Understanding causality in basic propositional logic

Question

I'm struggling through some logic in service of understanding proofs and I find that I'm able to do most of the problems but I'm having a hard time with some intuitions. This involves simple conditionals P->Q.

I understand that what's interesting is the formal analysis of P and Q where we understand the truth values of P->Q as being a consequence of the truth values of P and Q, and we can enumerate these truth values together in a truth table. It took me a moment to understand "vacuous truth," but I got there in the end. From a formal point of view it all makes sense.

Where I'm getting tripped up on intuition is the matter of causality in P->Q. We know that causality is not meaningful in the formal case - which is to say that P->Q is true in any case that it is true regardless of the causal relationship between P and Q - e.g. "If potatoes are roots, then Joe Biden is president" is formally true because the antecedent and the consequent are true, and "If Joe Biden is president, then I am Joe Biden" is formally false because the antecedent is true and the consequent is false.

But all of the examples used to illustrate conditionals in the books I'm reading rely heavily on causal connections to illustrate properties of conditionals. The common example used is "If I get an A on my final exam, then I will get an A in the class." The example is useful in explaining the vacuously true cases and the false case, and we rely on the notion of "someone having lied" as a representation of the truth value of a conditional. But saying something like "If basketball is a sport, then gardening is fun on sunny days" strikes me as something entirely different than lying or telling the truth - it's just semantic gibberish, and then I get really confused again. I would go as far as saying that it isn't true in a natural-language sense exactly because there is no causal connection, and establishing the causal connection is the fundamental intent of the sentence.

Where it really gets tricky for me is in actual proofs. Say I propose that "If 31 is prime, then 2222 is even." This conditional is formally true because the antecedent is true and the consequent is true. But I haven't really made anyone happy because now I'm expected to establish a causal relationship that connects the two statements - I can't just say "hey, look, my conditional is true! QED!" Somehow the importance of causal relationships got smuggled in under my nose and I can't see how it happened. I need to show - that is, I need to establish causally - how 31's primeness is somehow sufficient to guarantee the evenness of 2222.

I am probably missing something obvious and I apologize for the wall of text but this is legitimately keeping me up at night. I find logic fascinating especially because of how utterly stupid it makes me feel.

tl;dr: why does causality not matter in formal logic, but it does in applications of formal logic (i.e. mathematical proofs and in human semantics)?

