# Why Study Formal Logic?

Next week it’s back to the classroom for me, and I’m teaching intro logic again. I’ve been thinking a bit about what to do on the first day, especially in the “why you should take this course” department. There’s the obvious reason: it’s required (at least for philosophy and CS majors). So I’m really talking about “why you should want to take this course”. And here, the textbooks usually don’t do such a good job. First there’s the “you’ll learn how to think correctly and identify logical errors” line. The examples there are usually a valid and an invalid syllogism, examples that I suspect anyone with any chance getting a decent grade in the class can already identify as good and bad instances of reasoning. Second, there’s the “important applications in logical circuit design” story. But, honestly, any logic design course can cover the logic they need for combinational circuits in a week. Third, there’s the “taking this course will train your analytic and abstract thinking skills”. Ok, maybe, but that’s not really a good selling point.

So I’m looking for concrete, real-life examples where some of the things that you learn in a formal logic class are useful: examples that are relatively easy to describe, where it’s obvious that these are “really relevant” to whatever discipline they’re taken from, and where you can reasonably claim that you need to be able to deal with a formal language, understand relations and multiple quantification, or use logical methods like formal proofs or model-building techniques to avoid errors or solve a problem.

One of the examples I think I’ll use is SNOMED CT. That’s a health-care terminology database (aka an “ontology”) with over 300,000 concepts organized by over 1,000,000 rules. These rules could be formulated in a fragment of first-order logic (some description logic suffices, I’m not sure which). One example I’ve seen mentioned here is this: In SNOMED CT, an leg amputation is defined as a procedure with method amputation and procedure-site-direct lower limb structure; and a toe amputation as a procedure with method amputation and procedure- site-direct toe structure. Now SNOMED CT also knows that the toe is a part of the lower limb, so that if a procedure happens in the toe, it eo ipso happens in the lower limb. Therefore, a toe amputation is also a leg amputation. But of course you wouldn’t want a surgeon to take off your entire leg if you have a gangrened toe! On the other hand, if you have a pain in your temple, then since the temple is part of the head, you have a headache, and you do want SNOMED to know that. So here you need all kinds of logic: you need a formal language in which to express these concepts and relations, it needs to be expressive enough so that you can express everything you want to express, you need logical methods to tell you a) what follows from SNOMED (queries), b) wether SNOMED is consistent, c) where the errors are and how to remove them. (I learned about SNOMED CT from Frank Wolter’s talk at the Logic Colloquium, “Mathematical logic for life science technologies“.)

Of course, all of this is just a particular case of the various important applications of logic in AI and databases, but I thought it was a nice example that wasn’t just a toy database. Also, I like the “mistakes that logic helps avoid or correct” flavor.

I’d also like examples like that from philosophy and mathematics. For mathematics I was thinking of talking about Cauchy’s “erroneous” proof of the uniform convergence theorem, and pointing out the importance of the order of quantifiers. That has the problem that (as we know from Lakatos) Cauchy didn’t really overlook the necessary requirement of uniform convergence, and also it might be a bit too difficult (to explain in a short amount of time). For philosophy, I thought of maybe using Skorupski’s argument for the principle of moral categoricity from Ethical Explorations, which I found in a post by Doug Portmore on PEA Soup. I like it because it’s simple, and recent, and from ethics, which is often considered by students to be next to the opposite of logic( as far as courses are concerned).

Do you have other ideas? Better ideas? Ideas applying in other disciplines?

I think it would be nice to have an example where a famous mathematician or philosopher committed a more-or-less elementary logical error that can be diagnosed or avoided by formalization.

## 24 thoughts on “Why Study Formal Logic?”

1. Jacek Koba says:

The obvious answer to me is to understand your language better, or the language you are learning. It will not get you a cup of tea in the same way a phrasbook will, but if, say, the language you are learning is English, it will help you understand why what you want is not ‘the cup of tea’ but precisely ‘a cup of tea’. On the flipside, it will take you years and many cups of tea before you start asking yourself this question and looking for answers beyond texbook grammar.Questions about existence and uniqueness are not taken seriously before you hit 40 and the mortgage lender ups your rate, in the same way that studying Hamlet at school or university means no more than a mediocre story line.

