Teaching and Learning > DISCOURSE

On Elementary Formal Logic

Author: Paul Tomassi

Journal Title: Discourse

ISSN:

ISSN-L: 1741-4164

Volume: 4

Number: 1

Start page: 114

End page: 129

Return to vol. 4 no. 1 index page

Introduction

That the Subject Centre for Philosophical and Religious Studies (PRS SC) recognises the importance of formal logic within the Philosophy curriculum is ably demonstrated by the two Logic Workshops which have already taken place under those auspices, and, indeed, in the enthusiasm expressed at the second meeting for a third. Arguably, the same recognition is also apparent in the first issue of Volume 3 of Discourse1. However, Helen Beebee’s article on the subject can be read in a negative light. Indeed, like many no doubt, I was struck by the following remark:

The reader may suspect at this point that I am going to be suggesting that IFL courses be ‘dumbed down’. That is indeed exactly what I am going to do. (Beebee: 55)

This conclusion is drawn from two premises which are themselves inductive generalisations derived from descriptions of that author’s own experience of (i) the ‘fear and loathing’ which characterises the reaction of many students to the subject prompting an ‘ostrich’-like tendency to bury their heads in the sand (Beebee: 55) and (ii) the failure of many students to carry forward either the content or analytic methods which logic courses attempt to inculcate in them (Beebee: 55). In light of these considerations, the following recommendation is made:

… if we concentrate on what students actually learn from IFL courses, rather than what we attempt to teach them, it is clear that a majority of students do not get far beyond the basics anyway. For those students, cutting the harder material would not result in any loss at all; indeed, they may well benefit from the less intimidating nature of the course and actually learn more. (Beebee: 55)

I

There is much that one might quarrel with here. The pedagogical experience described above is ultimately anecdotal and may not accurately reflect everyone’s pedagogical experience. Speaking personally, I could not agree that a majority of students fall into the category described nor could I agree that teaching formal logic is costly to my department in virtue of having ‘significantly less pulling power’ (Beebee 2003: 53). At the University of Aberdeen, registrations for PH1010 Formal Logic 1 approximate the most popular Level 1 Philosophy course (PH1509 Moral Philosophy 1) with 20-30 registrations and, while Formal Logic 1 is technically compulsory for approximately 30 students, registrations now approach 130. Further, and more seriously, while in F.P. Ramsey’s famous dictum, logic itself is, in some sense, a normative science, one may feel that the recommendation to adapt the curriculum to the empirical unit of the fearful student qua lowest common denominator has the tail wagging the dog here.

The points engaging the nature of the reaction of the majority of students to formal logic and the cost effectiveness of such courses are, however, mere quibbles. We surely can safely predict that fear, loathing, head burying and failure to carry forward skills and content will constitute some part of every logic teacher’s pedagogical experience. Thus, the existence of the phenomena is not in doubt. The only issue is the degree to which such phenomena make themselves felt in any given cohort of logic students. Moreover, whether or not one agrees with the rationale for the curricular recommendation outlined above, that recommendation is at least valuable in promoting reflection upon fundamental issues as regards the nature of teaching and learning in formal logic courses at Level 1. Key among these is the question featured in the title of Helen Beebee’s article: why do we do it?

II

In part at least, the answer to that question must be historically constituted. We represent the contemporary phase of a modern, relatively new, tradition of formal logic rooted, above all, in the works of Frege and Russell. As is well known, the ancient Scottish universities were relatively slow on the uptake here and as I lecture at King’s College Aberdeen in a hall (KCF8) with a marble bust of Professor William Leslie Davidson, Professor of Logic at Aberdeen 1895-1926, prominently displayed on the wall behind me, the point is not lost to me that he would not have made available to his students any of the more sophisticated formal techniques, syntactic or semantic, which we make available to ours 2. Further, what was, eventually, accepted into Philosophy curricula at the ancient Scottish universities in general and Aberdeen in particular as the systems of logic proposed, in turn, by J.S. Mill, Alexander Bain, F. H. Bradley and Bernard Bosanquet were finally displaced was, speaking precisely, Elementary Formal Logic. Moreover, despite the equivocations of certain authors, this locution does not mean ‘formal logic: the easy bits’. Quite the opposite. This technical term of art denotes both propositional logic (PL) and quantificational logic (QL) through to, and including, first-order predicate logic with relations and identity. In one sense, that is why we teach what we teach.

