Intellectual Background

Anyone can benefit from good logical reasoning skills, and I hope to instill such skills in my students. However, having taught many courses on logic and reasoning, I have found that it can be hard for people to learn and improve such skills. The goal of the research I have been conducting over the past years is to develop more effective environments for students to learn logical reasoning skills.

Situated Cognition

My research is based on the assumption of situated cognition. To explain the view of situated cognition, consider the well known example of a hammer as given by the philosopher Martin Heidegger (Heidegger, 1962). When you use a hammer, you find yourself looking, and otherwise focusing, on the nail, and not the hammer. Thus, according to Heidegger, the hammer has become an ‘extension’ of one’s physical self, and is better seen as part of the agent, rather than as part of the external world the agent is manipulating. Notice that our language reflects this kind of thinking, as when we say that ‘I hit the nail on the head’.

More to the point, from the perspective of our brains, the intuitive boundary between ourselves and our environment as provided by our skin is merely that: a biological boundary that can be easily transcended through the brain’s re-interpretation of the incoming signals. Indeed, when I wear eye glasses, I don’t ‘see’ my glasses, but instead my brain will construct my experience in a way that regards those glasses as an extension of my eyeballs through which I experience the world beyond them. Thus, my glasses have become ‘transparent’: not just physically, but cognitively. So, while my glasses are not part of me as a biological being, they do have become part of me as a cognitive being.

Even if one is uncomfortable with the idea of extending one’s cognitive self beyond one’s biological self, one may still accept another basic tenet of situated cognition, which is that we often manipulate the environment in such a way as to perform cognitive tasks. For example, when we play Scrabble, we don’t just stare at the tiles and think of possible words, but instead we physically move the tiles around in the search for words. Or, when we play the well-known computer game Tetris, we rotate incoming pieces not because we already know where we want them to fit, but because by rotating them, we can more easily find a place for them to fit. The cognitive scientists Kirsh and Maglio call such moves epistemic moves: moves that aren’t part of the solution of a problem, but part of the problem-solving method (Kirsh and Maglio, 1992).

In general, then, we often use parts of our environment as tools to enhance our cognitive powers, and we don’t always perform cognitive tasks ‘in our head’ alone. Now, one of the more powerful of such tools seems to be language. For example, much of our knowledge is not in our head, but out in the environment, in the forms of books or other media. An even simpler example of such ‘external memory’ is leaving a sticky note on our desk. Moreover, thinking and reasoning about things is assumed to often involve the manipulation of linguistic expressions in our environment. Thus, we have specialized languages in many of our sciences to figure things out we could not with our bare naked brain: multiplication and division often requires the manipulation of symbols on a piece of paper using pen or pencil. And again, the writing down of symbol strings is not the passive result of a reasoning process already completed, but instead will cue the reasoner to proceed in one way or the other. Indeed, a paper doesn’t come about in one sitting during which we just ‘dump’ our already finished thoughts, but instead is the result of the constant addition, deletion, rewriting, and reorganization of parts, reflecting the gradual progression of our ideas and thoughts taking place through the very act of writing.

Taking this idea of the close relationship between language and reasoning even further, one interesting suggestion is that when we ‘think to ourselves’, we are really having some kind of internalized dialogue with ourselves as a debating partner. In other words, even when we think to ourselves, part of our brain is still interacting with its environment, which in this case is just another part of the brain that has internalized representations of interactions that would normally take place outside the skull. While this seems like a strange idea, it actually makes perfect sense on the view of situated cognition, which once again shifts the boundary of the cognitive self, but this time to exclude those parts of the brain that serve to represent the environment, rather than to interact with it.

One important implication of this idea, and of situated cognition in general, is that cognition is more likely to take place through dynamic sequences of concrete sensori-motor events, rather than through the manipulation of static and abstract internal representations, which is how cognition was more traditionally thought of. Of course, the external representations of our natural or specialized languages are abstract, and through those systems, our cognitive powers can be significantly enhanced, but the use of those symbols is still modality-specific, i.e.  Indeed, from an evolutionary point of view, linguistic systems are only a late ‘invention’, making it unlikely that they are central to the workings of the brain. On the other hand, sensori-motor control has been an important task of brains for millions of years, so it makes sense to build any further cognitive abilities off of those. Therefore, if linguistic systems are to really extend or enhance our cognitive powers, then they must ‘fit’ our brain and the rest of our sensori-motor system, just like a hammer is shaped so as to fit our hand. In other words, in order for linguistic tools to be effective, care must be taken to ensure that those systems are ‘cognitively ergonomic’.