2. kiju says:

Strawson identifies a quantifier-switch fallacy in Kant’s analogies, which seems to be a nice example for a fallacy that is easily detectable and explainable in a modern formalism — an one made by a great philosopher. (cf. Strawson: The Bounds of Sense, p. 137-8)

3. Gregory Wheeler says:

Hi Richard – Apropos medical informatics, you might consider looking at the WHO Family of International Classifications: http://www.who.int/classifications/en/,which EU member states are increasingly referencing in their laws. This is increasing the pressure to develop suitable electronic representations for the ICF (and ICD, which you can find as well) that will allow for updating of the rules as well as the ontology. Description Logic is the tool of choice here, and the reasons why worry over the basic meta-results in an FOL course can be animated by discussions of import and, also, to explain the demand for people with skills to cleverly find and move around within various fragments of FOL.As an aside, it looks like US health policy is lagging a bit behind EU policy w.r.t. incorporating standardized knowledge bases like the ICF and ICD (I don’t know what the story is in Canada), but everyone seems to be pointing in this general direction. So, I think the example can be reused and will grow in importance as time passes.

4. Richard Zach says:

Hm, it seems to me the problem in Kant that Strawson discusses is one of equivocation, plus possibly a modal fallacy.

5. Aldo Antonelli says:

Aristotle begins his Ethics with the claim that, since every action tends to some good, there is a good to which every action tends. Nice quantifier switch, no?Of course some inferences of this form follow validly from extra assumptions, so the issue is what other assumptions would/did make Aristotle’s arguments valid.

6. Greg Lavers says:

I too think understanding your own language better is something you can honestly say a first course in logic should help students with. In my first class, I put up three sentences which are instances of P, if Q , P only if Q, and if P, then Q. I tell them that two of the three are equivalent, one says something different. Although the sentences are composed out of words they know well, they can’t, in general, tell which two are equivalent. It is good for showing that there is something about language that they don’t fully understand. I also tell them that logic problems are more fun than sudoku.

7. Anonymous says:

A strange line of thought by … Mr. Timothy Williamson : “The larger purpose underlying my book Vagueness was to argue for realism like this: if realism is wrong about anything, it is wrong about vagueness (that premise was generally agreed); but realism is not wrong about vagueness; therefore it is not wrong about anything.” (italics not in the original…)

8. Bruno says:

There is also Logic Programming (e.g. Prolog). Studying logic opens a whole new way of thinking about computer programming…

9. Bruno says:

If you have a formal mathematical proof, you can apply algorithms to transform this proof. One example of such proof transformations is the system CERes for cut-elimination by resolution.

10. Bruno says:

Chris Fermüller used, in the Moscow-Vienna workshop, a nice real-world example to motivate the study of formal logic approaches to vagueness. The example involved law suits regarding insurance of the buildings destroyed in the 9/11 terrorist attacks. You could ask him the details and his slides.

11. Bruno says:

I am not so familiar with verification of hardware and software yet, but I guess you could find some interesting real world examples from this application area of logic.

12. Bruno says:

Matthias Baaz mentioned in his lecture on Proof Theory the case of Luckhardt and Bombieri. If I remember correctly, Luckhardt managed to prove a certain number-theoretical theorem by using techniques of Logic, and only a few years later Bombieri proved the same result by using number-theoretic techniques.

13. Bruno says:

In line with the “understanding your language better” reason, you could also mention the technological aspect of understanding language better: if you know it better, we can program computers to understand it better too. (Imagine google accepting queries in natural language, instead of just keywords…)

14. Bruno says:

Russel’s paradox within Naive set theory is a famous example of how things can go wrong if we just trust our intuition about seemingly plausible ways of reasoning such as using comprehension axioms in an unrestricted way.

15. Bruno says:

There are also connections between studies of Proof Complexity and the P = NP problem… This could be “sold” as an application of proof theory to theoretical computer science…

16. Bruno says:

Automated and Interactive Theorem Proving is another motivation for the study of Logic… I remember reading about a mathematical proof that had so many cases to be considered that it could only be completed with the help of automated theorem provers.

17. Panu Raatikainen says:

Did not Russell criticize Copleston about a quantifier switch in the proof of God: every event has a cause; therefore, there must be something which is the cause of every event. -Panu

18. Dirk Schlimm says:

I like to point my students toJames M. Bartlett (1964). Frege: On the Scientific Justification of a Concept-Script. Mind 73 (290):155-160.It gives Frege’s own motivation and justification for a formal logic and it puts the enterprise in an historical context.