Of course, the historical account sketched here underdetermines an exact curriculum: is proof-theory for first-order predicate logic with relations and identity thereby included? What of semantic methods at this level? This point brings us to the fundamental question; a question to which the remainder of Beebee’s paper can be understood as providing one putative answer, namely: within any Philosophy curriculum significantly informed by both the logical and philosophical works of Frege and Russell, what exactly is the minimal set of formal-logical learning outcomes necessary to enable students to successfully complete their degree programme?

Intellectual honesty in this regard requires an anecdote of my own. As the product of a curriculum at the University of Edinburgh that embraced elementary formal logic wholeheartedly, I had most initial exposure to Neil Tennant’s (1978) Natural Logic and the second edition of Benson Mates’ (1972) Elementary Logic, proof theory for the quantificational fragment of which was examinable and, invariably, examined—despite the delivery of a written petition against examining quantificational proof theory (signed by most other students in my year) to the logic lecturer. As a tutor and, later, lecturer at Edinburgh, logic teaching centred upon a set of notes (The Logic 1 Notes) prepared by my PhD supervisor, Dr John Slaney, as a supplement to E.J Lemmon’s (1965) Beginning Logic. These texts formed the basis of a 12-week module covering both propositional and quantificational logic through to first-order predicate logic with relations and identity, and, again, including proof-theory at that level. That course, Logic 1h, (and those course materials) was, and still is, compulsory for progression to Honours Philosophy at Edinburgh.

The fact that many students met the challenge presented by Logic 1h successfully should not be overlooked but despite one-to-one teaching, coaching and marking of extra homework each and every year, a number of individuals failed to meet that challenge and did not progress to Honours Philosophy. More than once, the former fact led to both men and women actually weeping with disappointment and frustration in my office—a genuinely distressing situation for all involved—while the latter fact no doubt altered career paths in ways which are not ultimately quantifiable. On arriving at Aberdeen, I anticipated, that the logic curriculum there would follow the same pattern as that at Edinburgh, St Andrews and elsewhere. However, I quickly discovered that the tradition at Aberdeen differed significantly. In essence, the familiar 12-week module was split down the middle into two six-week modules only the former of which was compulsory—a tradition that had endured at Aberdeen for many years. Colleagues anticipated that I would want to bring Aberdeen into line with Edinburgh, St Andrews and elsewhere here. Instead, I chose to confront the question which presently concerns us: what exactly is the minimal set of formal-logical learning outcomes necessary to enable students to successfully complete their degree programmes in Philosophy? And further: could that set of outcomes be delivered within a 6-week module?

III

What then is the bare curricular minimum? Like Helen Beebee, I recognised the fundamental importance of the logical analysis of natural language arguments and their translation into both propositional and quantificational form. Moreover, I also believed that a minimal grasp of traditional proof theory at the propositional level was essential. More specifically, I believed that a grasp of at least one strategic rule of inference was essential in order to characterise, and distinguish, categorical and hypothetical reasoning. Further, I also recognised the importance of achieving a balance between syntactic and semantic methods, and the fundamental importance of the ability to generate counterexamples. Thus, for example, the method of Comparative Truth-Tables appeared essential within any minimally apt curriculum. What remains? Again, it seemed to me that the ability to translate natural language arguments into quantificational form was essential and, indeed, that analysis of the validity of such arguments (even if only informally or via shallow analysis) was also essential. Thus, the notions of truth, validity and a QL-interpretation were unavoidable, again, even if only in informal guise. Finally, given that my methodology here cast formal logic as handmaiden to (if not servant of) the larger philosophy curriculum, an eye had to be kept to the formal constituents of courses in later years (which may vary both within and across curricula). Thus, where the Theory of Descriptions or Frege on sense and reference was germane, I taught numerically definite quantification. Where quantifier-switch fallacies were germane I taught multiple quantification. Were contemporary analytic ontology germane I would teach sortal quantification, and so on.

