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For instance,

"If $a \in \mathbb{Z},$ then $a^{5}>0$.

Now obviously the generalized conclusion is wrong because negative numbers belong to $\mathbb{Z}$ and a negative number to an odd integer power returns a negative number. However, $A \rightarrow B$ has some odd features about it.

$\begin{array}{c:c|c}A&B&A \to B\\\hline T&T& T\\ T&F& F\\ F&T& T\\ F&F& T\end{array}$

The conclusion as to whether a statement is true or false can't be interpreted from basic common sense, it has to be interpreted from a truth table of statements as that is the foundation for mathematical reasoning.

The hypothesis $a \in \mathbb{Z}$ may or may not be true depending on what you pick $a$ to be, so "if" $a$ belongs to natural numbers is already mystery, there are no other defined conditions besides what follows.

Then, it's obviously not always true that this implication $a^{5}>0$ is satisfied for $a$, but it is sometimes true. So, if A is true and B is false, then the statement is false, but the problem is it seems B could be either true or false depending on the specifics of the context.

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  • $\begingroup$ Is $a$ a free variable in this sentence? If so, the truth value is also variable. But if the context of this sentence tells us there is an implicit universal quantifier, $\forall a,$ then the sentence is false. One would need to see the sentence in context to know what it was meant to say. $\endgroup$
    – David K
    Sep 16, 2018 at 1:12
  • $\begingroup$ That is part of the problem as there is no context, this kind of information is all I was given. a could be a variable or a could be a constant, or it could be some mathematical object no one imagined, IDK. $\endgroup$
    – GaneGoe
    Sep 16, 2018 at 1:14
  • $\begingroup$ There's always context. Suppose you walk into a classroom to take a math test and the only thing written on the test paper is that formula and the words, "True or false?" Then the context is the mathematics class that you have been taking and the interpretation of such statements during that class. The context is not just what is written on the same piece of paper. $\endgroup$
    – David K
    Sep 16, 2018 at 1:31
  • $\begingroup$ It honestly could be any possibility as far as I know, there has been no information to determine which possibility it is. $\endgroup$
    – GaneGoe
    Sep 16, 2018 at 1:37
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    $\begingroup$ I don't really understand the question. It could be true or false. You already understand that. What more could you possibly want? What are you looking to get out of an answer? $\endgroup$
    – user14972
    Sep 16, 2018 at 4:10

1 Answer 1

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The statement $$ 3\in \mathbb Z\to 3^5 >0$$ is true since the premise and conclusion are both true. The statement $$ -3\in \mathbb Z\to (-3)^5 > 0$$ is false since the premise is true and the conclusion is false. The statement$$ -\pi \in \mathbb Z \to (-\pi)^5 >0$$ is true since the premise is false and the conclusion is true.

Your problem is you aren't looking at a statement, you are looking at an open formula with a variable. One way to interpret what you mean by asking if the formula is true, is that you are actually asking about the universal closure $$ \forall a (a\in \mathbb Z \to a^5 > 0).$$ In this case, the sentence is false, because it's not true for all $a$ in the domain. (We haven't said what the domain is, but let's assume it's $\mathbb R$ for definiteness since both the predicates make sense and are not trivial there.) On the other hand, if you don't mean that, then we don't have enough information to answer the question since truth values of open formulas only make sense relative to a variable assignment.

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