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An example of a logical (material) implication that is commonly used is: "If it is raining outside, then the ground is wet." The problem with this example is that it could be seen as a causal relationship, i.e. that rain causes the ground to be wet, possibly leading to confusion conflating implication and causality--a common mistake among beginners.

Q: Might a better example not be: "If it is raining, then it is cloudy?"

I think (hope?) that it is common knowledge that rain does not cause cloudiness. This alternative example means only that, at the moment, it is not both raining and not cloudy.

$~~~~~~~~Raining \to Cloudy ~~\equiv~~ \neg [Raining \land \neg Cloudy]$

More generally, for any logical propositions $A$ and $B$:

$~~~~~~~~A \to B ~~ \equiv~~ \neg [ A ~\land \neg B] $

This interpretation can be easily seen to be consistent with the usual truth table for logical implication:

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The rule of detachment (modus ponens) can also be easily seen to hold by this interpretation. When both $A$ and $A\to B$ are true (only on line 1), then $B$ must also be true.


From my comment below: Care must be taken to avoid examples of relations that may be construed as causal. Ideally, both antecedent and consequent should be stated in the present tense.

Mathematical statements, e.g. y=x+1, should be thought of as being stated in the present tense.

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    $\begingroup$ In Mathematics, the implication-refers-to-causation myth is overstated; the real issue with using the first example is simply that it is a propositional-logic (PL) example which does not actually correspond to typical maths statements. When students raise questions about implications, it is invariably about false antecedents or due to them forgetting that something like Px⟹Qx is implicitly universally quantified (∀x), that is, not realising that mathematical implications are predicate-logic implications rather than the PL implications they've just encountered in their Discrete Maths class. $\endgroup$
    – ryang
    Apr 2, 2023 at 5:56
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    $\begingroup$ Rain without clouds?. — If the students know calculus, which mine do, then I use "If $f$ is differentiable at $0$, then it is continuous at $0$." I have the students try to generate examples of that show all (possible) combinations T/F. Things seem clearer in mathematics than in nature/society. — I don't see an actual question being posed above. I suppose you're asking for reactions to your opinion, or for others to share their experiences. $\endgroup$
    – user1815
    Apr 2, 2023 at 13:20
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    $\begingroup$ Are you asking for more examples? I didn't see a question in the post. $\endgroup$ Apr 2, 2023 at 14:52
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    $\begingroup$ [continuing] That these toy examples are merely isolated conditionals that aren't about logical implication/entailment (that is, a conditional being true regardless of the meaning of the words/symbols used) is another reason they feel artificial/contrived. What I mean is that 1) the truth of the raining-cloudy implication depends on the particular interpretation/context, and that 2) it is not being specified as part of an argument (as a premise/conclusion) and thus not being used for inference/deduction. $\quad$ 4 types of 'imply'. $\endgroup$
    – ryang
    Apr 2, 2023 at 15:03
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    $\begingroup$ Who are the students that you’re referring to? Students in a formal logic course, or students in an intro to proofs course, or some other group of students? Different examples are suited to different pedagogical contexts. $\endgroup$ Apr 4, 2023 at 0:19

4 Answers 4

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My answer is a bit of a frame challenge to the question (and to the part of Stephen Gubkin's answer where real world examples are said to be poorly suited for teaching purposes).

I agree that "If it is raining, then it is cloudy" might be advantageous since, in contrast to "If it is raining outside, then the ground is wet", it does not suggest any relation to causality.

However, I think both examples are actually formulated in a way that makes it difficult to understand the point of implications. For me the problem, though, is not that they are "real-world" examples, but something else: the wording does not make it clear that the involved propositions in both implications depend on a free variable - but this is actually the point why implications are so useful in mathematics.

Here's what I mean by this in detail:

Implication between propositions without variables and with known truth values

An implication between two propositions whose truth values are fixed and known appears to be intuitively odd, no matter whether in mathematics or in real life. Consider the following examples, the first of which I copied from Stephen Gubkin's answer:

  • If the moon is made of cheese, then I am the Pope.

  • If the moon is not made of cheese, then water is wet.

  • If $5 = 11-10$, then $3 = 2+9$.

  • If $5 \not= 11-10$, then $2+2=4$.

