How to teach logical implication?

Question

One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences.

Question: What are good point of view, methods and tips to help students grasp the concept of logical implication?

To focus the question, I would like to restrict to math majors, although the question is probably equally interesting for other kind of students.

Answer 1

Various psychological studies have been done which show that most people (including university students, who are the most common subjects of psychological tests!) are very poor at grappling with the last two entries of the truth table for $A \implies B$ in an abstract context, but they are much better with it in a situation in which the consequences of falsifying the implication are understood.

When I taught (twice) a "transitions" course for undergraduate math majors, I gave them this essay question.

T3.1) a) You are shown a selection of cards, each of which has a single letter printed on one side and a single number printed on the other side. Then four cards are placed on the table. On the up side of these cards you can see, respectively, D, K, 3 and 7. Here is a rule: "Every card that has a D on one side has a 3 on the other." Your task is to select all those cards, but only those cards, which you would have to turn over in order to discover whether or not the rule has been violated.

b) You have been hired to watch, via closed-circuit camera, the bouncer at a certain 18-and-over club. In order to be allowed to drink once inside the club, a patron must display valid 21-and-over ID to the bouncer, who then gives him/her a special bracelet. In theory the bouncer should check everyone's ID, but (assume for the purposes of this problem, at least!) it is not illegal for someone who is under 18 to enter the club, so you are not concerned about who the bouncer lets in or turns away, but only who gets a bracelet. You watch four people walk into the club, but because the bouncer is so large, sometimes he obscures the camera. Here is what you can see:

The first person gets a bracelet.
The second person does not get a bracelet.
The third person displays ID indicating they are 21.
The fourth person does not display any ID.

You realize that you need to go down to the club to check some IDs. Precisely whose ID's do you need to check to verify that the bouncer is obeying the law?

c) Any comments?

Part a) is (up to psychological isomorphism) the infamous Wason selection task. Part b) is a real-world analogue designed to be closer to the students' experience. It makes perfect sense that we do not need to recheck the IDs of anyone who did not get a bracelet: we're trying to enforce the implication "If you get served drinks, you must have ID." People can understand that no one is going to get in trouble for the people that they didn't serve drinks to.

One can see relations here to one of the other (good) answers. For one, yes, it's good to think in terms of when $A \implies B$ is false: there's just one possibility and that's what we care about, so in every other case we make it true. But yes, I do introduce implication via the truth table. It can also be helpful to define it as "(not A) or B": that somehow seems less arbitrary, and gives them good practice seeing that the negation is "A and (not B)". But then we have the burden of explaining why we call this "implies"....and I've found that if you emphasize that the one possibility you need to exclude is that A is true and B is false, then it is not in fact so terribly hard for students to swallow. I follow up with the concept of "vacuously true", namely the implication is true because the hypothesis is false. This becomes a key proof technique later in the course: sometimes you need to begin to analyze an implication $\forall x \in A$, $P(x) \implies Q(x)$ by first figuring out for which $x \in A$ it is the case that $P(x)$ is true.

Answer 2

I find it helpful to introduce the negation of conditional claims simultaneously. For one, this better helps them to understand the "false implies false" case; but also, this helps them understand how to logically negate conditional claims (which is essential when they go on to learn proof techniques for conditional claims).

The classic "If it is raining, then I definitely have an umbrella with me" is my go-to. I say to the students: "I assert that conditional claim. How could you possibly call me out to be a liar?" They talk amongst themselves and realize that the only way to call me a liar is if they observe me walking around sans umbrella in the rain; all other situations do not yield a falsehood, so they must be true.

(Admittedly, this might be passing the buck to accepting the Law of the Excluded Middle, but I've found students are far more comfortable with "True or False?" than they are with "'false implies false' is true" :-) )

Answer 3

Historically, quantificational logic came before propositional logic.

So maybe start with quantified sentences instead. It is reasonable to say both "Every ten-foot-tall person plays basketball" and "No ten-foot-tall person plays basketball".

This leads to accepting both Ten-Foot-Tall $\rightarrow$ Plays-Basketball and Ten-Foot-Tall $\rightarrow \neg$ Plays-Basketball.

Answer 4

When I've taught propositional logic I acknowledge that this is a formalism that doesn't perfectly match the English usage, and use it as an opportunity to point out

1. The evaluation of $\rightarrow$ has to be purely a property of truth values, whereas "implies" in English involves the meaning of the statements, not just whether they're true or false. This is an important property of the propositional logic, and this is the first good opportunity to emphasize it. (My students usually agree that, if $\rightarrow$ has to be truth functional, the accepted interpretation is probably the best that can be done.)
2. They're in good historic company, and there are variations of propositional logic (like modal logic and relevant logic) which try to address exactly this issue, but they have to give up having everything be a function on truth values. This is a nice chance to indicate why there's more than one notion of formal logic.