The crux of it is that, ignoring context, we want logic to be as broad as possible; that is, we want to be able to form as many well-formed formulae as we can. If we were to restrict the use of implication to objects that are causally related, then we would have to know which objects are causally related before doing any logic at all, which is self-defeating as one of the points of logic is to elucidate those relationships. We would end up with countless different logics, one for each context. Although that's what happens in practice, this way round, we would need to have those a priori, — H. sapiens rex, Aug 13 '23 at 02:42
(cont) which is unreasonable. The alternative is to base logic purely upon empirical results, but then that would just be science and you still wouldn't have a single logic, but a multitude of them. — H. sapiens rex, Aug 13 '23 at 02:44
There is no causality in mathematics. There is no requirement that a proof establish a causal relationship. "$2222=2\times 1111$, so $2222$ is even" is a valid proof of "if $31$ is prime, then $2222$ is even." A mathematician might complain that the statement you're proving is not interesting, but they would not complain about the proof. — Alex Kruckman, Aug 13 '23 at 02:48
"If I get an A on my final exam, then I will get an A in the class" does not necessarily express a causal relationship either. Suppose the final exam is weighted to be only 5% of the grade, and you got 100% on all exams and assignments prior to the final. Then you'll get an A in the class regardless of your performance on the final. But "If I get an A on my final exam, then I will get an A in the class" is still a true statement. — Alex Kruckman, Aug 13 '23 at 02:53
Maybe the word 'causality' is the issue here, since that implies some sort of procedural relationship between objects - that is, one thing takes on some property and then that event propagates (at the speed of light?) changes to the properties of other objects. That certainly isn't happening. I'm struggling to find the right word - "connection" feels too vague, but it may do the trick. In the example I've established no connection between 31s primeness and 2222's evenness (though it may be possible to show the connection), but I've still submitted a complete proof of the implication? — Samuel Wainwright, Aug 13 '23 at 02:59
@SamuelWainwright correct; as Alex Kruckman said, mathematics doesn't deal with causality. That's physics. The only proof required of your implication is that $2222$ is in fact even. The word you're searching for is "relation" (mathematically at least), and yes it does include actually causal relationships, and much more. I think your problem may be that you're thinking too much in terms of physics, where it is true that interacting objects must have a causal relationship. Logic/mathematics is so much broader than that, though. Their appeal is that they can construct and treat objects that — H. sapiens rex, Aug 13 '23 at 03:11
(cont) have no basis in physical reality (and hence have no requirements re: causality) at all. Look at it this way: given a set of objects, there are infinitely many ways in which its elements can be related. Some of these relations are causal in nature; most are not. But it's all of a piece; the existence of non-causal relations does not preclude the application of logic to objects that do have causal relations. That's the power of logic: it can be applied to almost anything, real (causal) or not. It's a supremely broad foundation upon which to build knowledge. — H. sapiens rex, Aug 13 '23 at 03:19
Having thought about this a bit more, I think the right word to use here is correlation, and I also think it's important to incorporate variable predicates. My example isn't interesting because P is always true an Q is always true - so they inherently correlate. But variable predicates introduce an additional dimension: a predicate creates a world with constraints. Then what we're doing in proving an implication, say P(x) -> Q(x), is to show that in the world created by P(x) being true, Q(x) is always true. There's no causation there. I think I've wrapped my head around it. Thanks! — Samuel Wainwright, Aug 13 '23 at 09:55
@ryang yes, I've come to the same conclusion sat here thinking about it for a few hours. It really only starts to make sense to me when you invoke first-order logic. Then it's not a matter of causality as much as it is an observation of correlating facts given the constraints of the predicates. — Samuel Wainwright, Aug 13 '23 at 10:03
I know very little about this, but I think relevance logic is relevant here. I think material implication and classical logic in general are demonstrably not "perfect" for the reasons you mention and more, but they are practical and allow us to achieve a lot despite their shortcomings. — Nicholas Todoroff, Aug 13 '23 at 20:55
I agree with @NicholasTodoroff that you should learn about relevance logic. But I disagree about the relative value of relevance logic versus classical logic. Relevance logic seeks to incorporate a notion of causality, which you want, you but my attempts to understand causality have led me into philosophical swamps that seem not to be amenable to an adequate mathematical treatment. — Andreas Blass, Aug 14 '23 at 00:44
@AndreasBlass I think you may have misunderstood me; I was saying that while relevance logic is worth looking into/thinking about, classical logic has obvious advantages in practicality/ease of use. — Nicholas Todoroff, Aug 14 '23 at 01:03

Dan Christensen · Answer 1 · 2023-10-29T00:56:03.883

$A \implies B$ (material implication) should not be taken to mean that $A$ causes $B$. The standard interpretation in classical propositional logic is as follows:

$~~~~~~A\implies B~~~\equiv ~~~\neg (A \land \neg B)$

Or equivalently

$~~~~~~A\implies B~~~\equiv ~~~\neg A \lor B$

Example

Consider the implication: If it is raining ($R$), then it is cloudy ($C$).

$~~~~~~R \implies C$

This does not mean that rain causes cloudiness. In classical propositional logic, it means only that it is not the case that it is both raining and not cloudy.

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

Or equivalently

$~~~~~~\neg R \lor C$

Equivalent Truth Tables:

$R\implies C$ is false only if it is raining ($R$ is true), and it is not cloudy ($C$ is false).

Note that, as expected, if it is raining ($R$ is true) and it is true that $R\implies C$, then it must be cloudy ($C$ is true).

Note, too, that if $R$ is false, then it must be true that $R\implies C$ regardless of the truth value of $C$. This is the Principle of Vacuous Truth. Contrary to common misconceptions, we cannot then infer anything about the truth value of $C$.

The Principle of Vacuous Truth is rarely if ever used in daily discourse. Little consideration is usually given to the implications of propositions known to be false. It is, however, a valid method of proof in very technical arguments, e.g. in mathematical proofs.

But all of the examples used to illustrate conditionals in the books I'm reading rely heavily on causal connections to illustrate properties of conditionals. The common example used is "If I get an A on my final exam, then I will get an A in the class."

