Implications and Ordinary language

Question

I studied propositional logic, and everyday I see applications of what I learned on the internet, in mathematical books and miscelaneous resources.

One particular case is sentences in the form $a \rightarrow b$ , that is, material implications.
While I fully understand the mathematical logic approach to the study of those kinds of formulas, sometimes I get confused about what someone ( using implication sentences in the ordinary language ) is trying to say.

In ordinary language, I often see affirmations in the form of an implication. One example would be someone affirming that "the moon is made of cheese" $\rightarrow 2+2=4$.
[ Kleene, Mathematical Logic ].

I perfectly understand that if we choose the interpretation that gives the antecedent as false ( and the consequent as true ) we get that the whole implication is also true, but I would like to understand EXACTLY what he is trying to say when he AFFIRMS a sentence in the form of an implication.
Is he saying that under the commonly agreed interpretation of that sentence ( antecedent = false, consequent = true ), that implication becomes true?

Another example, totally different, would be someone affirming that $x \in \mathbb{N} \rightarrow x \in \mathbb{N}$ or $x \in \mathbb{Z}_{<0}$. Is he now affirming the same kind of sentence or saying something totally different?

When he affirms that sentence, is he saying that he finds the sentence to be a tautology? Why didn't he say that?

Then, with these two examples we have an affirmation of the same kind of sentence (implication) meaning completely different things, in one case he affirms to say that under the commonly agreed interpretation of the sentence, it is true. In another case, he affirms to say that there's no interpretation that makes the sentence false.

Am I missing something? There is really that ambiguity in ordinary language, when people are affirming sentences?
Can't I just take some general instance of the affirmation of a sentence in the form $a \rightarrow b$ in ordinary language and know exactly what he's trying to say, regardless of the propositions he chooses the substitute $a$ and $b$ for?

Thanks a bunch.

Answer 1

There are at least three senses in which "if P then Q" can be considered. It can be considered to mean that Q is at least as true as P.

1) In the first sense, any statement at all (Q) is at least as true as a known falsehood (P=f), and a known truth (Q=t) is at least as true as any other statement (P). This can be considered a trivial sense, because little else can be concluded from it, although the principle that anything at all can be concluded from a contradiction would be a less trivial example.

2) In the second sense, it is used as an assertion that this relationship holds. In this sense, is commonly used as a supposition, an assumption, hypothesis, or an axiom, for the purposes of examining its logical consequences. The meaning "implies" may be used in this sense.

3) Finally, it may also used to express a derived conclusion, that Q logically follows from assuming P (and other logical rules) It does not necessarily or always mean this, and "entails" is often reserved to refer to this sense.

"If P then Q" is true in each of these cases, and for logical purposes it is often unimportant why we claim it or how we know it is true. At other times, it is, and the difficulties come when one of these senses is confused with another. It is absurd to try to interpret the trivial case as an example of the third, and it also fails to interpret something as a conclusion if it is only a hypothesis. The particular meaning usually has to be determined from the context.

Answer 2

$P \implies Q$ in classical logic means that "$P$ is a sufficient condition for $Q$", or "$Q$ is a necessary condition for $P$". The definition $(P \implies Q) \equiv ($non$ P$ or $ Q)$ is not at all related to the usual concept of implication in natural language, because it is neither constructive, nor causal. Clearly, "I'm swimming $\implies$ I'm inside a liquid" is true (say, for example, water), whereas one has to be in a liquid before one can swim.

In intuitionistic logic, $P \implies Q$ is a constructive process : I suppose $P$, and I see if I eventually end up proving $Q$. If yes, then $P \implies Q$. This implication is still not causal! The same example above still holds.

In linear logic, $P \implies Q$ (where $\implies$ is to be undersood as the linear implication), is in some sense also causal. The linear implication conserves "information" : if $P \implies Q$, then by giving $P$, I end up with one $Q$ but I lost my $P$. So $P$ "transformed" into $Q$. It is constructive too.

Therefore, in classical logic, there's no other way to understand the implication rather than "non P or Q". In intuitionistic logic, you can see the implication as a "bridge" between two statements: proving that $P \implies Q$ means that there exists a bridge between the proposition "$P$" and the proposition "$Q$". Yet, you still don't know if $P$ is true. So even though the bridge might exist, it might be impossible to cross it via $P$, because $P$ might be false. The modus ponem is therefore easily understood: I need the bridge from $P$ to $Q$ to be true, and $P$ to be true in order to be able to cross it, and therefore end up on $Q$.

Answer 3

I suggest to you to see this post and also this.

I think that the following explanation by S.C.Kleene, Mathematical Logic (1967) [pag.10 - footnote 12] is the best "short" elucidation of it :

The ordinary usage certainly requires that "If $A$ , then $B$" to be true when $A$ and $B$ are both true, and to be false when $A$ is true but $B$ is false. So only our choice for $T$ in the third and fourth lines can be questioned. But if we changed $T$ to $F$ in both these lines, we would simply get a synonym for $\land$; in the third line only, for $\lnot$ . If we changed $T$ to $F$ in the fourth line only, we would loose the useful property of our implication that "If $A$ , then $B$" and "If $\lnot B$ , then $\lnot A$" are true under the same circumstances [...].

The truth-functional definition of propositional connectives is a "model" that in some cases "fit" quite well with our usage in natural language (negation, disjunction, conjunction) and not so well in other cases (conditional).

When we assert a sentence $\varphi$ we are expressing the fact that we "judge it" to be true.

Thus, asserting the conditional $\varphi \supset \psi$ means to "judge" it true.

When mathematicians (like Frege) introduced the truth-functional conncetive , they have in mind one characteristic property of the connective, viz., the rule of modus ponens. With this rule, we assert $A \supset B$ and $A$; in this case, the first assertion "exclude" the case when $A$ is true and $B$ false, while the second assertion "exclude" the two cases where $A$ is false.

Thus, we have only one possibility left : $B$ true, and this is what we expected.

The rule licenses us to infer $B$ from $A$ and $A \supset B$.

In our "ordinary" use of the language we seldom assert a conditional "if ..., then ___" when we know the antecedent to be false; but the "modelling" of mathematical logic fit quite well with the use in ordinary mathematics.

The very important "context" in mathematics is the following :

$\Sigma \vDash \varphi$;

in this case we say that $\Sigma$ entails $\varphi$. The condition validating the relation of "entailment" is that : every interpretation that satisfy (all the sentences in) $\Sigma$ will also satisfy $\varphi$; or, equivalently, there is no interpretation such that all of $\Sigma$ are true and $\varphi$ is false.

This "context" is commonly used when we assert that some thorem ($\varphi$) follows from a set $\Sigma$ of sentences, e.g.the axioms of a theory.

When $\Sigma = \{ \sigma \}$, a central result of classical logic is that : $\vDash \sigma \supset \varphi$.

This result establish a strict connextion between the conditional ($\supset$) and the relation of entailments ($\vDash$). The two are different relations, but the above link between them is so useful that we "accept" the "not perfect" fit of the conditional with our natural language habits.