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Event Report

Teaching Logic

University of Leeds, 22nd February 2002

Report by George MacDonald Ross

The Pedagogical Merits of Term Logic

Presentation and summary by Professor Peter Simons (Leeds)

Term logic is intermediate in strength and complexity between propositional and predicate logic; its only variables stand in for terms (names, horoi, termini) and everything else is a logical constant. The terms are unrestricted in that they may be semantically singular, plural, or empty (stand for one thing, or more than one, or none), they may be simple or complex, and they may be "proper" like 'Tony Blair' as well as "common" like 'Prime Minister'.

The kinds of logical forms dealt with by term logic may start with the familiar four categorical forms of syllogistic, but can range much more widely. Term logic has a long pedigree from Aristotle to the present, comprising most of the logic done in the West until the twentieth century, including much of the algebraic work of the nineteenth.

Among the chief pedagogical advantages of term logic is its naturalness. It is close to natural language (no bound variables, quantifiers as phrases not operators, logical and grammatical form closely similar), so minimal paraphrasing is required, special symbols can be wholly or largely dispensed with, and it has a straightforward intuitive denotational semantics requiring no set theory. It can be dealt with by a range of simple methods, including semantic diagrams, natural deduction proof methods and semantic trees or tableaux. Axioms can be used if wanted and it can be stand-alone or grafted onto propositional logic.

Term logic has a simple and accessible metalogic. It is sound, complete, and decidable with respect to the intuitive semantics; all its methods are finitistic, and one can script computer programs to show how it works. Being slightly more complicated than truth-functional semantics and introducing the idea of name denotation, it affords a good but uncomplicated initial illustration of model-theoretic ideas.

Term logic may be variously based. Most bases were anticipated by Leibniz, but one may take the basic forms to be equations, as did Boole, or subsumption, as did Schröder, or positive and negative existential statements, as did Brentano and Lewis Carroll, which incidentally allows one to raid Carroll's treasury of humorous and often tricky examples. Or one may like Lukasiewicz stay with the traditional Aristotelian forms.

Term logic can be readily extended in various directions, adding complex and constant terms, general quantifiers, modal and intensional functors. One may also add variable-binding quantifiers directly to it and subsume predicate logic, as did Lesniewski in his ontology. From then on, term logic merges into the mainstream.

So on the plus side, term logic is a natural, user-friendly "Logic Lite" for symbolophobes, affords greater historical connection to pre-20th century logic, liberates names from the straitjacket of post-Fregean singularity, comes with a variety of easy methods, and last but not least, is congenial to nominalists.

Its disadvantages are in many ways the converse of its advantages. It postpones students' encounter with variable-binding quantifiers, relational predicates and multiple generality, and therewith one of the chief tools and advantages of modern philosophy. Being unorthodox, it suffers a complete dearth of modern textbooks, and the metalogic of more complex extensions is little researched. Nevertheless, it is worth considering for its own interest and as a pedagogical stepping-stone between propositional and predicate logic.

Using a Networked IT Resource for Teaching Logic

Presentation by Dr. Roger Young (Dundee)

[Summary to follow]

An Electronically Enhanced Philosophy Learning Environment

Presentation and summary by Dr. Susan Stuart (Glasgow)

I will discuss the rationale for introducing into the classroom a particular sort of electronic equipment that encourages student engagement and interaction.

Every student in the class is given a handset similar to that of a TV remote control, of the sort used on the TV show "Who wants to be a millionaire?". The lecturer asks a multiple choice question, and each student presses the number on their handset that corresponds to their chosen answer. The answers are collected on a laptop which displays, via the room's projection system, a bar chart representing the distribution of the responses. The essential feature of the use of this equipment is that both students and lecturer get to know the distribution of responses and each student knows, in confidence, how their own response relates to that distribution. The element of anonymity encourages everyone to contribute and, unlike in face to face groups, each individual can express the choice they incline to, rather than the choice they would feel able to explain and justify to others.