At this point, it is worth pausing to emphasise that ensuring integration between the elements of formal logic courses and the wider philosophy curriculum within which such courses are properly contextualised is not only an essential virtue quite generally but is also one which will bring rewards to both staff and students. Moreover, there is no shortage of opportunity here. When introduced to material implication, students frequently ask whether the arrow represents causation? The fact that a negative answer is what is required here does not imply that there is nothing positive to be gained from exploring why that should be so: here there are prime opportunities to motivate the universal quantifier, modal operators, to talk about counterfactuals and so on. Again, some students will balk at augmentation thereby opening up the possibility of a discussion of Relevant Logic. And why do we bother distinguishing DNE from DNI? Well, consider Intuitionist Logic. Or what of bivalence? Consider, sea battles and three-valued semantics, realism and anti-realism. A lecture on Modus Tollens is an opportunity to talk about Popper, Falsificationism, and scientific reasoning more widely. Thus, the integrity between formal logic and philosophy can be reinforced while the ultimately philosophical character of formal logic itself is disclosed. Finally, if all that is just to state the obvious there is a further obvious fact to state, namely: tell students about the relationship between the formal logic course(s) and the philosophy curriculum. Students are unlikely to try to carry their logical skills forward if they are not aware that they will later be needed.

To return to the present proposition, the learning outcomes for PH1010, Formal Logic 1, can be seen vividly in the constitution of the one-hour class examination in which the course culminates. This consists of four equally weighted sections, A through D. Section A consists of two natural language arguments to be translated into PL wherein they are to be proved. Typical proofs range from seven to eleven lines and always involve at least one discharge rule so that Premise Introduction can be distinguished from the Rule of Assumptions. Section B consists of two PL sequents whose (in)validity is to be established by Comparative Truth-Tables. These are typically up to 8 rows in length (level of difficulty aside, the length of examination imposes a constraint here) and invariably involve one invalid case to which a counterexample must be constructed. Section C consists of two natural language arguments to be translated into the monadic fragment of QL and assessed informally or via shallow analysis for (in)validity. The elements of Section D are constituted on the wider basis of the Philosophy curriculum.

IV

A little reflection quickly reveals significant agreement between my answer to the question: what exactly is the minimal set of formal-logical learning outcomes necessary to enable students to successfully complete their degree programme? and that proposed by Helen Beebee. For example, both of Beebee’s ‘fundamental and overarching aims’ A1 and A2 are met:

A1 A general understanding of the logic of philosophical argument.

A2 A basic understanding of and competence in PL and QL.
(Beebee: 56)

All three ‘bald suggestions’ for core curricular content, (a) through (c), are also satisfied, though these, I will argue, require some qualification. Consider (a):

(a) Languages of logic. Basic PL and QL, with the emphasis on ‘basic’. Students don’t generally need to be able to construct a 16-row truth table, or learn how to say ‘there are at most two cats on the mat’ in QL, or to tell a symmetric relation from a reflexive one. (Beebee: 59)

To be clear, students do need to know how to construct a sixteen-row truth table. Whether or not such lengthy tables are actually examined is another question however. Further, taking seriously the thought that the upper end of the QL constituents within the curriculum should be dictated by the formal components of courses in later years opens up considerable flexibility. As noted above, where the Theory of Descriptions is germane, teaching numerically definite quantification will provide useful (and enabling) preparation. The ability to identify formal properties of relations can also be helpful. Transitivity failure may (or may not) be common in modal and counterfactual contexts but such failures are certainly common in everyday argument. And if, in a slightly more remote possible world, the definition of number were germane, then formal properties of relations would be essential. Consider (b):

(b) Proof Systems. Junk the truth-trees and concentrate on natural deduction. But don’t make them do proofs that are twenty-lines long. They don’t need to be able to prove theorems either. (Beebee: 60)

Speaking strictly, the Truth-Tree Method is a semantic method: the method of Semantic Tableaux. To omit semantic considerations from the teaching thereof is not to provide a course in proof-theory, if it were, students could master proof-theory without even knowing the names of the rules of inference. And while I agree that mastery of proof-theory can be exemplified in proofs of less than 20 lines, given that the three most fundamental logical laws: Identity, Excluded Middle and Non-Contradiction are all theorems, saying something about theorems seems unavoidable.

