical proof.

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