What is meant by "logically incorrect"?

Question

I was reading through Proofs: A Long Form Mathematics Textbook by Jay Cummings, and have gotten to the chapter on Logic. He makes a claim that I'm having trouble understanding:

And If I said "Socrates is a Martian and Martians live on Pluto, therefore $2 + 2 = 4$" then what I said was logically incorrect.

Clearly, "Socrates is a Martian and Martins live on Pluto" is a false statement, while "$2 + 2 = 4$" is a true statement, so by the truth-table definition of $P\ \text{implies}\ Q$, the implication

"Socrates is a Martian And Martians live on Pluto" implies "$2 + 2 = 4$"

is a true statement, and thus logically correct. I am struggling to see where I went wrong. Have I misinterpreted the idea of what it means for a statement to be logically correct, or have I made some other error?

Answer 1

And If I said "Socrates is a Martian and Martians live on Pluto, therefore 2 + 2 = 4" then what I said was logically incorrect.

1. I agree with your observation that this implication is (vacuously) true.

   However, the author is trying to say that the given argument is invalid (so, certainly unsound), since its form $$Ms\;\land\; \forall x\,(Mx\to Px)\quad\to\quad A\tag1$$ is invalid (i.e., not logically true), since it is false in certain interpretations where alternative definitions are assigned to addition, Socrates and Martian. In other words, its premises do not conjunctively logically entail its conclusion.

   Remember: for an argument to be valid, its corresponding implication needs to be true not just in our universe, but regardless of interpretation.

2. As implication $(1)$ is neither logically true nor logically false, we say that it is invalid but satisfiable.

   Although in this context the non-technical informal descriptions “logically incorrect” and “not logically correct” both accurately describe the given argument as invalid, the former is potentially misleading as it sounds like claiming that $(1)$ is logically false.

---

Reply to the OP's comment

Would the example given in the book of a logically correct statement "Socrates is a Martain, and Martians live on Pluto, therefore Socrates lives on Pluto" be a valid argument, since it is of the form $(P⇒Q) \land (Q⇒R) ⇒ (P ⇒R),$ which is a tautology?

Exactly! Or rather, your suggested argument form is the closest propositional-logic approximation of the given categorical argument's corresponding conditional $$Ms\;\land\; \forall x\,(Mx\to Px)\quad\to\quad Ps,$$ which is indeed valid (i.e., true regardless of interpretation).

Answer 2

Rather than assume what the author means, I consulted the textbook (p. $155$) and examined the excerpt in context...

"Logic is the process of deducing information correctly -- it is not the process of deducing correct information. For example,

1. Socrates is a Martian
2. Martians live on Pluto
3. Therefore, Socrates lives on Pluto

... is logically correct, even though all three statements are false. And if I said, 'Socrates is a Martian and Martians live on Pluto, therefore $2+2=4$,' then what I said was logically incorrect, even though the conclusion is correct."

By logically correct the author means the argument correctly deduces its conclusion from its premises. In other words, the argument is logically correct if its conclusion can be derived by applying the inference rules of some formal system to the premises. Hence, the argument is logically incorrect if its conclusion cannot be derived by an application of inference rules to the premises. Note this has nothing do with the actual truth values of the statements therein. This is precisely why the author makes the distinction between "deducing information correctly" and "deducing correct information." A logically correct argument does the former.

The first example concluding "Socrates lives on Pluto" is logically correct (even though it is not factually true) because the statement can be derived by applying inference rules (namely, those of first-order logic) to the premises. In turn, this guarantees the argument is valid, or in other words, it implies it's not possible for the premises to be true and the conclusion to be false.

The second example concluding "$2+2=4$" is logically incorrect (even though the conclusion is factually true) because there are no inference rules that can be applied to the premises which result in the conclusion. In other words, there are no inference rules that permit one to conclude $2+2=4$ based solely on the fact that "Socrates is a Martian" and "Martians lives on Pluto." A logically correct argument certainly exists for $2+2=4$, but it follows from a completely different set of premises.

Answer 3

The implication $\Rightarrow$ is not the same as the English language connective therefore. The former is only concerned with the truth values of the statements it is connecting, the latter does actually depend on the details of the statements.

Answer 4

None of the earlier answers seem to mention the distinction between language and metalanguage. The distinction may help understand the issue. Let $A$ be the statement "Socrates is a Martian" or preferably any incorrect statement that can be suitably formalized. Let $B$ the statement $2+2=4$. Then at the level of the formal language, the implication $A \to B$ is correct. On the other hand, at the level of the metalanguage, the implication "$A$ implies $B$" is an error of logical deduction. So it's really the difference between "$A\to B$" and "$A$ implies $B$".