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Can ($1=1$) $\implies$ (Einstein was born in 1879)? Both expressions are true and according to the $\implies$ truth table, a true statement implying another true statement is true. However I feel I must confirm this with you because intuitively 'imply' carries a causality connotation as in $1 = 1$ could imply $2=2$ but not Einstein's correct birth date. Are there ever mathematical contexts where $\implies$ has that sentiment?

Just to be clear: is (1=1) ⟹ (Einstein was born in 1879) mathematically true?

ryang
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BadUsername
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    No. Perhaps $(1=2) \implies$ (Einstein was born in 2025) and $(1=2) \implies$ (Einstein was born in 1879) both being logically correct implications illustrates this. – Henry May 26 '25 at 20:05
  • This question has nothing to do with physics - it's not enough that it mentions famous physicist in the body of the question. I've rolled back the edit. – Jakobian May 26 '25 at 20:49
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    Causality is a spatiotemporal notion. Mathematical notions are non-spatiotemporal. – Tankut Beygu May 26 '25 at 20:51
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    No................ – Mason May 26 '25 at 20:51
  • No. When you say $x$ causes $y$ do you mean $x$ happened and then later this caused $y$ to happen? Note that there is a timeline above. If $x$ implies $y$ then whenever $x$ happens $y$ happens but there isn't a timeline here. They happen simultaneously. – Mason May 26 '25 at 20:57
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    This is not the case for the usual meaning of mathematical implication, but there are other logics that try to capture this. – Nicholas Todoroff May 26 '25 at 20:58
  • I'm not convinced that causality is well-defined – CyclotomicField May 26 '25 at 23:44
  • Just to be clear then (1=1) ⟹ (Einstein was born in 1879) is mathematically true? – BadUsername May 26 '25 at 23:49
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    No. No indication of causality. Just get used it it. "Just to be clear then (1=1) ⟹ (Einstein was born in 1879) is mathematically true?" Yes: And here's the proof. Assume 1=1. Go to the national records and uncover a document of Einsteins birth. It will indicate the year of birth was 1879. So Einstein was born in 1879. Thus $(1=1)\implies $ (Einstein was born in 1879). QED. – fleablood May 27 '25 at 01:40

2 Answers2

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"Imply" does not indicate causality.

Consider for example: It is raining ($R$) implies it is cloudy ($C$).

$$R\implies C$$

This does not mean that rain causes cloudiness. It also does not mean that, historically, whenever it was raining, it was also cloudy, a counter-example being so-called sun-showers. It simply rules out the possibility that it is presently both raining and not cloudy.

$$R \implies C~~\equiv ~~ \neg (R \land \neg C)$$

Note that, if both $R$ and $C$ are true, then $R\implies C$ must be true.

Note, too, that if $R$ is false, it must be true that $R\implies C$ (the principle of vacuous truth).

In mathematical proofs based on classical logic, if proposition $Q$ can be derived from proposition $P$, we can write $P\implies Q$, which, as above, simply rules out the possibility that both $P$ is true and $Q$ is false.

$$P \implies Q ~~\equiv ~~ \neg (P \land \neg Q)$$

Dan Christensen
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Just to be clear: is (1=1) ⟹ (Einstein was born in 1879) mathematically true?

Einstein was born in 1879 isn't a mathematical proposition (although it can be framed as such, by first appropriately defining all its terms), so the given implication isn't mathematically true (or meaningful).

In the context of my household, the given implication isn't true: my son Einstein was actually born in 2023.

Thus, the given implication certainly isn't logically true.

However, the given implication is true in the context of real-world history, where 'Einstein' refers to that renowned physicist. Truth is relative.

(Yes, in all of the above, we are discussing propositional-logic implication, as defined by its truth table.)

intuitively, 'imply' carries a causality connotation

Are there ever mathematical contexts where ⟹ has that sentiment?

Conditionals in mathematics (e.g., the equation has at least 1 solution and the equation has at most 3 solutions ⟹ the equation has possibly 3 solutions) are underpinned by the material conditional, and never signal causality—or modality or counterfactuality. They form the backbone of deductive reasoning. Futhermore, they are generally quantified, so are richer than what truth tables alone can capture.

In any case, I don't fully get the persistence of the myth that implication in everyday language connotes causality: is the statement her tardiness implies that she doesn't care signalling that its antecedent causes its consequent?

ryang
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