Question

What is the defferenece between "the justification of line 6" and "the application of xx on line 6"? For example, we can find that in the line 6, the application is wrong because we need to use PC rather than ¬I, but I have no idea about if "the application of ¬I on line 6 is false" equals to "the justificayion of line 6 is false". I mean, we can indeed prove that the conclusion in the line 6 is correct.

So the statement "the justification of line 6", Does this statement become correct by only considering the correctness of this conclusion, or is it necessary for both the process and the formula symbols on the right to be correct for the statement to be correct?

2 If an error occurs in an earlier line, can we conclude that all subsequent lines, line x included, are incorrect? For example, if line 1 is incorrect ,can we conclude that "the justificayion of line x(x>1)"and "the application of xx on line x" are both false?Or is it that if there's an error in the previous formula application, it only affects subsequent "justification", not the inference "formula applications"?

To summary, What I want to know is the criteria for determining the correctness of these two statements "the justification of line 6" and "the application of xx on line 6", and whether errors in the previous application and justification would affect the subsequent assessments.

Answer 1

Understanding the 'justification of line 6' and the 'application of xx on line 6' involves differentiating between the validity of the logical process and the truth of a conclusion.

Step-by-step explanation:

To evaluate the correctness of statements such as "the justification of line 6" and "the application of xx on line 6", we need to distinguish between the truth of a conclusion and the correct logical process leading to that conclusion. Justification refers to the reasons given in support of a belief, suggesting it be more likely true, which includes offering logical support or evidence. In contrast, application refers to the use of a logical principle or rule in a specific instance.

When we say "the application of ¬I on line 6 is false", it suggests that the rule was not applied correctly, which can invalidate the subsequent reasoning, even if the conclusion of line 6 happens to be true. The statement "the justification of line 6 is false" generally means that the reasons provided to support the conclusion of line 6 are inadequate or faulty, regardless of whether the conclusion itself is true or falsified by the incorrect application.

Errors in earlier lines of an argument do have the potential to invalidate all subsequent lines, as they can perpetuate faulty reasoning or incorrect premises. However, an error in the application of a rule to earlier lines only affects the justification if it directly impacts the logical structure or premises upon which subsequent reasoning is built. Simply put, errors in initial applications may not necessarily affect all subsequent formula applications, but they can undermine the logical integrity of the arguments that follow.