Originally Posted by

**RogueDemon** The following are conceptual questions pertaining to modern symbolic logic. I have attempted to answer both of them, but I am unsure as to whether my answers are correct and sufficiently explanatory. I would really appreciate it if someone can look them over and tell me if there's anything about them that I should remove or add. The explanations must be brief. Thanks in advance.

1. Is the material conditional a necessary logical connective in our system? Justify your answer with an explanation that considers the role of the material conditional in both symbolization and derivations.

Attempted Answer:

Yes, the material conditional is a necessary logical connective in our system. Consider the statement, "If you study, then you will pass." This statement describes not two but three things: whether you will study, whether you will pass, and, most notably, the relation between those two things. The material conditional is a symbol of that relation, and without it, symbolizing the above statement would be impossible. Furthermore, some arguments and theorems cannot be proven valid by direct or indirect derivations unless assumptions for conditional derivations are made, and thus in order to complete those derivations, the material conditional is necessary.

2. Our derivation system is complete in that every valid argument and theorem which can be expressed in our system can be proven valid with a derivation. Show, in an organized manner, that every valid argument and theorem can be proven with an indirect derivation.

Attempted Answer:

An indirect derivation proves arguments and theorems by showing that if they were false, then contradiction and thus absurdity would follow. Since contradictions cannot be true, it follows that if the negations of arguments and theorems entail contradiction, then those negations must be false, and thus that the negations of those negations - i.e., the arguments and theorems - must be true.

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Also, given that a set of sentences {P, Q, R} is logically inconsistent, I am to determine whether the following arguments are valid:

1. P. ~Q. .: R

2. ~(Q -> R). .: ~P

3. S v R. ~S v Q. .: ~(P & Q & R)

4. S & P. R -> ~S. .: ~R v Q

I know that logical inconsistency implies that there is no truth value assignment for which all the members of the set are true, but how would I apply that to the solving of this problem?