I have been using this equipment in an Introductory Logic course with a class of about one hundred students, and intend to use it in the forthcoming term with an introductory Philosophy of Mind course. There have been a number of noticeable results. For example, if the students are to answer the questions in a way that will be helpful to them, they have to reflect more on what they have learnt and how they are learning. Reflection of this sort helps them to identify their weaknesses, or, at least, what they perceive to be their weaknesses, and this, in turn, affects the direction that I take in my teaching. I am no longer second-guessing or making unwarranted assumptions about their progress. The students have also reported that using the handsets builds their confidence in a variety of ways. It gives them the opportunity to see that it's not just them having a particular difficulty, that there are others 'in the same boat', but it also promotes discussion with their neighbour when they are asked to discuss a question for a minute or two before voting. I would argue that using handsets in this sort of learning environment is an excellent method of formative assessment, both for students and teachers. It allows each to see how they are progressing at a time when something can still be done about it, and long before the post-course questionnaire and examination results are in, when it is too late to go back.

Overcoming Fear of Formal Notation, and Making Logic More Attractive

Facilitator: Bernhard Weiss (Lampeter)

Was fear of formal notation, or 'dyscalcula', a kind of dyslexia? It was pointed out that some dyslexic students coped well with formal notation. The problem might be that some students tried to read formulae like a text, whereas others saw a structure. Trees made the structure evident, and were less confusing than complex bracketing.

Some notations were easier than others, and perhaps different students took more easily to different kinds of notation (in one case the problem was solved by using Polish notation - whereas others found it more difficult).

Logical notation was no harder than GCSE maths - but some students had been put off by bad maths teaching at school, and not all students had passed GCSE maths.

It was suggested that humanities students might be unused to situations where there were right or wrong answers, and might by frightened of being mistaken.

The problem might be mainly psychological and motivational - students didn't know what logic was for, and departments were not always clear why logic should be compulsory.

If students complained that logic cannot be applied directly to philosophical arguments, they could be told that it is like military drills, which are different from what you do in battle, but none the less necessary training.

Another view was that logic shouldn't be taught as improving the mind or helping one to think logically - rather, it is the science of the structure of thought.

An approach used in one department was to start by treating logic as an enjoyable game, in order to get students over the hurdle of fear (a disadvantage was that this might make logic appear all the more pointless).

Students might be encouraged to use formal notation by insisting on their writing everything out in full. They would soon see that symbols save time.

Another view was that logic is self-standing, and that it should not be treated as a language to be translated into English.

The use of examples might be helpful, but they had the disadvantage of not being topic-neutral.

One reason why an understanding of logical notation was essential was the practical one that it is often used in the books and articles which students are required to read. Students would have a greater motivation to master the symbolism if it were used more often by non-logic teachers.

While it was agreed that the simplest notation should be used for beginners, there was no strong feeling that it was important that all teachers in a department should use the same notation. It was noted with amusement that there was a DIN standard logical notation in Germany (not that anyone observed it).

The Way Forward

Facilitator: David Mossley (Leeds)

There was some discussion of textbooks and logic software. It was noted that the PRS-LTSN was under some pressure from above to focus on reviews of teaching materials, rather than discussion of teaching methods. The consensus was that, while it would be useful to know more about other people's experience of using the better known materials, logicians preferred to write their own hand-outs.

There was demand for information about the evaluation of the experimental methods described by Susan Stuart. This would be made available when the evaluation had been completed.

David Mossley would set up a mailing list, so that those present (and anyone else who wished to join) could discuss problems to do with the teaching of logic.

Participants would like to know more about what was available on the internet - even if what was available would probably be used only as an optional extra. The PRS-LTSN saw it as part of its function to provide links from its website to appropriate pages, and it would welcome any suggestions and reviews. (One participant mentioned pages devoted to fallacies and to logic puzzles.) However, it was important not to reduplicate effort unnecessarily, and substantial links were already provided by gateways such as HUMBUL and SOSIG in the UK, and the Voice of the Shuttle in the US.

Some concern was expressed at the loss of teacher control when students used the internet. It was pointed out that the PRS-LTSN had been responsible for the Internet Philosopher (part of the Virtual Training Suite of the Resource Discovery Network), which was a DIY tutorial to train students to make proper and effective use of the internet. It was suggested that the PRS-LTSN might produce a variant specifically for use by logicians.

The PRS-LTSN intended to organise more workshops on teaching logic, preferably at a range of venues around the country. It was noted that many participants had travelled a long distance to attend, and participants were invited to act as local organisers in their own region.

Possible topics suggested included the following: