Symbolic Logic: An Accessible Introduction to Serious Mathematical Logic
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In recent times, I have found myself expanding a set of notes into a full-fledged logic textbook. This page reproduces the Preface from the current version. In addition parts corresponding to courses in sentential logic (Phil 200), predicate logic (Phil 300) and then that plus basic metalogic (Phil 400) are available in the PDF format. This is work in progress, and so subject to change. I am happy for anyone to use this material -- I request only that you forward me comments, positive or otherwise.
Sentential Logic1 formatted for single sided printing
Sentential Logic2 formatted for double sided printing
Predicate Logic1 formatted for single sided printing
Predicate Logic2 formatted for double sided printing
Symbolic Logic1 whole text formatted for single sided printing
Symbolic Logic2 whole text formatted for double sided printing
Summer 2008
Preface
There is, I think, a gap between what many students learn in their first course in formal logic, and what they are expected to know for their second. Thus courses in mathematical logic with metalogical components often cast the barest glance at mathematical induction, and even the very idea of reasoning from definitions. But a first course also may leave these untreated, and fail as well explicitly to lay down the definitions upon which the second course is based. The aim of this text is to integrate material from these courses and, in particular, to make serious mathematical logic accessible to students I teach.
Accessibility, in this case, includes components which serve to locate this text among others: First, assumptions about background knowledge are minimal. I do not assume particular content about computer science, or about mathematics much beyond high school algebra. Officially, everything is introduced from the ground up. No doubt, the material requires a certain sophistication --- which one might acquire from other courses in critical reasoning, mathematics or computer science. But the requirement does not extend to particular contents from any of these areas.
Second, I aim to build skills, and to keep conceptual distance for different applications of 'so' relatively short. Authors of books that are entirely correct and precise, may assume skills and require readers to recognize connections and arguments that are not fully explicit. Perhaps this accounts for some of the reputed difficulty of the material. In contrast, I strive to make arguments almost mechanical and mundane (some would say "pedantic"). In many cases, I attempt this by introducing relatively concrete methods for reasoning. The methods are, no doubt, tedious or unnecessary for the experienced logician. However, I have found that they are valued by students, insofar as students are are presented with an occasion for success. These methods are not meant to wash over or substitute for understanding details, but rather to expose and clarify them. Clarity, beauty and power come, I think, by getting at details, rather than burying or ignoring them.
Third, the discussion is ruthlessly directed at core results. Results may be rendered inaccessible to students, who have many constraints on their time and schedules, simply because the results would come up in, say, a second course rather than a first. My idea is to exclude side topics and problems, and to go directly after (what I see as) the core. One manifestation is the way definitions and results from earlier sections feed into ones that follow. Thus simple integration is a benefit. Another is the way predicate logic with identity is introduced as a whole in Part I
. Though it is possible to isolate sentential logic from the first parts of chapter 2 through chapter 7, and so to use the text for separate treatments of sentential and predicate logic, the guiding idea is to avoid repetition that would be associated with independent treatments for sentential logic, or perhaps monadic predicate logic, the full predicate logic, and predicate logic with identity.Also (though it may suggest I am not so ruthless about extraneous material as I would like to think), I try to offer some perspective about what is accomplished along the way. In addition, this text may be of particular interest to those who have, or desire, an exposure to natural deduction in formal logic. In this case, accessibility arises from the nature of the system, and association with what has come before. In the first part, I introduce both axiomatic and natural derivation systems; and in the second, show how they are related.
There are different ways to organize a course around this text. For students who are likely to complete the whole, one might proceed sequentially through the text from beginning to end. Though chapter 3 makes conceptual sense where it is, pedagogically it is best if attempted after chapter 6. I am currently working within a sequence that treats sentential logic, quantificational logic and metalogic over an academic year in three quarters. The first course covers chapter 1 with the first parts of chapter 2, chapter 4, chapter 5, and chapter 6. The second covers the rest of chapter 2, chapter 4, chapter 5, and chapter 6 along with chapter 7. And the third takes up chapter 3 with the rest of Part II. Other organizations are possible.
Answers to selected exercises indicated by star are provided in the back of the book. Answers function as additional examples, complete demonstrations, and supply a check to see that work is on the right track. It is essential to success that you work a significant body of exercises successfully and independently. So do not neglect exercises!
Naturally, results in this book are not innovative. If there is anything original, it is in presentation. Even here, I am greatly indebted to others, especially perhaps Bergmann, Moor and Nelson, The Logic Book, and Mendelson, Introduction to Mathematical Logic. I borrow freely from what they and others have done. In particular theorems for exercises, while they stand on their own (!) have genetic antecedents in texts I have encountered over the years, including especially these places along with Forbes, Modern Logic. I thank my first logic teacher, G.J. Mattey, who communicated to me his love for the material. And I thank especially my colleague Darcy Otto for many helpful comments. In addition I have received helpful feedback from Hannah Roy, along with students in different logic classes at CSUSB. I expect that your sufferings will make it better still.
This text evolved over a number of years starting from notes originally provided as a supplement to other texts. The current version represents a complete rewrite and conversion to LATEX2e. This revision afforded the opportunity for many improvements. However, insofar as the conversion required retyping the text entirely by hand from what was a relatively stable manuscript, I would be surprised if it has not introduced a number of new typographical errors. I apologize for these in advance, and anticipate that you will let me hear about them in short order! The current version includes Part I and Part II. Both chapter 10 and chapter 12 are not complete. Rather, they contain material which could be developed in the direction of completed chapters. That material is included as it may be of interest, and Part II remains coherent even apart from those chapters.
I think this is fascinating material, and consider it great reward when students respond "cool!" as they sometimes do. I hope you will have that response more than once along the way.
T.R.
Summer 2008