Logical Equivalence and Corresponding English Statements

Question

Consider the statement, "If it is Tuesday, then it is raining"; in propositional logic, the statement would read as, "$p \implies q$." Now, in accordance with the rules and definitions prescribed in logic, we have a plethora of logical equivalences. Taking the our statement, we can rewrite it as $ \neg p \vee q$, and in English, "It is not Tuesday or it is raining. By setting up a truth table, we can prove that these statements are equivalent, hence, they have the same meaning. However, if I was to be read the two statements in English, I wouldn't suspect that they have the same meaning in everyday language. My question is, does the fact that the two statements in logic have the same meaning necessarily imply that the have the same meaning in everyday language? Because I honestly don't see how they convey the same meaning.

Answer 1

Technically speaking no they do not have to be equivalent. In your particular case it depends on the interpretation given to the "or" conjunctive in the English Language. Many people (and I am not saying it is wrong or right) interpret the statement "$A$ or $B$" as "$A$ is true, or $B$ is true, but not both at the same time". Using this interpretation, both statements are not equal. Nevertheless if we all agreed that "$A$ or $B$" means "$A$ is true, or $B$ is true, and it is possible that both are true" then the statements would be equal since the interpretation of $\vee$ and "or" would then be the same.

Answer 2

It is strictly by definition, $p\Rightarrow q$ is true if and only if the consequence $q$ is true, or the antecedent $p$ is false. You can see it in the truth table that defines the implication. That is, $p\Rightarrow q$ is true if and only if either $\neg p$ is true, or $q$ is true; i.e., if and only if $\neg p\lor q$ is true..

Or you can simply look at the truth tables. The truth table of $\neg p\lor q$ is the same as the truth table of $p\Rightarrow q$: true if $p$ and $q$ are false; true if $p$ is false and $q$ is true; false if $p$ is true and $q$ is false; true if $p$ and $q$ are both true: $$\begin{array}{c|c||c} p & q & p\Rightarrow q\\ \hline 0 & 0 & 1\\ 0 & 1 & 1\\ 1 & 0 & 0\\ 1 & 1 & 1 \end{array}\qquad\qquad \begin{array}{c|c|c|c} p & q & \neg p & \neg p\lor q\\ \hline 0 & 0 & 1 & 1\\ 0 & 1 & 1 & 1\\ 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 1 \end{array}.$$ The final columns are identical, so the two formulas take the same truth values given the same truth inputs: that is, they are propositionally equivalent.

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This may seem counter-intuitive, particularly with respect what is sometimes meant by "or" in standard usage: sometimes we speak of "this or that" when we really mean to say, "this or that, and not both". Logically, though, "or" is taken to mean "this or that or possibly both."

Answer 3

While it’s perhaps not immediately evident, the statement It is not Tuesday or it is raining does at least imply the statement If it is Tuesday, then it is raining even in everyday language: if the first statement is true, and if today really is Tuesday, then it must in fact be raining. The second probably does not imply the first in most people’s everyday language, because in everyday usage if ... then is normally taken to imply some connection $-$ perhaps not a truly causal connection, but something along those lines. Material implication ($\to$) is definitely not the same as everyday if ... then, so a formal equivalence between statements involving material implication may not translate into an equivalence between the apparent everyday counterparts.

The same problem arises to some degree with disjunction ($\lor$). In everyday usage the word or is often closer to exclusive or ($\veebar$) than to ordinary inclusive $\lor$, especially when preceded by either: It was either John or Charles; Apologize to your sister, or leave the table!

More generally, there’s a problem translating between formal and everyday language. Consider the statement Touch me, and you’ll lose some teeth! On the face of it that is $p\land q$, where $p$ is you touch me, and $q$ is you will lose some teeth, but the purely formal version closest to the real sense is $p\to q$.

Answer 4

Think about it this way: if they're not equivalent, then there must be some instance when one is false and the other isn't.

Let $A$ be the statement "If it is Tuesday, then it is raining".

Let $B$ be the statement "It is not Tuesday or it is raining".

When is $A$ false? The statement $A$ is false exactly when it is Tuesday AND it is not raining.

When is $B$ false? the statement $B$ is false when it is Tuesday AND it isn't raining.

We can now conclude that $A$ is false only when $B$ is false and $B$ is false only when $A$ is.

Therefore $A$ and $B$ must be equivalent.

EDIT: In what I did above, I assumed that a disjunction of two statements is true if, and only if, at least one of the statements is true. If the disjuction of two statements requires that one of the statements is true and the other is false for the disjunction to tbe true, then the statements are not equivalent.