Monday, February 20, 2012

Wacky Student Time: Conditionals

Over the last four or five years, I've noticed that an increasing proportion of students demonstrate an increasingly difficult time understanding how conditionals work. To the point where a huge proportion of them now seem incapable of understanding sentences of the form "if P, then Q". They act as though it means the same thing as plain old "P and Q".

This causes problems. For one thing, it makes it hard for them to understand, in a general way, what in the hell is going on with things. It's as though they see the world as a series of fundamentally unconnected, independent occurrences. They see two facts sitting there by themselves, and it doesn't occur to them that they might have anything to do with each other—that they might be somehow connected. (And I find it very hard to believe that this accurately reflects how they see the world. It can't be that they don't see a connection between the drinking and the drunkenness, for example. They can't see the drunkenness as unconnected with the drinking. Can they? They can't.)

For another thing—and this is where the confusion really makes itself known—it causes problems when they try to explain how the philosophers we study defend arguments in which they employ conditional premises. Which is almost all of them. And it causes real trouble when they try to defend a conditional that comes up in the context of an argument in the form of modus tollens. In that case, they're trying to explain why some person thinks that "P and Q" is true, when they know that the very next thing is to explain why the same person thinks Q if false. They tie themselves into highly confused, contradictory knots.

Relatedly, students often find it hard to explain how to defend a premise when the student him- or herself believes that the premise is false. That is, they find it hard to explain why someone would think that P is true when they think (or maybe they even know) that P is false. I think this is closely related to the conditionals thing. They don't realize that you can use P even if you don't believe that P. Maybe there are interesting hypotheticals involving P; maybe Socrates believed that P; maybe reasonable people disagree about P. But they can't see it, or at least they act like they don't. If they believe that P is false, they find it impossible to consider what things would be like if it were true, or why someone else might disagree with them.

I think this is pretty strange. Right? I don't really remember when I learned that if/then sentences don't just mean that P is true and so is Q; that they assert some kind of connection between P and Q. But I think I don't remember because it's so obvious that this is what it means that I didn't find it noteworthy (or else it happened so long ago that I forgot about it). So, I guess the question is, how can these students be so oblivious to something that's at once so obvious and so crucial to understanding how the world works?

--Mr. Zero

26 comments:

Anonymous said...

I think it's more about how logic is taught than about students' obliviousness. You couldn't function in the world if you didn't get the basic gist of conditionals.

With philosophy, a lot of it is just word games to students (esp. non phil majors, which are the majority of students I teach), so they'll make mistakes often if they're not paying attention or don't particularly care or aren't thinking through the subject matter. But it's not that they don't 'understand' conditionals in general.

Talk to someone about something they care about (sports, parties, videogames, art, chemistry etc) and they'll be able to think through complex logical relationships. I think the issue is getting them to connect more with the material to see that they already know a lot about conditionals; they just haven't thought about it abstractly a lot.
*This is for students who are actually trying.

Anonymous said...

You're argument evidences some invalid assumption's. You say that "If they believe that P is false, they find it impossible to consider what things would be like if it were true, or why someone else might disagree with them". However, but some of they believe P is true. So it is shown to be invalidated, when he writes that he believes that P is false. Q.E.D.

Anonymous said...

Quick points:

1) To reason from a premise one does not endorse is a skill, or at least that's what some of the psychological literature suggests.

2) In ordinary English we don't often say "If P, then Q" unless we mean to convey something stronger than the material conditional. I've found this trips up a lot of students. You can tell them about denying the antecedent, they'll nod, recite it, and then botch the exam question. If fallacies have names, they've probably tripped up a lot of people.

I've had a lot of success in intro classes by explaining the principles (analogies to programming help for some), giving some examples, and then having them work through some basic problems in groups: tell me if it's valid, tell me if it's sound, etc. Drilling it in groups and then coming back as a class to discuss it seems to help them get it.

Anonymous said...

Terminology also trips up some people *a lot*. For them, logic and philosophy are like learning a new language and then we demand they construct and parse sentences in this new language after we just gave them the vocabulary.
--Or they're like English class where the topics are familiar but the approach is abstract and unfamiliar.

For instance, I don't think well with abstract, empty letters: ("If P then Q unless R for any person S at time T"). I can think through concrete examples faster to extract a logical concept.

Because all the students we teach know that a cat can't be alive and dead inside a box at the same time (and this is what makes quantum physics weird and cool). And they all know that if you win the lotto, you'll be rich, but just because you're rich doesn't mean you won the lotto.

Then again, US edu in general has dropped the ball on teaching this kind of abstract thinking and the critical thinking that could get a student to easily consider a line of argument whose conclusion they don't agree with. So it's not entirely on our shoulders.

*Occasionally, I find a student tripped up because they think that at the end of class there will be a "right" answer to whether (e.g) ethics are duty-based or utility-based, so there's that, too. They ask, "Ya, but was Aristotle right?"

**And some students think philosophy is like first grade art class where no matter what you say/think, it's special and unique and deserves a gold star.
They use the phrases "my opinion" and "I believe" a lot.

