Preface for Instructors

The rationale for this textbook is entirely pedagogical, based on my experience teaching logic in Canada, Nigeria, and the United States during the past thirty-something years. I have found myself dissatisfied with the dichotomy of approaches to teaching introduc­tory logic.

By contrast, my approach in this textbook is to introduce rules of inference in the con­text of natural arguments, i.e., ones that have actually been offered by historical agents in real-life argumentative contexts, and to apply the formal techniques we Ieam to such arguments.

To this end, I take a natural deduction approach, bucking the recent trend of relying solely on the method of truth trees (semantic tableaux) for determining validity. The advantage of truth tree methods is readily seen: one can determine whether or not a given argument is valid by a systematic procedure, and, at least in statement logic and unary predicate logic, this is a decision procedure; whereas the natural deduction method does not determine invalidity, for which one has to appeal to supplementary methods such as truth tables, argument diagrams, or the method of counterexamples. But it seems to me that if the fundamental goal of an elementary logic course is to try to hone students’ rea­soning skills, the natural deduction method has the advantage that it is based on “rules of inference” that correspond fairly closely with the ways we intuitively reason, or at least to rules that are implicit in our reasoning. So although I include chapters on truth trees in the text, I delay their introduction until after students have got a good handle on the rules of inference of natural deduction.

Much attention has been paid here to the order and pace of the introduction of the material. Instead of hitting students with a large batch of rules of inference that are too much to be taken in at once, it introduces them gradually, encouraging rather than de-em- phasizing correspondence with their own logical intuitions and competences.

In this connection, a distinctive feature of the approach taken here is the appeal to a Chrysippean notion of validity, according to which an argument is valid if and only if denying its conclusion is incompatible with accepting all its premises, and otherwise invalid. This definition is the appropriate one for natural arguments, and is distinguished from formal validity, according to which an argument is formally valid if it has a valid argument form. An argument form, on the other hand, is valid if and only if there is no argument of that form which has all true premises and a false conclusion.

In Predicate Logic, a similar appeal to context allows a neater solution to problems of existential import than usually given in introductory texts. Existential import is implicit in certain contexts, but not in others. So universal statements do not logically imply exis­tential import, but in certain contexts they can be taken to presuppose it. This leads to a very natural treatment of penevalid arguments, ones whose validity depends upon one of the universal premises being taken to have existential import. Thus we symbolize A- and E-statements in such a way that no existential import is involved; but if the validity of an argument depends upon an existential import that is implicit in the context, we treat this as an implicit existential premise that must be made explicit. In such a case, the argument will be valid only on that assumption.

There are several other pedagogical innovations, all of them minor, but with a cumu­lative effect that is significant. Among them are:

a) The method of presenting conditionals and indirect (or reductio) proofs, i.e., proofs which depend on an unasserted or dischargeable assumption (here termed a sup­position), is adapted from those of Hurley and Hofstadter, with an indentation for each supposition, and for each subsequent line depending on this supposition until it is discharged.

b) The text exploits the structural similarity between these conditional and indirect proofs in formal logic and natural arguments based on suppositions. The method of diagramming natural reductio arguments is indebted to Alec Fisher’s The Logic of Real Arguments, but the systematic treatment of the correspondence between con­ditional and indirect proofs in formal logic and natural suppositional arguments is original with this text.

c) New logic diagrams, inspired by those of Charles Dodgson (Lewis Carroll), with debts to John Venn and George Boole for their interpretation. These diagrams gen­eralize very smoothly from arguments involving 2 predicates to those involving 4 or more, and much more naturally than the traditional Venn diagrams. Although these are presented as modifications of CarrolTs diagrams, the modifications are signif­icant, and make them easier to use than CarrolTs or Venn’s, as my students have attested. A comparison of Venn’s, Carroll’s, and my diagrams is given in Appendix 3.

d) The rules for generalizing from an instance of a universal quantification (UG) and for instantiating an existential quantification (EI) are simplified by the introduction of the notion of arbitrary individual names: i, j, and ê are arbitrary names used only for EI, and u, v, and w are different arbitrary names used in UG: they are introduced in anticipation of UG, and universal generalizations may only validly be made from instances involving u, v, or w.

e) The supposition of arbitrary names for the UG and EI proofs corresponds to the sup­position of specific examples or cases for the sake of specificity in natural arguments. The distinction of suppositions for the sake of specificity from suppositions for the sake ofargument is original with this text.

There is more material in the textbook than can be comfortably managed in a 12 week semester. This allows considerable scope for adjustment to the different tastes and approaches of instructors by taking different tracks through the material. One track could emphasize the application of logic to natural argument and forego truth trees and logic with identity, finishing with relational logic. Or one could set aside treatment of the complexities of natural argument, omit truth tables in favour of truth trees, and also leave out the theory of the syllogism and argument diagrams. Another option would be simply to remain content with a thorough introduction to logic that stopped short of relational logic, but possibly got as far as logic with identity. These options are outlined in the following table, though clearly many other choices are possible depending on the instructor’s own preferences.