(c) Meta-Logic. Students do need to understand the semantic notion of validity, and it is certainly worth getting them to understand that the rules of inference are truth preserving. (After all, this is what gives the rules their normative character.) But they don’t need to even know what soundness and completeness are: the chances are that most of them will never even come across these expressions again. (Beebee: 60)

For both PL and QL, the proof establishing that within such systems every syntactically valid sequent is semantically valid is a proof of Soundness. A proof of Completeness shows that, despite the co-existence of syntactic and semantic methods, only one notion of logical consequence is under investigation. Explaining to students of formal logic that proofs of both results are available is essential and reassuring (though not as reassuring as decidability); working through a proof of either result will require a separate module on Metalogic.

Finally, the emphasis upon the fundamental importance of the identification, reconstruction and logical analysis of natural language arguments and their translation into propositional and quantificational form is intended to obviate the need for any independent, auxiliary or alternative course in ‘Informal Logic/Critical Thinking’ (Beebee: 60-62). Given the nature of the normative logical standards presupposed by any such enquiry, the best student of informal logic must be the student who has mastered the basics of formal logic3.

 

V

If the foregoing picture (PH1010 Formal Logic 1) constitutes an acceptable answer to the question I have canvassed as fundamental here then the way is then open to augment what has been achieved with a further six-week module (PH1306 Formal Logic 2) which builds upon what has gone before but which, specifically:

i. presents a complete picture of PL proof theory, including both primitive and derived rules, within which students may run amok to 30 lines or more (though few such sequents require such lengthy proofs).
ii. provides students with proof theory for quantificational logic through to and including relations and identity, and
iii. continues to ensure an adequate balance of syntactic and semantic methods at both levels by including coverage of the Truth-Tree Method—a semantic method which, unlike the method of Comparative Truth-Tables, easily extends to sequents of QL—to deprive students of that method is not only to deprive students of any effective semantic method for monadic QL but is also to prohibit easy access to high-quality introductions to metalogic such as Boolos and Jeffrey (1989) and, indeed, recent introductions to modal logic and modal philosophy such as Rod Girle’s (2001) Modal Logics and Philosophy or Beall and Van Fraassen’s (2003) Possibilities and Paradox: an introduction to modal and many-valued logic.

In combination, the content of PH1010, Formal Logic 1, and PH1306, Formal Logic 2, outstrips most traditional conceptions of elementary formal logic curricula (few, if any, of which would have included semantic tableaux while those that did might well involve no account of proof-theory) and requires more by way of examination—two hours of examination in formal logic represents a 33% increase over the traditional 90-minute Class or Degree examination. Further, in my experience at least, the confidence-building effect of mastering the Formal Logic 1 curriculum naturally motivates registration to Formal Logic 2—approximately 60% of students at Aberdeen continue to pursue their logical studies immediately while others take the course the following year. Thus, all but the weakest students are likely to learn more logic than was originally required by the elementary formal logic curriculum.