A further consequence of this idea is that transfer of knowledge from one domain to another is likely to involve being able to apply similar sequences of concrete sensori-motor events from one domain to the other. This supports the well-recognized importance of analogical reasoning, in which one learns domain-independent skills not through domain-independent symbolic descriptions, but rather through the recognition that what one does in one specific domain is similar to what one does in another. Indeed, we are all familiar with the fact that many students can’t immediately work with the general theorems or abstract principles to solve specific problems, but are often much more helped by seeing some specific examples of the use of those principles, even if applied to a different domain. Put another way: our ‘know-that’ may turn out to involve much more ‘know-how’ than we probably think,  from which it further follows that even learning something as abstract as logic, is still mostly a matter of practice, not unlike learning to ride a bike. Indeed, good logicians have learned to produce dynamic sequences of reasoning ‘moves’ or patterns.

The Teaching of Logic

Traditionally, logic is taught through the use of a logic system that uses linear strings of abstract logic symbols. Immediately, we see how from the perspective of situated cognition, this is less than ideal: Students learning such a system will have few things to compare it to, and to many it will feel like learning a new, alien language. Indeed, from personal experience, I find that many students have a hard time learning this new language and, even more interestingly, often do worse in logical reasoning using this abstract system than they would if relying on their reasoning skills learned at that time in their lives! I also have the impression that many students are treating the system of logic as just that: a system. The fact that it is about logic seems to escape some of my students. Instead, I see some of my students playing with the symbols and rules of the system as if it is just a puzzle of how to transform certain symbol strings into others. Thus, if these students are learning any logic, it would only be in so far as it -indirectly- requires logic to solve the puzzle, i.e. to them, the system of logic has become just another logic puzzle, meaning that one might just as well teach logic using any other logic puzzle.

Indeed, I am sure that quite a few logic instructors will have their students solve logic puzzles, if only to make the course a little more interesting. However, while logic puzzles are certainly engaging, and require the students to use their logic skills, they rarely engage in any kind of meta-cognition during the puzzle-solving process or afterwards. They rarely have to state their justification, or reflect on their reasoning afterwards (which is often lost anyway, as no record of their reasoning was required). Indeed, with logic puzzles, the emphasis is easily more on getting the answer, rather than on how the answer was reached. In short: doing logic puzzles does little to help one to study logic.

To actually study logic, many modern logic books and instructors discuss some kind of formal proof system: a system where one writes down sentences in the formal language of logic as mentioned earlier, and where one derives sentences from other sentences using formally defined inferences. Thus, contrary to logic puzzles, users really are forced to state their reasoning. And, by using abstract symbols that can mean anything, logic is to be highly abstract and subject-independent. However, there are some serious drawbacks to this approach as well:

1. It is based on the traditional logic system using linear strings of abstract symbols, the problems of which are already discussed above: many students don't really grasp the fact that logic is abstract, but instead see a concrete system of symbols. Again, it is as if the logic system becomes a logic puzzle itself, and a really hard and boring one at that.
2. Many formal proof systems are characterized by having exactly one Elimination and Introduction rule for each logical symbol. However, while mathematically elegant, such systems are not very user-friendly, as they often lacks basic inference such as DeMorgan, Disjunctive Syllogism, or even Modus Tollens.
3. Most systems do not clearly distinguish important logical reasoning techniques from much more trivial and relatively unimportant inference rules, and instead puts them side by side. For example, ~ Intro represents the centrally important Proof by Contradiction, and v Elim represents the important Proof by Cases reasoning technique, but their counterparts ~Elim and v Intro are really rather uninteresting and trivial.
4. In fact, some of the inference rules are nothing but an unfortunate consequence of the particular notation used, and indeed have very little or nothing to do with logical reasoning! For example, the fact that P & Q is seen as a syntactically different statement than Q & P, and hence requires either a commutative principle or - even worse - a whole series of & Elim and & Introduction rules, is nothing but a peculiar consequence of the linear nature of the representation used. In Existential Graphs, for example, no such distinction exists: P and Q are both true, and no artificially induced 'order' on the two claims comes into play.
5. Proofs themselves are often represented in such a way that they are sometimes hard to grasp. For example, proofs are usually depicted as long, linear sequences of sentences, and even where subproofs are used to provide a bit more structure to proofs, they are often still listed sequentially. However, much reasoning can happen in parallel. For example, when doing a Proof by Cases, one explores several possibilities, and any kind of order in which one does this is unconsequential, so listing them sequentially does not help the user to grasp the nature of the proof.

