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Teaching Validity with a Stanley Thermos

Andrew Chrucky on logical arguments which hold water. And coffee.

I teach critical thinking and logic, and I would like to share with Philosophy Now readers a useful way of understanding the concept of deductive validity by an analogy to a thermos flask. I know that some people find it difficult to distinguish the truth of a logical argument’s premises from the validity of an argument. They think that a valid argument has all true statements, and an invalid one a false premise. Nothing could be further from the case, but to understand the difference between truth and validity we need to introduce the idea of an argument form, for it is the form which is the vehicle of validity, not what is put in the form. An argument form does not contain statements (but statement forms), so there is nothing in the form to be true or false. Yet the form has the property of validity, which is the property of truth preservation. This is to say that a valid argument form will never simultaneously allow the premise forms to be filled with true statements and the conclusion form to be filled with a false statement.

Some students apparently have a hard time keeping the idea of an argument form distinct from the idea of the content which may fill the form. I needed some concrete, even dramatic, method of bringing this distinction across.

Well, one day in the Spring semester of 1990 I was driving to my class with a very large thermos bottle – given to me by my brother-in-law as a Christmas present – resting on the passenger seat next to me. This was not an ordinary thermos but a very large Stanley thermos, made of stainless steel, with the capacity to hold about a quart of coffee. It may have been the first time I was using it, so I was thinking about it. Also I was thinking about my class and how I was going to reinforce my previous lectures about the soundness and validity of deductive arguments. I remember thinking of pouring truth into the thermos and how truth would come out later. But I also thought of pouring falsehood into it with falsehood pouring out later. And I was disappointed. Then I thought, “In what way could the thermos be modified so that it acts like the mechanism of a valid argument form?” Once the question occurred, the answer was simple. If falsehood is poured into it in the morning, in the afternoon one poured out either truth or falsehood. I mulled over this idea as I drove, and by the time I arrived for class, I was prepared.

Upon entering the class, I placed this monster of a thermos on top of my desk, and told the students that this was not an ordinary thermos but a Stanley thermos – a very peculiar thermos. Unlike other thermoses which are not totally efficient, this thermos was 100% efficient: if one poured hot coffee into it in the morning, one was guaranteed getting hot coffee in the afternoon. But like some other thermoses, it was not efficient with cold coffee. Indeed it was so inefficient with cold coffee that if cold coffee was poured into it in the morning, one could not predict what would happen to the coffee by the afternoon: either hot or cold coffee would come out. So the Stanley thermos was heat preserving, but not cold preserving.

After characterizing the Stanley thermos, I asked my students to think about all the possibilities. I pour hot coffee into the Stanley in the morning, what do I get out in the afternoon? Hot coffee, of course. I pour cold coffee in the morning, what do I get out in the afternoon? We don’t know – either cold or hot coffee. Now I asked them to think. Suppose in the afternoon I pour hot coffee from the thermos, what can I deduce about the state of the coffee that was poured in? Well, it could have been either hot or cold, so we don’t know. But what about the case of pouring out cold coffee in the afternoon? What was poured into it in the morning? It must have been cold coffee, of course.

Having gone over the properties of the Stanley thermos, I told my students that the Stanley thermos was like a valid argument form. I asked them to substitute for ‘hot coffee’, ‘true statements’; and for ‘cold coffee’, ‘false statements’. The coffee poured in the morning are the premises; the coffee poured out in the afternoon is the conclusion. And I told them also to regard an argument with at least one premise being false as containing false premises. With these instructions, I proceeded.

“In a valid argument, if the premises are true, the conclusion is … ?” I asked. “True,” came the answer. “If one of the premises is false, the conclusion is … ?” “You can’t tell,” came the answer. “If the conclusion is true, what can you tell about the premises?” “Nothing, they could be true or false,” came the answer. “If the conclusion is false, what can we tell about the premises?” “At least one of the premises is false,” came the answer.

Of course things did not proceed this smoothly. I had to remind the students to think of the Stanley thermos to get things straight.

Next, I told them that the argument forms were like thermoses: some thermoses were ordinary thermoses that preserved heat and cold with some but not perfect efficiency, some were defective thermoses of poor efficiency, and some were Stanley thermoses with perfect efficiency for hot coffee. Ordinary and defective thermoses were not to be trusted: pour hot coffee into them in the morning and you could get cold coffee in the afternoon. But never with Stanleys – you were always guaranteed a hot cup of coffee in the afternoon provided, of course, that hot coffee was poured into them in the morning. Next, I posed the following problem for my students. Suppose you came into a room containing a bunch of similarlooking thermoses and had to find the Stanley thermoses. How would you do it? All other thermoses may dissipate heat, so that it is possible to pour hot coffee into them in the morning and get lukewarm or cold coffee in the afternoon. But not with Stanley thermoses. Pour hot coffee into them, and you get hot coffee in the afternoon. The problem is that sometimes the non-Stanley thermoses will also preserve the heat of hot coffee. So what will distinguish them?

The simplest procedure is to pour out whatever is in these thermoses and pour fresh hot coffee into all of them in the morning, and check all of them in the afternoon. Those that have lukewarm or cold coffee are not Stanley thermoses. So we have at least this method for eliminating some non-Stanley thermoses. But the remaining thermoses may still all be or include non-Stanley thermoses. So this method would not enable you to single out Stanley thermoses from other brands. This situation is analogous to coming across an argument and wanting to check it for validity. To do this, abstract the form of the argument (analogous to pouring the old coffee out), and fill in the form with all true premises and a false conclusion. If you can do this with an argument form, then you know the argument form is invalid. It is like filling a thermos with hot coffee in the morning (true premises) and getting cold coffee in the afternoon (false conclusion). If this happens then the thermos is not a Stanley.

I remember everyone, including myself, getting a kick out of this presentation, and I hope that the idea of validity was dramatized in an effective way.

Andrew Chrucky runs the Meta-Encyclopedia of Philosophy on the Web (www.ditext.com) and is a US Editor of Philosophy Now.