VI

To be precise, I have argued that any minimally adequate introduction to formal logic should ensure that students who successfully complete such a course are thereby enabled to:

If my claim on behalf of this set of learning outcomes is correct, the question of how such outcomes are to be delivered in the contemporary context remains to be answered. As noted above (Section 1), the existence of the ‘fear and loathing’ to which Helen Beebee draws attention is undeniable but ways to ameliorate such phenomena are available. The accessibility of the curriculum is clearly essential but successful formal logic courses also require friendly human faces4. Methods of assessment are equally crucial here. Like myself, many logic teachers will have inherited rather than designed assessment procedures. The format traditionally adopted in the ancient Scottish universities consists in the submission of weekly handwritten homework exercises marked by the Tutor and returned in tutorial. Summative assessment is completed by Class (or Degree) Examination. At Aberdeen, that regime was productively amended as follows: weekly homework is retained as a purely formative element of assessment: written comments are made, proofs corrected and so on but no numerical mark is recorded—successful completion of the course depends upon the submission of the exercises not upon marks awarded. Model solutions to exercises are issued in hard copy after the last tutorial of the week and are then posted on the web (the complete set of solutions in electronic form to all exercises in Logic is available free from myself to lecturers using the text).

Even so, the contemporary context in which such learning outcomes are delivered is perhaps as challenging as it has ever been. Most of my students have jobs (some full-time)—a factor which inevitably interferes with attendance at class—while others face any of a plethora of life/work issues including family-related commitments (which may involve acting as carers) or disability-related issues. Given the cumulative character of teaching and learning definitive of formal logic courses, attendance-gaps pose a significant challenge, especially so as regards Arts students who may have explicitly intended to avoid all further contact with mathematical reasoning by pursuing Philosophy at university. On the other hand, successful completion of a course in formal logic course can be enormously rewarding for anxious students—not only in terms of acquisition of transferable skills but also with respect to self-esteem and personal development.

At Aberdeen, the response to this challenge has focussed on the use of electronic media to enhance the accessibility of the curriculum by providing alternative means to manifest the learning outcomes associated with homework exercises and, potentially, examinations in PH1010 Formal Logic 1. More specifically, with the help of the staff of the Learning Technology Unit at the University of Aberdeen5, websites featuring a number of Exercises from Logic in electronic form were designed and made available to students. This strategy alleviates attendance-based problems at least in so far as entirely open 24/7 access to homework exercises is thereby provided. Further, the project also provides the opportunity to manifest the learning outcomes associated with PH1010 measurably in ways alternate to traditional paper-based methods. A student’s level of success during completion of the exercise is monitored throughout—this is visible on screen as a percentage and can easily be recorded. Thus, completion of assessment on-line can readily constitute homework submission.

Completion of exercises on-line promotes autonomy and self-reliance in the learning process but the opportunity to gain immediate feedback and support via completion of exercises on-line is equally crucial here. Indeed, with respect to non-attendance, that is the heart of the issue. Therefore, a key aim of the project, particularly as regards the first exercise, Logic Exercise 1, was to supply unprecedented levels of feedback across a range of different question types5. For that purpose, LTU staff exploited QuestionMark Perception to support both logic-based question banks and a wide range of branching feedback mechanisms capable of delivering both single feedback items and combinations thereof 7. From the first, both questions and items of feedback were self-consciously designed to enable quick identification of common traps and pitfalls. A further guiding principle was the need to eliminate guesswork-based approaches to completion of exercises. Thus, for example, while the opening question of Logic Exercise 1, on the identification of arguments in natural language, initially invites a ‘yes/no’ answer, the next question invites the student to identify (up to) three reasons (from five) for their first answer whether or not that answer was correct.

In all honesty, while I believe that the collaboration with the LTU has provided a useful, self-diagnostic tool promoting a genuinely accessible curriculum for PH1010 Formal Logic 1, descriptions of the electronic media involved are no substitute for actually engaging with them. Therefore, for demonstration purposes, the following website (for Logic Exercise 1 only) was constructed:

https://qmp1.abdn.ac.uk/q4/perception.dll

 

Access to this website is free: use 'logic' in both the username and student ID fields. Further, a comprehensive evaluation report based upon a voluntary pilot of the scheme with the PH1010 Formal Logic 1 class (2002-2003) is available at:

http://www.abdn.ac.uk/diss/ltu/projects/21phil.htm

I should also state that this project (which was demonstrated at the second Logic Workshop, King’s College, University of London, May 7th 2004) is presently the part of a consultancy between myself and the Subject Centre for Philosophical and Religious Studies whose support for formal logic in general and this project in particular I gratefully acknowledge.

