Question about the meaning of logical implication

Question

So, I came across the following equivalence:

$$(p \to q) \equiv (\neg p \lor q)$$

Which I had been knowing for a while, but I never quite understood. I understand when it's used this way: I assert $p \land (p\to q)$, it follows that $q$. In other words, asserting $p \to q$ is like saying "if $p$, then $q$".

But I fail to understand it the other way: if $p$ and $q$ are both true, then $p \to q$. So if Eminem's name is Marshall and the Sun is bigger than Earth, can I rightly state that Eminem being named Marshall implies that the Sun is bigger than the Earth? If this is true, does this mean that logical implication doesn't describe causality?

From my (little) understanding, there's a substancial difference between $\vdash$ and $\to$, but I think I can't quite grasp it. Maybe $\vdash$ is the causality connective I'm looking for?

Answer 1

You are right that logical implication does not describe causality.

In natural language $p\rightarrow q$ means "whenever $p$ is true, it follows that $q$ is true".

So if $q$ is always true, then $p\rightarrow q$ is always true. For example, the following statements are both true:

• If Eminem is called Marshall, then the Sun is bigger than the Earth.
• If Eminem is not called Marshall, then the Sun is bigger than the Earth.

Lets look at the symbols you mention:

The symbols $P\vdash q$ mean that from a set of propositions $P$ you can derive the statement $q$, using deduction rules such as modus ponens. So $\vdash$ doesn't care whether anything is true, only whether it is provable.

Meanwhile, $p\rightarrow q$ is a proposition in a formal language. The symbol $\rightarrow$ is a connective, which we would like to play nicely with the truth-values we are going to assign to the propositions. It is often introduced using a truth-table.

The closest thing to a "causality" connective is probably $ \vDash$ (though I would not normally describe it as a connective). If we write $P\vDash q$ we mean that if all propositions in the set $P$ are true in a given model, then the proposition $q$ is necessarily true in that model too.

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There are connections between these symbols. For example the Deduction Lemma lets us exchange $\vdash$ and $\rightarrow$ in some circumstances.

$$ \{p\}\cup P \vdash q \iff P\vdash p\rightarrow q$$

The interplay between $\vdash$ and $\vDash$ leads to the discussion of completeness and soundness in logic, and may be of interest to you.

Answer 2

So, I came across the following equivalence:

$$(p \to q) \equiv (\neg p \lor q)$$

IMO more intuitive would be:

$$(p \to q) \equiv \neg (p \land \neg q)$$

Example

Consider the implication, it is raining ($R$) implies it is cloudy ($C$)

$$R\to C$$

This does not mean that rain causes cloudiness. It also does not mean that it is always cloudy when it is raining (e.g. so-called sunshowers). It simply rules out the possibility that it is currently both raining AND not cloudy (present tense). This is entirely consistent with the usual truth table** for implication in classical logic:

Note the following:

• When $R$ is true and $R\to C$ is true (line 1), then $C$ is true. (The Rule of Detachment)
• When $R$ is false (lines 3-4), then $R\to C$ is true regardless of the truth value of $C$. (The Principle of Vacuous Truth.) This form of argument is rarely if ever used in daily discourse since we rarely consider that the implications of a proposition known to be false. It is, however, routinely used in very technical arguments, e.g. in mathematical proofs.

You also wrote:

So if Eminem's name is Marshall and the Sun is bigger than Earth, can I rightly state that Eminem being named Marshall implies that the Sun is bigger than the Earth?

Correct. In other words, it is not the case that both (1) Eminem is named Marshall AND (2) the Sun is not bigger than the earth.

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** Text version of truth table:

R C R=>C

T T T

T F F

F T T

F F T