16

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?

Skinny Kevin
  • 195
  • 4
    Personally, I would just ignore that sentence or not worry about it too much. Perhaps a more detailed development of logic would give you a collection of axioms such as "($P \implies Q$ and $Q \implies R$) $\implies$ ($P \implies R$)" that can be used to derive new statements from given statements. And perhaps the author is saying that there is no such axiom that can be applied in this example to take you from the hypotheses to the conclusion. But, more detail would need to be given in order to really make sense of this. – littleO Jul 19 '23 at 05:22
  • 2
    My two cents is on "not sound," the author probably means it's valid but not true. – Bertrand Wittgenstein's Ghost Jul 19 '23 at 05:42
  • I follow this author on twitter and I'm sure he'd try to answer your question if you asked him. – Matthew Leingang Jul 19 '23 at 18:19
  • Note that both "Socrates was a Martian and all Martians lived on Mars, therefore Scorates lived on Mars" and "Socrates was a Martian and all Martians lived on Earth, therefore Scorates lived on Earth" are both logically correct in that the conclusion is a correct application of logical deduction from the premises. – Henry Jul 20 '23 at 09:51
  • 3
    @seldon I don't think the author means that (and even so does not clearly mean that). See the other comments. – Trebor Jul 20 '23 at 11:29

4 Answers4

62

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.

RyRy the Fly Guy
  • 5,950
  • 1
  • 11
  • 27
  • Related technical term: Logical consequence/entailment. – preferred_anon Jul 19 '23 at 22:42
  • To emphasize a significant point, "Socrates lives on Pluto" isn't correct, it is the deduction of "Socrates lives on Pluto" that is logically correct. To make an analogy, if your kid makes a mud pie, you can bake it in an oven for the right amount of time at the right temperature, but it is still inedible. – JonathanZ Jul 20 '23 at 03:17
  • @ikegami Note the 2nd argument concluding "$2+2=4$" has a conclusion that is in fact always true. For this reason, it's not possible for the premises to be true and the conclusion false. Hence, by definition, the argument is valid. However, the conclusion is clearly not a logical consequence of the premises "Socrates is a Martian" and "Martian's live on Pluto." Therefore, validity and "logical correctness," as defined by the author, are not synonymous. If it is "logically correct," then it will be valid, but the converse does not necessarily hold. – RyRy the Fly Guy Jul 20 '23 at 19:46
  • 2
    @RyRytheFlyGuy I sort of disagree that "2+2=4" is always true. In boolean algebra say, where "+" means bitwise or, then 2+2=2 and not 4. Of course it is true using the standard definitions of "+" and "2" and "4" - but if we're using those definitions then we really have to include them in the premises of the argument. If we do then the argument becomes valid. – Carmeister Jul 20 '23 at 21:57
  • @Carmeister the author, Jay Cummings, refers to the conclusion $2+2=4$ being "correct" despite not following from the premises, so i think the standard definitions of $+,2,4$ are what is being used in the excerpt I'm referring to. In general, though, i agree with what you're saying. – RyRy the Fly Guy Jul 20 '23 at 22:07
  • 1
    @Carmeister Yes. Re: Ryry’s latest comment: from the substance of the excerpt, the author is trying to say that 2+2=4doesn’t logically follow from the premises even when (rather than “even though”) it is correct (for example, under the common interpretation of +, which unlike and or not etc. is not a logical operator, and thus whose specific interpretation does not influence a sentence’s validity). – ryang Jul 21 '23 at 06:50
  • Or in English, "logically correct" = rational. It does not imply that the statement is true or factual, only that the inference process is logical and rational. – iBug Jul 21 '23 at 07:09
  • @iBug Tangential comment (sorry!): in my mind, 'logical' is a strict subset of 'rational': sorting my phone contacts by closeness of relationship, and arranging my wardrobe according to the colours of the rainbow, are not logical per se, but most definitely rational. -) – ryang Jul 21 '23 at 09:42
4

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).

ryang
  • 38,879
  • 14
  • 81
  • 179
  • 2
    There is no "then" here, it's "therefore:" $\models$/$\vdash$. Philosophically speaking, the two are different. – Bertrand Wittgenstein's Ghost Jul 19 '23 at 05:21
  • 1
    Is the word "therefore" even used when logic is formalized? I hadn't thought so. – littleO Jul 19 '23 at 05:25
  • 1
    @BertrandWittgenstein'sGhost In fact, the word "therefore" in the given sentence is incorrect (I ignored it as a minor misake), because it indicates that some premise is true, yet the word "if" patently leaves this open; in other words, it's either "if P, then Q" or "P; therefore Q", never "if P, therefore Q". – ryang Jul 19 '23 at 05:26
  • You are correct. I thought "therefore" was in OP's post. – Bertrand Wittgenstein's Ghost Jul 19 '23 at 05:32
  • @BertrandWittgenstein'sGhost Yes "therefore" is verbatim from the OP's quotation, and I chose not to address it in the post so as not to distract from the main point. Thanks for allowing this clarification. – ryang Jul 19 '23 at 05:34
  • 2
    @BertrandWittgenstein'sGhost That was my mistake on the use of the word if. In the book the quotation only included therefore. I have updated the question to fix that mistake. – Skinny Kevin Jul 19 '23 at 05:53
  • On that note, would the example given in the book of a logically correct statement "Socrates is a Martain, and Martians live on Pluto, therfore Socrates lives on Pluto." Be a valid argument since it is of the form (P => Q ^ Q => R) => (P => R) which is a tautology? – Skinny Kevin Jul 20 '23 at 01:11
1

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.

Arno
  • 3,788
  • 1
    The statement "A, therefore B" just means that A is true and that A implies B; in other words, every 'therefore' statement implicitly contains an implication/conditional and additionally asserts the truth of its premise/antecedent (as well as its conclusion/consequent). – ryang Jul 19 '23 at 16:26
1

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 $B$".

Mikhail Katz
  • 42,112
  • 3
  • 66
  • 131
  • 1
    RyRy the Fly Guy's answer has posted the full context behind/around the OP's Question; isn't it clear from their excerpt that the author is simply trying to distinguish between truth in a particular context and validity (truth regardless of context)? Using your symbolisation, A→B (A implies B) is true (under our usual interpretation) but invalid; this is independent of whether we are at the object-language or metalanguage level. – ryang Jul 20 '23 at 14:33
  • @ryang, I am not sure I agree with your analysis of RyRy's answer. I think my answer is compatible with his and is just a more concise version of his answer. The issue of "validity" is just not part of the formal language. It seems to me that analyzing the quality of logical inferences and validity of proofs is part of the meta-language analysis. – Mikhail Katz Jul 25 '23 at 08:50
  • We may be talking at cross purposes here, since none of what I said suggests that validity is part of the object language and since whether or not it is part of the object language is an independent issue from the point I made. Anyhoo. – ryang Jul 25 '23 at 13:14
  • @ryang, perhaps I misunderstood your comment, but you seemed to say that the formal sentence $A\to B$ is true but invalid. What I object to is talking about being invalid in the context of the formal/object language. It seems to me that it has to do with an analysis of logical inferences at the level of the metalanguage. – Mikhail Katz Jul 25 '23 at 13:17