Conclusion

In essence, the present proposal has argued for a compromise between, on the one hand, proponents of the elementary logic curriculum traditionally conceived and, on the other, those who propose significant revision/dilution of any such curriculum. Given, from the former viewpoint, the Formal Logic 1 curriculum proposed here might appear to settle for less than the traditional syllabus required. In mitigation, the following points should be borne in mind. First, if the minimal set of formal logical learning outcomes has been identified correctly then the less settled for is nonetheless adequate with respect to the Philosophy degree curriculum. Second, it is no part of the present proposal to reduce in any way the provision of formal logic within the curriculum. Indeed, as noted in Section V, in combination, the content of PH1010, Formal Logic 1, and PH1306, Formal Logic 2, outstrips most traditional conceptions of elementary formal logic and requires more by way of examination. Thirdly, while Formal Logic 2 is not compulsory on this model it does not follow that the course should not be strongly recommended to students. Again, as noted, my experience is that the confidence-building effect of mastering the Formal Logic 1 curriculum will naturally motivate registration for Formal Logic 2.

Those who subscribe to more revisionist views will appreciate not only that the present proposal constitutes an accommodation of such views but also that in its explicit concern with the fundamental importance of the identification, reconstruction and logical analysis of arguments framed in natural language, the proposal in effect seeks to motivate formal logic on an ultimately informal basis. Moreover, by way of amelioration of the kind of pedagogical phenomena, which tend to despondency in staff and students alike, this proposal seeks to supply, via electronic media, a support mechanism that recognises the character of the contemporary learning environment in general and the problem of non-attendance in particular. Finally, even if, as I have already acknowledged, the nature of and relations (if any) obtaining between formal logic and Informal Logic/Critical Thinking merits another paper, the following thought is worth perusal. Those institutions which have been or are now considered to be of the very highest quality as regards Philosophy are most unlikely to eliminate formal logic from their curricula in the foreseeable future. Given the highly competitive character of the graduate job market generally and postgraduate awards in particular, a question of intellectual conscience naturally arises: with respect to the ability to compete with their peers in such pressing matters, would any further revision/dilution of the elementary formal logic curriculum constitute a service or disservice to students of Philosophy?

p.tomassi@abdn.ac.uk

Endnotes

  1. Beebee, H. (2003) ‘Introductory Formal Logic: Why do we do it?’, Discourse learning and teaching in philosophical and religious studies, Volume 3, Number 1, pp. 53-62
  2. Indeed, the logic text adopted after Davidson (as used, for example, in academic year 1927-1928), was An Introductory Text-Book of Logic by Sydney Herbert Mellone, then its seventeenth edition. The aim of the work is: “… to connect the traditional doctrine with its Aristotelian fountainhead …” (Mellone 1916: v). The scope of the work is “… intended to stop short of giving what is supplied in Professor Bosanquet’s Essentials of Logic.” (Mellone 1916: v). For Mellone, “The most important works in which [Logic] has been developed are those of Herschel, Whewell and John Stuart Mill.” (Mellone 1916: 6). However, ‘formal logic’ is properly defined as “… the logic which the mediaeval writers developed out of such acquaintance with Aristotle as they possessed.” (Mellone 1916: 6).
  3. Some readers may feel that if this is not a conclusion too far, it must, at least, be one that is too quick. I acknowledge that the relationship between these subjects (if, indeed, these are independent subjects) or between these skills (if, indeed, these involve independent skills) and formal logic, merits a paper of its own. However, I would make the following points here. If informal logic and/or critical thinking are genuinely skills or subject-matters ultimately independent from both formal logic and Philosophy then the case for such courses can ultimately be made out by non-logicians and non-Philosophers. In other words, Philosophy would appear to have no special responsibility for any such course. But, if that is so, a further case must now be made out as to how (and why) any such course relates (at all) to formal logic? Ex hypothesi, no such course would have any obvious special significance for either Philosophy in general or formal logic in particular. Thus, proponents of such a view would talk themselves out of the debate.
    The alternative, that the two (informal logic/critical thinking and formal logic) share some (subject-matter and/or skills-based) point of intersection is prima facie more plausible: both formal and informal logicians might be interested in argument reconstruction, for example, or the identification of fallacies. But now my earlier point looms large. Are we to say that fallacies are inferences that fail to establish their conclusions? If so, by which standard? Could the conclusion be false while the premises are true? If so, we avail ourselves of the semantic (modal) definition of validity. Can we construct counterexamples to such inferences? If so, we avail ourselves of a notion of logical form. If not, why settle for less when formal logic patently offers more? Further, consider argument reconstruction. Minimally, this must involve identification of reasons given for a conclusion or conclusions. But isn’t the notion of a reason irreducibly normative? Is this a good reason or a bad reason for that conclusion? Again, we seem to require recourse to a notion of validity here. If not, how are good reasons distinguished from bad, and, indeed, good arguments from bad? Considerations of this kind prompt my conclusion in the main text. Further, while proponents of informal logic/critical thinking may propose criteria definitive of good argument which are less than central to classical logic, such as relevance of premises to conclusion, it does not follow that formal logic has no insight to offer here. To make any judgement on that matter, however, appears to presuppose some familiarity with Relevant (and Relevance) logic.
  4. In my experience, ensuring a gender balance among teaching staff can be a significant factor here.
  5. See http://www.abdn.ac.uk/diss/ltu/.
  6. This particular feature of the electronic media in question would constitute much of the basis of my response to the objection that any existing programme or package might serve the present purpose adequately.
  7. See http://www.questionmark.com/uk/