Again though, partly philosophers' and partly society's fault for not giving students a good working definition of philosophy. And partly students' fault if they're not paying attention or don't care.

Anonymous said...

Interesting. And sad. I have noticed these inabilities, but not their increasing frequency. It seems the problems have something in common: an inability to imagine. It's as though both their current belief state and the state of the world are frozen or crystalized or something such that they are necessarily that way. There is no other way they could possibly be.

I think understanding both of these as a failure of imagination allows one to invent exercises designed to overcome them. So, the inability to imagine in this context cannot be some generalized inability to imagine at all--or the world really is as bad as I sometimes think it is. Anyway, surely they have wishes, desires, predictions, etc. about the future and therefore can understand possibility v. actuality. The rambling point here, I think, is that perhaps there is something about being in philosophy class, class in general that shuts off their mundane abilities. So, an answer is to try to get those re-engaged. Maybe...
[renefan seems] The renefan seems to be on the fritz again.

Anonymous said...

I would conjecture that it might be the content of P and Q that is the culprit here, not necessarily the conditional itself. I don't think you need buy into the Evolutionary Pscyh module stuff to find plausible that philosophical conditionals might cause problems for the average student. I would try starting the students off with a few softballs. For example,

- If Jones is U.S. president, the Jones is a U.S. citizen.

Most of my logic students go along with this even though the antecedent is clearly false.

Anonymous said...

Students have all sorts of problems with conditionals. I haven't experienced too many cases of the one you mention, but there is one that has been increasingly common over the past couple of years.

I've read papers where students grant that if p, then q and also grant that p (and so, q). So it seems like these students understand how modus ponens works. However, these same students will go on to say something like "but it's just a hypothetical statement, so it doesn't show anything." I really don't know what these students think conditional statements are saying.

Anonymous said...

I have noticed this when I try to teach modus ponens. Students seem to have less difficulty with statements involving "all" and "only", though. This has led me to wonder whether syllogistic logic better approximates natural human reasoning than does propositional logic; I wonder if this has been explored by psychologists.

Anonymous said...

It's difficult to convince students that a conditional relation is not a relation of causation, but one of implication.

Anonymous said...

to follow up on Anon 11:30 AM: I always try convincing my students that they tacitly know the stuff or believe some of the stuff I am going to teach them before I teach it by using mundane examples. I do it to convince them that philosophy is real, and also to give them some initial confidence that maybe they will be able to understand what I am about to teach them. I don't know if it works :(

Anonymous said...

Since I just spent an entire class period today trying to get students to understand what an "implication" is, I am really, really relieved to see that others have had similar problems. I ended up starting with a similar softball approach as anon. 12:44, and then working up to more complex relations of implication (this is an intro to philosophy, not a logic class, so I need them to be able, at minimum, to speculate on the necessary consequences of a particular belief).

My experience has been that their first impulse is, as you suggest, to simply deny that an antecedent they don't like is true, instead of trying to anticipate the Q. Or, more often, they suggest that this not-P actually IS the Q. (The other day I said something like "Let's say you don't believe in souls; you just think that bodies are all there is--what other kinds of things would you think? The first response was "well, you'd need to have a soul.") And yes, it gets really difficult when we start seeing modus tollens arguments in the text. In those cases, I find that the response is almost always to claim that the philosopher "is just going in circles."

I'm pretty sure the answer is to show them examples of the ways in which they use this style of reasoning in their own lives, and then draw the analogy. But from this side of things, it is so difficult to get your mind around these sorts of arguments not making sense. In other words, teaching is hard.

Cincinnatus C. said...

I'm completely with 4:21.

It has been a number of years since I've taught first-year students, but there's a similar weird thing--essentially the same, I think--that I used to find striking as a TA in intro classes: a lot of students don't get that an author might write something in a book that the author doesn't think is true. Many students would report that according to Descartes there is an evil genius deceiving us about everything. Some would report that Descartes later contradicts himself by saying that God wouldn't allow us to be wrong about everything. In other words, many people can't handle dialectical writing, which lives on conditionality.

I've often gotten the feeling that many people don't just take whatever they read to be taken to be true by the author, but to actually *be* true. But if the Spinozan believe-first-then-evaluate view of belief-formation is true, then, well, in a sense, of course they do; what's distinctive of these troubling people is that they don't get to the "evaluate" part.

Anonymous said...

I think 12:31 is right that the problem is connected with a failure to imagine appropriately. But there's a flipside to this. Yes, if P is false, then we might need to pretend that P is true in order to determine whether "If P then Q" is true. But once students learn that lesson, then they sometimes get the idea that philosophy is all about imagining wacky things just for the sake of imagining wacky things. Only the chosen few understand that (a) we *sometimes* need to imagine that falsehoods are true--but (b) simply imagining that a falsehood is true isn't always philosophically productive.

Anonymous said...

Over the last four or five years, I've noticed that an increasing proportion of philosophers demonstrate an increasingly difficult time understanding how conditionals work.

After some research, this is totally reasonable. Lots of smart people have been unable to determine what a conditional is or how conditionals work. I'd guess that the students are in the same boat.