Answer 5

I have come to understand this to be analogous to an order relationship among the truth values. P -> Q should be understood to mean "Q is at least as true as P", or "Q is not less true than P".

So, any statement at all (Q = t or f) is at least as true as a known falsehood (P=f), and a known truth (Q=t) is at least as true as any other statement (P= t or f).

This can be presented in connection with the most common instance of valid reasoning, modus ponens, and distinguishing two parts of sound reasoning: true premises, and valid reasoning. What P -> Q to be true means is that it is logically valid: Using it will not introduce a false conclusion. If we know or assume that P is true, and "if P then Q" is valid, then it is safe to conclude Q. We don't often consider the case of when P is false. However, if know or assume that "If P then Q" is valid, but P is false, that combination of facts tells us nothing about Q, and it could equally well be true or false.

Answer 6

My approach is like that of others but I like to use math instead of everyday language. I get them to agree that we want this statement to be true: "if x is a perfect square then x is not prime" simply because x=y*y is a factorization. Then we use various x's to get the different lines of the truth table.

Answer 7

Let me add the tricks I used already to the interesting ones proposed in the other answers. They are partly, but not completely redundant.

First, implication is often best understood in quantified propositions; this has already been pointed out by Matt F., but I am not too found of allegedly real-life assertions, which usually sound rather artificial. I rather use statements like: "every integer $n$ that is divisible by 6 is also divisible by 3" or "for any integer $n$, if $n^2$ is even then so is $n$". One of the best of this kind is "for any integer $n$, if $n$ is a multiple of 4 then $n^2$ also is a multiple of 4": you have all three cases where implication is true by taking $n=2,3,4$.

The second angle is to give an example where one wants to state and use an implication without knowing the truth value of its terms. Here ironically, an allegedly real-life assertion does the job quite beautifully: assume you are investigating a murder that took place in London Saturday and that Colonel Mustard is a suspect. You can say confidently that "if Colonel Mustard is guilty, then she was in London Saturday" even if you do not know whether she is guilty nor whether she was in London Saturday. What is especially good in this example is that it is a good way to introduce contrapositives: the statement "if Colonel Mustard was not in London Saturday then she is not guilty" shows convincingly that the contrapositive has the same truth value than the original implication. I learned this example from Viviane Durand-Guerrier, a researcher in educational studies in mathematics in Montpellier.

Answer 8

One of the things that really helped me (not to learn it, but to appropriately apply it), back when I originally learned this myself was the equivalence between $p\implies q$ and $\neg p \vee q$. I constantly reminded myself of it when manipulating logic statements. Make sure the students are aware of this equivalence.

What it did was give me a basis of comparison to something I already understood well. It gave me something to check in case I was worried I have made a mistake. It made negation of implication a lot easier to understand as well.

Answer 9

To supplement Brendan's idea: I like to connect quantified statements to unquantified in the following way: Assume a statement like "If X is a dog, then X has a head". Now, once you have found this to be true, you might want the truth not to depend on X. Thus replacing X for anything like "my car" should still give you a true statement. However, "If my car is a dog, then my car has a head" should be true then.

A second remark: At this point one should make clear, that the truth of statements is not given by nature, but something we define. So the question should not be which truth value is right, but which one makes more sense.

Edit: third remark: There are statements which feel "more true" than others, althoug they are of the form "A=>B" with A, B being wrong. Compare "If I am the pope, then I am a woman." and "If I am the pope, then I live in Rome". For me (not the pope, male, not living in rome), all parts of the statements are wrong. However, the second feels true, whereas the first one feels wrong. So it is worth a discussion if and how one should calculate truth values of conditional statemens from the truth values of their parts.

Answer 10

1. Avoid real world analogies. They confuse students because real world means natural language. Instead remind them of implication and its truth table whenever you write a mathematical proposition on the board.

2. Remind students to work with the definitions. Over and over again.

3. Introduce the notion of a vacuous truth (or vacuous implication) as soon as possible, I prefer the first class of the semester (even if it's before anything related to propositional calculus), and point it out explicitly whenever it comes up.

Answer 11

I am really impressed by the answers given by others here; I will definitely keep them in mind when teaching freshmen next semester. But I also have my 2 cents to add, since I haven't seen anything like that in them. I realize that this is only some kind of vague intuition, and it would probably confuse a lot of students, but it might as well help some. (It did help me at some point, at least.)