19. Steve Awodey says:

The best reason to study logic is that it represents one of the great intellectual achievements of the 20th century: solving the classical problem, over 2000 years old, of the nature of logical truth. We can now say exactly why some truths are logical (analytic, apriori, etc.), and can verify by a range of different methods when a sentence has this property.As for applications, not only does much of modern computer science depend on this breakthrough in logic, but much of the old philosophy — more than most philosophers like to admit — is rendered irrelevant by it. It is simply no longer plausible to claim that there is metaphysical knowledge, of the kind historically asserted, without the examples once provided by unanalyzed logical truths.It’s easy to overlook the consequences of the development of modern logic, because they are so deep that one hardly notices how much the landscape above them has shifted.

20. Nicolas says:

I’m going to be teaching a “Basic Logic” course as a philosophy elective for non-philosophy students; we’re using The Logic Book, but they won’t be tested on the metatheory chapters.Anyway, I had my first class last week and I had a slide called “Why Study Logic?” which tried to show how this abstract topic was relevant to the myriad departments represented among my undergrads.So I pointed to the following ideas that logic helps one understand:-The distinction between syntax and semantics, form and content, object language and metalanguage, etc. (this pops up everywhere); -The role axioms play in theorem-derivation (and not just in mathematics);-Logical notations in general, often deployed informally in a number of disciplines;-The structure and meaning of arguments, so you can evaluate them quickly in science, politics, the humanities, even at cocktail parties and family dinners;-The real meaning and extent of logical ideas that have entered the popular consciousness, such as Gödel’s proofs, too-often used and abused by cranks of all stripes;-The relationships between logic, computability, Turing machines, artificial intelligence, and so on. Students really get a feel for e.g. Searle’s “Chinese room” thought experiment when they have some hands-on experience with logic.I don’t have good examples of famous fallacies, unfortunately, but the discussion thread had some good ideas I’ll try and implement over the semester…

21. Maciek Godek says:

I’ve observed during my teaching practice , that students frequently make certain mistakes. For instance, given p->q, they tend to infer ~p->~q. Another example is that when they translate entailments from natural language to formal logics, often fail with inversions, mistaking antecedent with consequence.(Perhaps the cause isn’t the lack of capability to reason properly, but the fact, that they sometimes don’t know what they’re doing)Anyway, apart from formalisms, logic is the theory of proper reasoning.It is a good question: what makes a reasoning proper or improper. Albeit the answer is problematic, it seems that some forms or rules just feel right — and the formal system is just a tool to catch and express this rightness.In addition, the study of formal logic helps to understand the abstract concept of rule, in its purest form. It provides rules of applying rules. At this point, it might be encouraging to show some everyday examples of the rules we live by, and to point out our natural (transparent) capability of apprehending rules.Sincerity requires us to present not only the advantages of formal logics, but it’s limitations as well. I think it is important to explicate the assumptions that underly the propositional and predicate calculus: that it isn’t necessarily natural to think of sentences as of something that is just either true or false (there’s much more to it!), and that the timeless object-predicate approach employed by predicate calculus, however powerful, is far from our everyday experience.On the other hand, it is an interesting experiment to see how far we can get with this approach.There are, indeed, many other advantages of studying formal logic — especially for those who are interested in artificial intelligence and computation theory. It is also a good question, to what extent our mental activity can be implemented on a computer.I’m not sure if the medical database example is a good choice. Obviously you can try if you want, but my mind switched off on that part of your post.(It seems that among all the non-technical university departments, law students are best motivated to study formal logic, because it’s application is graspable to them)

22. Greg Frost-Arnold says:

I know this post is long dead, but I have recently had a thought about how I might try motivating intro logic next term: start with a batch of results from the psychology of reasoning literature showing how bad humans can be at reasoning. The Wason selection task is of course the most famous, but I get the impressions that there are a lot of other results out there too. (I haven’t collected them myself yet; a lot of the overviews intended for the non-expert (=me) have a bunch of statistical reasoning errors… which is not so helpful for FOL.)

23. styvrt says:

Hi Prof. Zach,I hold a bachelors in Electrical Engg. and a Masters in Applied Maths, but am interested to apply to interdisciplinary Logic programs. I am aware of such programs in Berkeley, UCI-lps and CMU; this is to ask if you could mention more such programs which would admit students with background like me.Also, such departments require long writing samples. Since I possess a limited background in logic & only informal random readings in philosophy, I’m not sure what to write in such samples. Is it fine if I come up with a personal viewpoint on, say, religion or transhumanism (or may be an expository article on Fuzzy implication based on a course I had)?Kindly help. Thanks in anticipation.

24. grenzenlos says:

I would say that the main argument in order to encourage students to study logic is to know the structure and meaning of arguments. I would basicly emphasis that logic is a toll that makes us realize of all the fallacies that people make in everyday life. It is easy to find one, just open a newspaper and abracadabra!