IMHO that is a common mistake by authors. To avoid confusion, antecedents and consequents should be stated in the present tense with no suggestion of causality.

Ethan Bolker · Answer 2 · 2023-08-13T19:40:09.553

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My answer speaks to

Where it really gets tricky for me is in actual proofs.

You are not alone.

I think that when reading and writing actual proofs implication really does express causality. The author provides reasons for the reader to believe the truth of some mathematical statement based on some previously established true statements.

That's a psychological assertion about belief. I tell my students that a correct proof is one that convinces me that you have convinced yourself of something for good reasons.

Since the the formal first order logic definition of implication deliberately avoids causality, that formality should not be part of the curriculum in a course where people are first learning to read and write proofs. (That sentence is an example of an implication that captures a causal relationship.)

When I teach such a course I deliberately leave formal first order logic out of my syllabus.

On rare occasions I have encountered a proof that's simpler when you admit the truth of a statement like "If I am the king of England then the moon is made of green cheese."

edited Aug 13 '23 at 19:40

answered Aug 13 '23 at 19:23

Ethan Bolker

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I totally agree with your characterization of correct proofs. I totally disagree that this characterization means that implication expresses causality. Mathematical implication expresses a containment: $p\to q$ means simply that whenever $p$ is true, $q$ is also true, regardless of any other relationship or lack thereof between $p$ and $q$. And I think that failing to teach students what mathematicians mean when they write "if...then..." in an intro to proofs class does them a disservice (e.g. it will lead to endless confusion about vacuous truth, questions like the OP's etc.) – Alex Kruckman Aug 13 '23 at 21:03
@AlexKruckman I think we need to disagree about some of this. Perhaps "causality" isn't the right word for what implication is in a proof. But I think it's more than "containment". A mathematician would never say "$2$ is prime, therefore $2$ is even". I would hate to see it in a student's proof. The way to avoid student confusion about vacuous truth is not to discuss it when teaching about proof. If you must have it in your discrete math syllabus do it at the end of the semester. Treat formal logic as a branch of mathematics, not as the underlying tool for mathematical reasoning. – Ethan Bolker Aug 13 '23 at 21:46
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@EthanBolker In my opinion, mathematicians don't say "2 is prime, therefore 2 is even" not because they disagree with it but because an implication without variables is awfully contrived and non-useful. The (mathematical) link between the antecedent and the consequent is not 'causality' but the background definitions (axioms) and theorems (non-axioms) that are typically implicit. – ryang Aug 14 '23 at 00:46
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@EthanBolker I'm not saying one should teach formal logic in an intro to proofs class. But things like vacuous truth are not purely "formal" - they're fundamentally baked into mathematical language, and students need to learn to understand and use this language correctly. Learning how to property understand "if/then" is just as important as learning, for example, that when mathematicians say "or", they mean the inclusive or, not the exclusive or. – Alex Kruckman Aug 14 '23 at 14:52
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I can't imagine a situation in which writing "$2$ is prime, therefore $2$ is even" would be the clearest way to express things when writing a proof. So I would also complain if a student wrote this. But if a student were considering the proposition "if $p$ is prime and $p\neq 2$, then $p$ is odd", I would expect them to be able to see that the special cases "if $2$ is prime and $2\neq 2$, then $2$ is odd" and "if $9$ is prime and $9\neq 2$, then $9$ is odd" are true statements. This is the kind of issue at stake here. – Alex Kruckman Aug 14 '23 at 14:57
@AlexKruckman The proper mathematical use of "or" is indeed crucial right from the start. But spending early class time on properly understanding "if/then" to include the confusing case is pedagogically wrong (in my opinion). (Also: I think my example on the evenness of $2$ was poorly chosen.) – Ethan Bolker Aug 14 '23 at 14:58
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Sorry to hammer on this - I'll drop it after this comment (it's just a pet peeve of mine that so many people view a fundamental and natural part of mathematical reasoning as somehow exotic or an unfortunate side effect of "formal logic"): In my pedagogical opinion, if you don't address the confusing case when teaching, students will continue to be confused. Their confusion may not surface immediately, because they may notice at first that there's something to be confused about. But there will be knock-on effects down the road. – Alex Kruckman Aug 14 '23 at 15:05

Understanding causality in basic propositional logic

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