So let us assign a "$0$" to false statements and a "$1$" to true ones (this is a common convention at least here in Poland). Now "$p\implies q$" is true iff $p\le q$ (note that in the first formula we treat $p$ and $q$ as propositional variables, and in the second as numeric variables, which is clearly an abuse of notation!). This way one can view a material implication as a way of saying that "one sentence ($p$) implying another one ($q$) means that the latter one must be at least as true (whatever that means!) as the former one". In other words, when going from the antecedent to the consequent, we cannot "lose knowledge", only gain it. (Now this is really stretching things from philosophical point of view, and logicians would probably torture and kill me for that; but then again – this is only a (vague) intuition I'm talking about).

The way I present this to students (if I do!) is more or less this (with a wink): "So just like $p\land q$ is somehow similar to (but different from!) $p\cdot q$ – so that even we call $p$ and $q$ the "factors" of the conjunction [at least this is what we do in Polish; we also sometimes call a conjunction a "logical product", and we do similar things with alternative (i.e., use words "logical sum" and "summands")] – in a similar vein, $p\implies q$ is somehow similar to $p\le q$. But you know, better forget about it, since "truth" is not "one" and "falsehood" is not "zero" anyway, so what I've just told you is more or less a lie anyway."

Answer 12

I think the intuition for implication is aided when you consider it in the context of universal quantification.

A claim of the form $(\forall x)(A(x) \Rightarrow B(x))$ can be translated roughly as every $x$ that has property $A$ must also have property $B$. Most of the use of conditional claims used in practice have this form implicitly assumed.

For example, using Brendan Sullivan's example which I agree is good for explanation, when someone says, "If it rains, I use an umbrella," the implicit reading is that this is a universal principle with the logical structure of $(\forall t)(\text{it rains at time } t \Rightarrow \text{I use an umbrella at time } t)$. Intuitively, we grasp that whatever happens when it is not raining cannot affect the truth value of the universal implication.

The challenge in teaching this, of course, is that universal quantification is often not introduced until well after logical implication, but maybe that should be done differently.

Answer 13

Here is an example I sometimes use to motivate the truth table for implication. The example taps in to our ability to recognize cheaters.

Suppose you go to the vending machine. The price of a soda is one dollar.

1. Suppose you put a dollar in the vending machine and receive a soda. Do you feel cheated? No!

2. Suppose you put a dollar in the vending machine and do not receive a soda. Do you feel cheated? Yes!

3. Suppose you do not put a dollar in the vending machine but receive a soda anyways. Do you feel cheated? No!

4. Suppose you do not put a dollar in the vending machine and do not receive a soda. Do you feel cheated? No!

Answer 14

If you will eventually be teaching the basic methods of proof (conditional proof, proof by contradiction, etc.) in your course for math majors, you might consider starting with the truth tables for NOT, AND and OR and postpone the truth table for IMPLIES until they understand some of those basic methods of proof. Then they should be able to understand a proof of ~A => [A => B] (see below) which corresponds to lines 3 and 4 of the usual truth table for IMPLIES, and IIUC is usually where problems seem to arise. (Screenshot from my proof checker.)

Answer 15

In my opinion, the truth-table definition of material implication is disturbing because "if ..., then ..." is used in mathematics in two distinct ways (and no similar distinction exists for the other connectives like NOT, AND, and OR):

1. "If $p$, then $q$" means that you can start out with $p$, make some deductions, and end up with $q$. Or more precisely, it's a sentence you can interact with in the following two ways: (a) You can conclude "if $p$, then $q$" by starting out with $p$ and then deriving $q$. (This is often called a conditional proof or direct proof.) (b) You can make use of a sentence "if $p$, then $q$" if in addition you know $p$, to conclude that $q$ is true. (This is called modus ponens.)
2. As the truth table definition.

These end up being the same given the rest of classical logic as background, of course, but they are conceptually distinct. I think what is necessary is to connect these two concepts. In other words, for the truth table definition to make sense, and for a logical connective to be called an "implication" or "conditional", it has to be shown equivalent to the deductive sense (1) above. In mathematical logic we have the distinction between $\{p\} \vdash q$ and $\vdash p\to q$, which are shown to be equivalent by the deduction theorem (and its converse), but I believe it is possible to show this same kind of thing without bringing in all of the scary formal notation from mathematical logic.

I have a longer essay that implements the above strategy for explaining the material implication.

Answer 16

This may add to the detailed discussions and explanations above. I had problems with mathematical implication until I read Keith Devlin's "Introduction to Mathematical Thinking". Hopefully what follows is a faithful summary of some of the ideas in the book.

1. Implication is not the conditional
2. Truth tables can help

1 We learn and experience that most things happen because of causality, or our interpretation of it. $A$ happens then $B$ happens. This gets expressed in different ways: $A$ happens so $B$ will happen, if $A$ happens then $B$ will happen, etc. These different expressions all get conflated and some people learn or are taught they all are represented by the expressions:

"if $A$ then $B$"

or

$A \implies B$

All of the above are wrapped up in the the idea of implication.

