Translating "therefore" in propositional logic and using contradictions to prove an argument.

Question

So I have the statements:

A Football player is either running at the oval or sitting in the library. If he is running, then he is not reading a book. The player is reading a book. Therefore, he is sitting in the library.

This is what I got so far:

Running at the Oval => R

Sitting in the Library=>S

Reading a Book=> B

Football player is running on the oval or sitting in the Library.

R V S

If Football player is running on the oval, then he is not reading a book

R→¬B

Football player is reading a book therefore he is sitting in the library.

From what I gather using "therefore" implies it's a material implication so:

B→S

If this is the case can I safely use the following to describe the statements given:

(R→¬B) V (B→S)

Answer 1

"Football player is running on the oval or sitting in the Library.

R V S

If Football player is running on the oval, then he is not reading a book

R→¬B"

That works.

You didn't symbolize the sentence that the player is reading a book (ignoring the potential linguistic issue that a person reading a book is not a "player" of a sport at that time at least). That would be "B". So, the premises are:

(R V S)

R→¬B

B

Now, can you deduce the conclusion S?

...

Spoiler:

B is true. So, $\lnot$B is false. (R→¬B) is true, so that's (R$\rightarrow$F). (R$\rightarrow$F) is only true when R is F. (R$\lor$S) is true, so we have (F$\lor$S). But, that is only true when S is true. So, the person is sitting in the library. Thus, the argument is valid. Supposing it were invalid would be a contradiction. So, the argument is valid.

Answer 2

1. The word ‘therefore’ $(\therefore)$ translates neither as ‘implies’ $(\implies)\;$ nor as the material conditional/implication $(\rightarrow).$

   The sentence “$P$ is true; therefore $Q$ is true” $(P;\;\therefore Q)$ has a different meaning from the sentence “if $P$ is true, then $Q$ is true” $(P{\implies} Q):$ the former implicitly contains the latter (which doesn't indicate whether $P$ is true), and additionally asserts that $P$ is true. “$P{\implies} Q$” can be just means that “$P\rightarrow Q$” is true.

2. In an argument, the word ‘therefore’ indicates that the statements before it are (true) premises, and asserts that the statement after it is the conclusion. So, your given argument contains $3$ premises $$R \lor S\\ R \rightarrow\lnot B\\ B$$ and a conclusion $$S.$$ It is valid if and only if its conclusion is logically inevitable, given its premises.

3. For our example, start by further assuming that $S$ is false. Then, by Premise 1, $R$ must be true. Then, by Premise 2, $B$ must be false. But by Premise 3, $B$ is true. We have derived the absurdity $$B \land \lnot B,$$ which must have arisen from the 4th assumption; thus, $S$ must instead be true.

   Hence, the argument's 3 premises logically (because we have been reasoning abstractly) entail its conclusion; that is, the argument is valid.

Answer 3

Using a direct proof and a simplified form of natural deduction, we have the following proof (screenshot from my proof checker, where '|' = OR):

Answer 4

If the player were not sitting in the library, then he would be running at the oval. But if he were running at the oval, then he would not be reading a book. However, we are given that he is reading a book. That's a contradiction, so our initial premise must be false.

Our initial premise is that he was not sitting at the library. Since we now know that this premise is false, it follows that he was sitting at the library.