Conceptual Questions for Modern Symbolic Logic

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?

Re: Conceptual Questions for Modern Symbolic Logic

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**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?

With respect to your question number 1 - You haven't told us what other connectives beside the material conditional you have in the system you are working with. In most systems you have other connectives that you can use to define the material conditional. If you have negation ("~") and you have either "or" ( 'v' ) or "and" ( '&' ) for example, then you can define the material conditional.

With respect to your last question you should be able to construct truth tables to show that the arguments are valid

Does that help?

Re: Conceptual Questions for Modern Symbolic Logic

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**RogueDemon** 1. Is the material conditional a necessary logical connective in our system?

What exactly is *your* system?

The answer depends on what connectives are considered primitive in your system. E.g., Hilbert systems are sometimes considered only with $\displaystyle \to$ and $\displaystyle \neg$, and $\displaystyle \lor$ and $\displaystyle \land$ are expressed through $\displaystyle \to$ and $\displaystyle \neg$. Other times, $\displaystyle \to$, $\displaystyle \land$, $\displaystyle \lor$ and the contradiction $\displaystyle \bot$ are considered primitive, and the system has the corresponding inference rules or axioms for all of them.

Recall that in regular logic, $\displaystyle A\to B$ is equivalent to $\displaystyle \neg A\lor B$. So, when $\displaystyle \neg $ and $\displaystyle \lor$ are not expressed through $\displaystyle \to$, implication can be considered as a contraction. E.g., $\displaystyle (A\to B)\to C$ would really be $\displaystyle \neg(\neg A\lor B)\lor C$.

Not only formulas with $\displaystyle \to$ can be systematically translated to formulas without $\displaystyle \to$, but also inference rules for $\displaystyle \to$ can be derived from other rules. E.g., Modus Ponens becomes "$\displaystyle \neg A\lor B$ and $\displaystyle A$ derive $\displaystyle B$" (derivable from disjunctive syllogism and double negation introduction). So, every *derivation* of a formula with $\displaystyle \to$ can be systematically translated into a derivation without $\displaystyle \to$.

More later.

Re: Conceptual Questions for Modern Symbolic Logic

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**RogueDemon** 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.

If there is a direct (or, in fact, any) derivation D of A, then there is a superfluous indirect derivation of A. Namely, assume ~A, derive A using D, conclude a contradiction from A and ~A; therefore, the assumption ~A was false and so A holds. Clearly, such derivation does not add anything meaningful to D and can be reduced to D.

Interestingly, even when indirect derivation (i.e., reasoning by contradiction) seems to be essential, it can often can be mechanically transformed into a direct one. Namely, this is true for theorems of the form $\displaystyle \exists x\,A(x)$ and $\displaystyle \forall x\exists y\,A(x,y)$ where A does not have quantifiers and some conditions on atomic formulas hold. When theorems have more complicated form, reasoning by contradictions is sometimes essential. In propositional logic, any theorem of the form ~A has a direct derivation, and for any theorem A, there is a direct derivation of ~~A.

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**RogueDemon** 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

You check if the implication ~(P & Q & R) & P & ~Q -> R is a tautology. The assumption ~(P & Q & R) means that {P, Q, R} is inconsistent. This is the same as making a truth table for P & ~Q -> R to see if the formula's truth value is T in all rows except where P = Q = R = T (those rows do not matter).

Re: Conceptual Questions for Modern Symbolic Logic

Thanks a lot for the help! (Happy)

For the argument P. ~Q. .: R, would the following be correct?

P. T T T T F F F F

~ F F T T F F T T

Q. T T F F T T F F

R. T F T F T F T F

Since the TVA in the fourth column is T T F, the argument is invalid.

When making the truth table, do I have to make it for P & ~Q -> R or can I leave out the & and -> and make it as I did above?

Re: Conceptual Questions for Modern Symbolic Logic

Also, would I have to somehow do a shortened truth table for the arguments with four letters, such as 3 and 4? If these kinds of questions are asked on the exam, they would only be worth 0.5% each (2% total), so I shouldn't take too much time to solve them.

Re: Conceptual Questions for Modern Symbolic Logic

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**RogueDemon** For the argument P. ~Q. .: R, would the following be correct?

P. T T T T F F F F

~ F F T T F F T T

Q. T T F F T T F F

R. T F T F T F T F

Since the TVA in the fourth column is T T F, the argument is invalid.

When making the truth table, do I have to make it for P & ~Q -> R or can I leave out the & and -> and make it as I did above?

What you have above is fine. In this particular problem, you only need to remember to exclude columns (I said rows in post #4 because I am used to vertical truth tables) where P = Q = R = T.

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**RogueDemon** Also, would I have to somehow do a shortened truth table for the arguments with four letters, such as 3 and 4?

I usually try to find a falsifying truth value assignment. For example, for the argument

S & P. R -> ~S. => ~R v Q

to be invalid, we must have S & P = T, R -> ~S = T and ~R v Q = F. The latter means Q = F and ~R = F, so R = T. To have R -> ~S = T we must have ~S = T and S = F. But then S & P = F. Therefore, the argument is valid. In this case, we did not have to consider any alternatives. If you are lucky, there may be two or three cases to consider, which is much faster than creating a truth table. This method works especially well when assumptions are conjunctions and the conclusion is a disjunction, negation or an implication.