We all feel comfortable, or we internalise that when $A$ happens, $B$ will certainly happen because $A$ caused $B$. This is implication.

However, we are left with an uncertainty about what happens when $A$ doesn't happen.

To deal with this uncertainty in mathematics we need to abandon the idea of causality and fall back to a reduced idea of the conditional.

In Devlin's terms

"implication is the conditional with causality"

or the equivalent

"the conditional is implication without causality"

So what?

Well we can choose to define the conditional for every possible (4 ways) of combining $A$ and $B$ being True or False. I will break off point 1 here then come back.

For point 2 I understand these things in truth table terms, not linguistic terms.

2. In parallel, if we think of simple logic (I am not sure of the exact terminology - propositional logic?) we can combine two statements $A$ and $B$ with a variety of operations: and, or. We learn (usually passively) even if we can't express the ideas formally, about the meaning of compound statements $A$ OR $B$, $A$ AND $B$ whose truth value depends on the truth value of $A$ and $B$ and we can define these in truth tables. In more concrete terms, most people are comfortable with TRUE OR FALSE = TRUE and TRUE AND FALSE = FALSE and so on for all possible combinations.

Again, so what?

Well, $A \implies B$ is also a compound statement whose truth value depends upon the truth value of $A$ and $B$. We can now define the truth table values for False $\implies$ False to be TRUE. That's weird and uncomfortable.

There are two ways to deal with this first uncomfortable weirdness.

Either, claim that

$A \implies B$

is equivalent to

$\neg A \lor B$

derive the truth table values and demonstrate no contradictions,

alternatively (and this is how I deal with the situation - Devlin does not write this), we do not think about the result (truth value) of a compound statement for various "input" values as being True or False, but just ask whether the logical argument is valid (in place of true) or invalid (in place of false).

For example if A: x = 3, is False and B: x < 7 is False

then

$A \implies B$ in this case is a valid argument. In this case: if x $\neq$ 3 then x $\geq$ 7 is valid and hence true.

Rather than the truth table values being some arbitrary definition of the "meaning" of $A \implies B$, the conditional contains and is consistent with our hazier understanding of implication or if ... then, where A "causing" B is wrapped up in our understanding.

So back to 1. My understanding is this

$A \implies B$ is the conditional without causality, and all written or spoken expressions such as "implies" or "so" or "when" or "sufficent" or "necessary" I translate into the form $A \implies B$ ($A \to B$ ???) and work from there.

I would recommend Devlin's book (I have no commercial interests or connection) and the free Stanford/Coursera course by Devlin that covers the same ground.

Answer 17

I use a specific example: If I'm in the rain then I'm wet. I also emphasize that we're talking about whether it's possible for the statement to be true.

Now think about the four cases:

true/true: If you are in the rain then it's possible for you to be wet so this is a true result.

true/false: If you are in the rain then it isn't possible for you not to be wet so this is a false result.

false/true (which I think is the most confusing): If you aren't in the rain, is it possible that you're wet? The answer to that is yes because you may have gotten wet by other means. That makes this a true result.

false/false: If you aren't in the rain it's possible that you aren't wet so this is also a true result.

Answer 18

My favour example is the implication, "If it is raining, then it is cloudy."

This does not mean that rain causes cloudiness. (There is no causality in mathematics.) In the usual sense of if-then statements in mathematics, it means only that it is not the case (present tense) that it is both raining and not cloudy. Using this definition, we can easily construct the truth table:

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

1. $~~T~~T~~~~~~~~~T$
2. $~~T~~F~~~~~~~~~F$
3. $~~F~~T~~~~~~~~~T$
4. $~~F~~F~~~~~~~~~T$

Where $R = $ "It is raining", and $C = $ "It is cloudy" (both in the present tense).

(We can also formally prove the above "definition", $(A\implies B) \iff \neg (A \land \neg B)$ from "first principles" here. Only 19 lines.)

This is the so-called material conditional. (Other forms of the conditional may be required for past and future tenses and causality. See Wikipedia)

As required, if it is true that it is raining ($R$), and it is true that if it is raining then it is cloudy ($R\implies C$), then $C$ must be true (see line 1).

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

This is the so-called Detachment Rule.

To disprove that $R\implies C$, it is sufficient to prove that $R$ is true and $C$ is false (see line 2).

Interestingly, if it is not raining ($\neg R$), then it is true that $R\implies C$, regardless of the truth value of $C$ (see lines 3 and 4).

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

Note that we cannot infer from this statement that $C$ is true, or that it is false since $A$ itself is false.

This is the so-called Principle of Vacuous Truth. It is rarely if ever used daily discourse since we don't usually give much consideration to implications with a false antecedent. It is, however, a valid and useful method of proof in mathematics.