Alternative interpretation of logical implication

Question

Given any two unrelated true statements P and Q (for example, P=1+1=2 and Q=Trump is the 45th US president), then from the mathematical logic point of view "P implies Q" is true.

However, such a statement does not make sense because of lacking "logical force".

Is there an alternative way to define "P implies Q" so that "logical force" is a must for the implication to be true?

Is there a theory under which "P implies Q" means "P is a reason for Q"?

Answer 1

The implication in mathematical logic, $X \implies Y$ isn't normally used - or meant to be used - to mean one statement logically follows from the other. Instead, it's used to describe properties of mathematical objects. Let's take a simple example:

Let $X \subset \mathbb{Z}$ be such that: $x \in X, y \in X \implies x + y \in X$ (in other words, closed under addition).

In this definition, we define a set, and stipulate the elements are such the sum of any two elements of the set is also in the set. This implication isn't a logical conclusion we've somehow reached. It simply describes a property the elements of $X$ have by definition.

It doesn't make sense to demand "logical force" - because we don't usually use it in such a way.

Sometimes we use it to mean logical force, for instance: "$x > 0, y < 0 \implies x > y$" - and that's a true statement - but it's more of an informal shorthand than the typical use.

Answer 2

Alternative interpretation of logical implication

Given any two unrelated true statements P and Q (for example, P=1+1=2 and Q=Trump is the 45th US president), then from the mathematical logic point of view "P implies Q" is true. However, such a statement does not make sense because of lacking "logical force".

Indeed, the statement $$P\Rightarrow Q\tag1$$ lacks "logical force"—and your question doesn't match its title "logical implication"— because what you have provided offers no basis for claiming any of the following:

• "$P$ implies $Q$" is a logical validity
• "$P$ implies $Q$" is true regardless of interpretation
• $P$ logically implies $Q$
• $P$ logically entails $Q$
• $Q$ is a logical consequence of $P$
• $P\models Q$.

Since $Q$ ("Trump is the 45th US president") is a synthetic rather than analytic statement, the truth of statement $(1)$ is also non-analytic, thus "lacking logical force".

More to the point: if we modify the interpretation such that only those whose family name starts with 'A' are eligible to run for president, then statement $Q$ is no longer true, so statement $(1)$ is no longer true; this means that $P$ does not in fact logically imply $Q.$

On the other hand, "every city is big and Bangkok is a city" does logically (and analytically) imply "Bangkok is big": $$\big(\forall x\,(Cx\to Bx)\big)\land Ck\quad\models\quad Bk.$$

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Addendum re: Todor's answer

The implication in mathematical logic, $X \implies Y$ isn't meant to be used to mean one statement logically follows from the other. Instead, it's used to describe properties of mathematical objects. This implication isn't a logical conclusion we've somehow reached. It simply describes a property the elements of $X$ have by definition. It doesn't make sense to demand "logical force" - because we don't usually use it in such a way.

Although mathematical implication is generally weaker than logical implication, once the relevant mathematical axioms (i.e., definitions) are made explicit, that is, specified as assumptions in the implication's antecedent, the former becomes the latter.

Sometimes we use it to mean logical force, for instance: "$x > 0, y < 0 \implies x > y$" but it's more of an informal shorthand than the typical use.

The statement $$x > 0\land y < 0 \implies x > y$$ implicitly means $$\forall x\:\forall y\:\big(x > 0\land y < 0 \implies x > y\big);$$ its "logical force" is due to universal implication, which, to be clear, is also generally weaker than logical implication.