Bibliography

  1. Beall, J.C. and Van Fraassen, B.C. (2003) Possibilities and Paradox: an introduction to modal and many-valued logic, Oxford University Press, New York.
  2. Beebee, H. (2003) ‘Introductory Formal Logic: Why do we do it?’, Discourse learning and teaching in philosophical and religious studies, Volume 3, Number 1, pp. 53-62.
  3. Boolos, G.S. and Jeffrey, R.C. (1989) Computability and Logic, Third Edition, Cambridge University Press, Cambridge.
  4. Girle, R. (2001) Modal Logics and Philosophy, McGill-Queens University Press.
  5. Jeffels, P. (2003) ‘Formal Logic 1: Evaluation Report, April 2003’, http://www.abdn.ac.uk/diss/ltu/ .
  6. Lemmon, E.J. (1965) Beginning Logic, Thomas Nelson and Sons Ltd, London and Edinburgh.
  7. Mates, B. (1972) Elementary Logic, second edition. Oxford University Press, New York.
  8. Mellone, S.H. (1916) .An Introductory Text-Book of Logic, eighth edition, William Blackwood and Sons Ltd, Edinburgh and London.
  9. Slaney, J.K. (1983, 1988) The Logic 1 Notes, University of Edinburgh, Departmental publication.
  10. Tennant N. (1978) Natural Logic, Edinburgh University Press, Edinburgh.
  11. Tomassi, P. (1999), Logic, Routledge, London & New York.

Web Addresses

Logic Exercise 1:

  1. https://qmp1.abdn.ac.uk/q4/perception.dll  Note: Access to this item is free: use 'logic' in both the username and student ID fields.

  2. Home of the software used for Logic Exercise 1: http://www.questionmark.com/uk/
  3. LTU (Learning Technology Unit) pages: http://www.abdn.ac.uk/diss/ltu/
  4. Patricia Spence (LTU Manager): e-mail: p.spence@abdn.ac.uk Tel: 01224 273924
  5. Paul Tomassi/Logic: The Solutions Book: e-mail: p.tomassi@abdn.ac.uk Tel: 01224 272375 Note(s): a complete set of solutions in electronic form to all exercises in Paul Tomassi’s Logic are available free from myself to lecturers using the text.

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This page was originally on the website of The Subject Centre for Philosophical and Religious Studies. It was transfered here following the closure of the Subject Centre at the end of 2011.

 

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