I've been working in mathematics for about 10 years now and still don't have any good intuition for any of those four examples. Actually, I find the last two just as non-telling as the first two. The intuitive problem with all four implications is that the truth values of their premises and conclusions are fixed and known - they do not depend on any free variable.

Propositions that depend on a variable:

Implications make much more sense intuitively if one uses them for propositions that depend on a variable and whose truth values can change depending on that variable. Examples:

  • If an integer $n$ is divisible by $4$, then it is also divisible by $2$.

  • If a function $f: \mathbb{R} \to \mathbb{R}$ is differentiable, then it is continuous.

  • If $H$ is a subgroup of a finite group $G$, then the cardinality of $H$ divides the cardinality of $G$.

  • If a Banach space $E$ is infinite-dimensional, then every algebraic basis of $E$ is uncountable.

The same works for "real-life" examples: as soon as they depend on a variable which might make the involved proposition true or false, things get much more intuitive:

  • If a planet supports the evolution of Carbon-based life, then there exists liquid water on it.

    Or, if you want to make the variable more explicit:

    If a planet $P$ supports the evolution of Carbon-based life, then there exists liquid water on $P$.

  • If it's raining on a street $S$, then $S$ gets wet.

  • If it's raining at a place $P$, then there are clouds over $P$.

The last two example are the ones from the question on which I commented at the beginning of this answer. The way I worded them now is, of course more technical - but I think this is actually an advantage for the following reasons:

  1. It makes it very clear and explicit that there is, in a sense, "uncertain knowledge" involved in the implication and that the uncertainty stems from the fact that we do not know the value of the variable. This illustrates the very point of a (true) implication: while there might be uncertainty about the truth values of the premise and the conclusion, the implication tells us that at least a certain relation between them is always true.

  2. In such an example one can easily illustrate all four combinations of truth values of the premise and the conclusion by inserting different values for the variable.

  3. Implications between propositions that depend on a variable is actually how implications occur in mathematical results, so those examples are a good way to prepare the students for this.

  4. The transition from everyday language to more formal logical expressions can be smoothened this way: We may start with the sentence "When it rains, it is cloudy", then let the students notice that there is an implicit assumption in this sentence that we talk about a specific location (since the whether is not the same everywhere), and then make this assumption explicit by denoting the location by a variable.

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    $\begingroup$ @StevenGubkin: Yes, I agree. The part of your answer with which I don't really agree is the claim that real world examples were poorly suited for implications in most cases. I think this mainly happens when there's no variable over which one can quantify (e.g. you're "moon made of cheese" examples). I'll edit to clarify that I'm referring to this specific part of your answer. $\endgroup$ Apr 5, 2023 at 19:57
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    $\begingroup$ @JochenGlueck I disagree that all examples of "useful" implications are of the form $\forall x: P(x) \implies Q(x)$. For instance, we might usefully prove a theorem of the form "If there exists a largest twin prime $p$, then [some inequality is true]." Here both the antecedent and consequent are both "plain facts", but the theorem could still be useful as part of an attack on the twin prime conjecture. We still prove the theorem the same as always: assuming we have a proof that an integer $p$ is the largest twin prime, we attempt to prove the inequality. $\endgroup$ Apr 5, 2023 at 20:32
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    $\begingroup$ I guess the point is that stating an implication between statements with known truth values is useless (so feels "wrong" to say), but as long as the truth values are currently unknown the implication seems useful. $\endgroup$ Apr 5, 2023 at 20:47
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    $\begingroup$ @StevenGubkin: You are of course right. I don't think we disagree that much on the topic - we might just have a somewhat different focus. And I think it might indeed be a good idea to explain implications by discussing different types of examples where the truth values of the antecedent and consequent are not known - with the case where the propositions depend on a variable being one prominent special case. $\endgroup$ Apr 5, 2023 at 21:25
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    $\begingroup$ Jochen's answer complements my three comments under Dan's "question", as does this answer that I just posted at mathematics.SE. $\endgroup$
    – ryang
    Apr 7, 2023 at 18:34
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I personally think all of these real world examples do a pretty poor job of helping students form an understanding of implication which is both accurate and useful in forming arguments.