They'd probably be better off if we could figure some of this out! :)

Anonymous said...

I think that the holy grail of philosophy teaching is finding a way to help students understand the logical structure of views that they do not themselves accept, so that they can see what objections should bug them if they held those views, what further implications they should accept, and so forth. It is dark and sad that they are trapped within their own perspectives. And I fear that the folks who are in charge of instructing them have the least experience of what it's like to not be able to do this.

Anonymous said...

Cincinnatus C. writes:

"I've often gotten the feeling that many people don't just take whatever they read to be taken to be true by the author, but to actually *be* true."

This is my observation as well. But then something else happens in class. I assert "P" and students ask, "Are you saying P is true or only that you believe that P is true?"

Anonymous said...

1. Tell students the truth: there are a number of different conditional relationships: causal, definitional, stipulative, subjunctive, etc. Illustrate them.

2. Tell students the truth: the hook is a truth-functional interpretation of all such conditionals defined by the negation of the case that the antecedent is true and the consequent false.

3. Tell students the truth: the resultant material conditional yields two lines of the truth-table that show that false antecedents break the implication down, and so entail anything.

4. Tell students the truth: the truth-functional interpretation is competent to write binary code that makes their #@&*%$! smartphone work.

4. will impress them. One truth they can hold in their hands. And they know the value of that.

Hanuman said...

If it's true that Aristotle claimed that the mark of an educated mind is to be able to entertain a thought without believing it, then this is presumably a very old pedagogical problem. I know I've seen these problems a lot. The most memorable consequence: My saying, "I had a teacher who used to say that if we intend capital punishment to deter crime, we should hold executions in stadiums and make school kids watch," was once taken as evidence that I believe that schoolchildren should be made to watch executions. Oops.

One approach I've found helpful is to point out the connection between 'All F are G' and 'If something is F, then it's G'. This helps students see that 'If p, then q' is not "just a hypothetical statement that doesn't mean [i.e., assert] anything." Another approach is to say that conditionals are like promises: If I promise you to buy you beer if you help me move, I haven't said that I'll buy you beer, and I haven't said that you'll help me move. I'm just saying that there's a certain connection between the two things. But of course there are important disanalogies between promises and conditionals. (The promise analogy does help students understand the truth table for the material conditional, though.) I got the promise thing from an article in Teaching Philosophy some years back.

Anonymous said...

The connection between conditionals and all/only statements is tricky though. The problem with students failing to understand conditionals with false antecedents comes back in a big way with "All Fs are Gs" when there are no Fs. I have a hard time convincing some grad students of the truth of these.

Anonymous said...

If presenting the truth table for the material, it is also good to give an example where it is actually used in natural language, e.g., when we would say someone had lied to us in saying that they will take us to the park if it's sunny (a friend mentioned this way of doing it to me once). Only one, right? When the antecedent is true and the consequent is false. This kind of glosses over the messy details in a way, but it does help them not just dismiss the material conditional as a useless invention.

Anonymous said...

Most readers will probably have heard this example many times before, but I've found it the most effective way of getting students to appreciate why conditionals in classical logic are only false when the antecedent is true but the consequent false. Here goes (cribbed from Hurley's Logic book):

Suppose I tell you, "If you get an A on the final, you'll get an A for the course."

If you get an A on the final, and an A for the course, have I lied to you? (No.)

If you don't get an A on the final and you still get an A in the course, have I lied to you? (Still no.)

If you don't get an A on the final and you don't get an A for the course, have I lied to you? (No again.)

But if you do get an A on the final and you *don't* get an A for the course, then I've lied to you.

Reid Atcheson said...

Math grad student chiming in. I see this error constantly when grading, students use the implication arrow to mean "therefore," and it's not just a commonly accepted idiom that math people just do either because they will turn right around and explicitly use the word "implies" for the exact same intended meaning.

It's such a common error that it is often ignored, I even see professors doing it. I think such bad habits are learned. Students suffer though when they are eventually asked to actually prove conditional statements (many theorems in math are of the form: A if and only if B), and since their notion of material implication is warped they end up trying to prove something totally different and failing.

Anonymous said...

A perhaps related phenomenon is that intro students, when presented with a counterintuitive view and some arguments for it, will not respond "huh, interesting", but "what a load of foolish nonsense!"

I've found that it is only the best students who can hold a thought like, "sure I think view X is false, but that is a good argument for it, how interesting"

Anonymous said...

One thing that was mentioned in a previous thread really caught my eye about what to do when you already have enough publications: spend time thinking about/showing in your materials how the publications fit into an innovative research program. (Thinking mainly of R1-type jobs here.) I don't really have a sense how common this kind of coherence is in the market and if people thinks it makes a big difference. Any thoughts?

Anonymous said...

Anon11:35--

Tell them to give you all the flowers that are on the table (make sure there are no flowers on the table) and after they've given you nothing, ask them whether they failed to do what you told them to do.

They didn't--they gave you "all" the flowers because there are no flowers.

That's worked for me in the past with at least some students.

Anonymous said...

Amusing and highly disturbing...I have encountered this as well