Finally, many logic textbooks now come with software, providing an interface for the user to construct formal logic proofs. For example, I have used the program 'Fitch', packaged with the book Language, Proof, and Logic, for years in my classes. However, while Fitch is probably one of the best commercial software packages for logic on the market, it still suffers from some serious problems:

1. The interface requires more user-actions than necessary, with the result that the interface cannot support the normal ‘pace’ with which humans like to reason. Thus, the interface is not cognitively ergonomic.
2. Indeed, the user often spends more time dealing with the interface, than with the actual logic problem at hand. That is, the interface is not cognitively transparent either.

I suspect that many of the other software programs suffer from similar problems, due to the rigidity of the traditional systems. This is why I have been looking for an alternative.

LEGUP

LEGUP (Logic Engine for Grid-Using Puzzles) is a computer interface application in which the user is asked to solve different types of grid based logic games or puzzles, of which Sudoku is a well-known example. However, rather than simply entering the values into the grid, the user must supply a reason for each step of the process. This ensures that the user is solving the puzzle with logical reasoning, rather than lucky guesses. And, if the user makes a mistake, this mistake can be traced back to a specific step in their reasoning process, so that -hopefully- the user can learn from his or her mistake.

LEGUP has the following important features:

1. Intuitive and Easy-to-use Graphical Interface: In LEGUP the users makes changes to the puzzle board, and selects a graphical representation of a rule pertaining to that puzzle as their justification.
2. Concrete and Engaging puzzles: LEGUP users reason about concrete and engaging puzzles.
3. Proof (or Reasoning) Trees: A history is made of all the moves made by the user. This tree is of course a kind of logic proof: a record of the user's reasoning. The user can go back and make changes to their reasoning. The Proof Tree can also be saved, and thus worked on later, and the reasoning can be shared by others. Finally, the tree depicts reasoning as sequential where it needs to be, but parallel where it can.
4. Different Puzzle Types leading to Abstract Logical Reasoning Principles: LEGUP supports different puzzle types (Sudoku, Tree-Tent, Might-Up, etc), but the interface has the same format every time, thus revealing abstract and general reasoning techniques such as Proof by Cases and Proof by Contradiction. Indeed, the visually striking branches and dead ends of the tree make the student naturally focus on, and comprehend the nature of, important reasoning techniques of proof by cases and proof by contradiction.

Simply put, LEGUP combines the best of the ideas of doing logic puzzles, and of using a system of rules to justify one's reasoning. Indeed, by combining them, LEGUP overcomes the drawbacks of either of these approaches used by themselves.

I predict that LEGUP will make for a more effective for learning logical reasoning than traditional logic systems do, because instead of forcing users to state their reasoning in a hard-to-use symbolic language, users can focus on the reasoning, and readily provide justifications (point 1), and also because instead of being engaged with abstract and boring symbols (even though their abstract nature is often poorly understood), users in LEGUP are engaged with concrete, meaningful, and fun puzzle environments (point 2).

Similarly, LEGUP goes far beyond simply doing a logic puzzle. LEGUP requires users to indicate their reasoning and justification in a format that is friendly to understand, and that can be reviewed or discussed later (point 3), and by different puzzles sharing a common interface, LEGUP users will discover important abstract reasoning principles (point 4). That is, LEGUP allows students to perform some serious reflection and meta-cognition on logical reasoning, and not just solving the puzzle.

The innovative part of LEGUP is therefore mostly, and 'simply', the integration of two popular approaches to teaching logic: logic puzzles and formal proofs.

Future Work

Features of the LEGUP interface that we are thinking of implementing at some point in the future are:

1. Annotations: Anyone who has ever done a Sudoku puzzle will probably have used the technique of putting little numbers in the corner of cells to indicate that this is a possibility, rather than a necessity. This is of course a perfect example of situated cognition. We hope the LEGUP interface to support these kinds of annotations as well.
2. Dynamic Rule Application: When a rule is applied, the interface doesn’t just show the ‘before’ and ‘after’ of that rule, but dynamically visualizes the application of the rule to the given situation.
3. Artificially Intelligent Tutor: The tutor that is able to provide hints, demonstrate and explain different problem-solving techniques, and generate on-the-fly problems specifically pertaining to the student’s “logical reasoning problem areas” as perceived by the tutor.