For instance,

  • False implies False "If the moon is made of cheese, then I am the Pope".
  • False implies True "If the moon is made of cheese, then the moon is not made of cheese".
  • True implies True "If 1 + 1 = 2, then in nineteen ninety eight the Undertaker threw Mankind off hеll in a cell and he plummeted sixteen feet through an announcer's table."

These all highlight the fundamental lack of "relevance" between the antecedent and consequent of an implication. There have been attempts to fix this such as https://en.wikipedia.org/wiki/Relevance_logic.

Since these kinds of examples are just as "valid" as the more "sensible" ones you are coming up with using rain and clouds, they must also be accepted by the student for them to have a truly robust model of material implication.

My own hot take is to introduce introduction and elimination rules without mentioning the truth table. These rules are motivated by the following interpretation:

Interpretation: The implication $p \implies q$ is true exactly when given a proof of $p$, we can produce a proof of $q$.

Introduction rule: to argue that $p \implies q$ is true assume we have a proof of $p$ and argue $q$ relative to that assumption.

Elimination rule: A proof of $p$ and a proof of $p \implies q$ together constitute a proof of $q$.

These are the things which are useful for actually doing and using mathematics. The truth table is really not. It is a distraction.

I would focus on using and proving sensible mathematical implications for a long time before discussing truth tables. Things like "For all integers x, if 15 divides x then 5 divides x".

I would also define absurdity by $\bot \implies p$ for all statements $p$. This definitely requires some motivation (such as $1=0$ implying both other absurd statements like $2=0$ as well as true statements like $0=0$: the proof of the first is to multiply both sides by 2 and the second by 0).

You can also mention that this definition of absurdity permits a uniform treatment of the proof of quantified implications. For instance in the proof of "For all integers x, if 15 divides x then 5 divides x" we take an arbitrary integer $x$ and then argue that $5$ divides $x$ under the assumption that we have a proof of $15$ dividing $x$. If $x = 45$ this proof will do a beautiful job of showing exactly how to take the proof that 15 divides 45 (namely demonstrating that for $k=3$ we have $45 = 15k$) and turning it into a proof that $5$ divides $x$ (namely $ j= 3k = 9$ will work to show that $45 = 5j$). However if $x = 17$, and we still want "the same argument to work" we need to accept that a false premise can lead to the conclusion we desire.

After getting a lot of experience proving honest mathematical theorems using the introduction and elimination rules I would introduce truth tables as a kind of "algebraic shadow" of our reasoning processes. It is an arithmetic of truth, but does not fully capture the subtleties.

Then the rows of the truth table become fairly intuititive: $\bot$ implies everything, if a statement is "true" then it ``proves itself'', so a valid proof of $q$ starting from an assumption of $p$ is to just state $q$, and it should be evident that there is no valid way to argue a false conclusion from a true premise.

I also think that relying on excluded middle as heavily as you do in your proof is a mistake. I try to avoid excluded middle whenever possible, and I would certainly not want to wed it to my fundamental understanding and motivation for implication.

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    $\begingroup$ For sound pedagogical reasons, truth tables and the equivalence (A => B) <=> ~(A & ~B) are often used to define logical implication in introductory texts. It seems to me that their formal justification using formal rules of introduction and elimination are usually far beyond the scope of such texts and the abilities or motivation of many students. I myself used these "definitions" for many years before realizing they could be derived from "first principles." Also, I see nothing wrong with relying on proofs by contradiction. I know of no good reason to avoid them. $\endgroup$ Apr 3, 2023 at 16:23
  • $\begingroup$ If you are interested in formally deriving these "definitions" from first principles, see dcproof.wordpress.com/2017/12/28/if-pigs-could-fly $\endgroup$ Apr 3, 2023 at 17:49
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    $\begingroup$ @DanChristensen We certainly have very different "taste" in logic, which is fine. For my, I would hate to prove that $15$ divides $x$ implies $5$ divides $x$ by showing that assuming $15$ divides $x$ and $5$ does not divide $x$ leads to a contradiction. It would be much more natural to proceed "directly". $\endgroup$ Apr 3, 2023 at 19:03
  • $\begingroup$ I avoid proof by contradiction for many reasons, but chief among them is that it destroys the constructive character of the proof. I can prove that there is an irrational real number by showing that assuming there is not leads to a contradiction (namely that the reals would be countable). I think this proof is much less informative than just directly demonstrating an irrational number such as $\sqrt{2}$. $\endgroup$ Apr 3, 2023 at 19:05
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    $\begingroup$ @DanChristensen No, would not penalize! Would definitely point out when it is being "overused" however. $\endgroup$ Apr 3, 2023 at 21:32
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I don't like the whole example of "It's raining, which means the ground gets wet.", because:

  • It might rain but the ground is covered, so it doesn't get wet.
  • The ground gets wet, even when it's not raining, because you spilled a glass of water on it.
  • ...

Therefore you might better give an example which is less doubtful, like:

I am located inside this classroom, so I am located inside this school building.

As there is no way being inside the classroom without being at school grounds, this is much clearer.

If somebody says "But I can still be on school grounds without being inside this classroom", then you can react that this is a perfect example of $A \Longrightarrow B$ without $B \Longrightarrow A$.

In top of that, this example show a link between logic and set theory, which might be useful at later ages.

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    $\begingroup$ Also a good example because there is no suggestion of causality. And both antecedent and consequent are in the present tense. $\endgroup$ Apr 12, 2023 at 11:19
  • $\begingroup$ If we are insisting on sticking merely to incidental truth/implications rather than analytic implications, then those moon is made of cheese and circles are round examples straightforwardly show that does not fundamentally indicate influence/causality/prophecy, but rather, is essentially a truth-table-row-matching logic gate that sometimes happens to coincide with causation. And once we are willing to actually consider analytic implications, then that is a reasoning relation becomes more evident. $\endgroup$
    – ryang
    Apr 12, 2023 at 15:25
  • $\begingroup$ @ryang Have such "analytic implications" as you call them been shown to be useful in applications? $\endgroup$ Apr 12, 2023 at 17:20
  • $\begingroup$ @DanChristensen Do mathematical theorems have applications? $\endgroup$
    – ryang
    Apr 12, 2023 at 17:32
  • $\begingroup$ @ryang Yes. "2+2=4," for example, is a theorem with many applications. It is a theorem in the sense that it can be derived from axioms using the rules of classical logic. $\endgroup$ Apr 12, 2023 at 17:48
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I understand your question to be about the efficacy of replacing the language "A implies B" with "not A or B." In other words, is there benefit in teaching to avoiding $A\rightarrow B$ and reflexively converting to the conjunctive normal form $\neg A\vee B$? The reasoning goes that "implication" carries too much experiential baggage by intelligent agents such as students, and humans get mired in "causation versus correlation" and other matters irrelevant to logic.

I am trained as a mathematician, and only recently learned how people in artificial intelligence make fundamental use of what they call "proof by resolution" in the context of logical agents. See, for example, chapter 7 of Artificial Intelligence: A Modern Approach, fourth edition by Russell and Norvig. They start with a knowledge base $KB$, tantamount to a set of axioms in propositional logic or first-order logic, and convert everything to conjunctive normal form. Then to decide if a statement $\alpha$ is entailed by $KB$, in symbols $KB\models\alpha$, one adds $\neg\alpha$ to $KB$ and applies proof by resolution, which amounts to nothing more than algebraic cancellations. For example, if $KB=\lbrace A,A\rightarrow B\rbrace$, the conjunctive normal form is $\lbrace A,\neg A\vee B\rbrace$. If we want to decide if $B$ is entailed by $KB$, we work with $\lbrace A,\neg A\vee B,\neg B\rbrace$. The we crank the gears of proof by resolution, which in this simple-minded case amounts to "algebraically" canceling $A,\neg A$, and then in the next step algebraically cancelling $B,\neg B$ leaving "absurdity" or $\bot$.

The perspective of working exclusively with conjunctive normal forms, setting up a query $\alpha$ as a proof by contradiction (add $\neg \alpha$), and then going through algebraic cancellations of appearances of $x,\neg x$ is something that is amenable to artificially intelligent agents, who probably have no experiential baggage related to the implication, causation vs correlation, etc. A student is somewhere between an artificially intelligent agent and the "expert system" human teacher.

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