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Sentential Logic Primer

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Sentential Logic Primer Sentential Logic Primer Richard Grandy Daniel Osherson Rice University1 Princeton University2 July 23, 2004 [email protected] [email protected] ii Copyright This work is copyrighted by Richard Grandy and Daniel Osherson. Use of the text is authorized, in part or its entirety, for non-commercial non-profit purposes. Electronic or paper copies may not be sold except at the cost of copy- ing. Appropriate citation to the original should be included in all copies of any portion. iii Preface Students often study logic on the assumption that it provides a normative guide to reasoning in English. In particular, they are taught to associate con- nectives like “and” with counterparts in Sentential Logic. English conditionals go over to formulas with → as principal connective. The well-known difficul- ties that arise from such translation are not emphasized. The result is the conviction that ordinary reasoning is faulty when discordant with the usual representation in standard logic. Psychologists are particularly susceptible to this attitude. The present book is an introduction to Sentential Logic that attempts to situate the formalism within the larger theory of rational inference carried out in natural language. After presentation of Sentential Logic, we consider its mapping onto English, notably, constructions involving “if . then . .” Our goal is to deepen appreciation of the issues surrounding such constructions. We make the book available, for free, on line (at least for now). Please be respectful of the integrity of the text. Large portions should not be incorporated into other works without permission. Feedback will be greatly appreciated. Errors, obscurity, or other defects can be brought to our attention via [email protected] or [email protected]. The provenance of revisions will be acknowledged as new versions are pro- duced. Richard Grandy Daniel Osherson iv Note to students So . We’re going to do some logic together, is that it? OK. We’re on board. We’ll do our best to be clear. Please forgive us if we occasionally let you down (wandering into impenetrable prose). In that case, don’t hesitate to send us (polite) email. We’ll try to fix the offending passage, and send you back a fresh (electronic) copy of the book. Now what about you? What can we expect in return? All we ask (but it’s a lot) is that you be an active learner. Yes, we know. Years of enforced passivity in school has made education seem like the movies (or worse, television). You settle back in your chair and let the show wash over you. But that won’t work this time. Logic isn’t so easy. The only hope for understanding it is to read slowly and grapple with every idea. If you don’t understand something, you must make an additional effort before moving on. That means re-reading the part that’s troubling you, studying an example, or working one of the exercises. If you’re reading this book with someone else (e.g., an instructor), you should raise difficulties with her as they arise rather than all-at-once at the end. Most important, when the discussion refers to a fact or definition that appears ear- lier in the book, go back and look at it to make sure that things are clear. Don’t just plod on with only a vague idea about the earlier material. To facilitate this process, read the book with a note pad to hand. When you go back to earlier material, jot down your current page so you can return easily. Now’s the time to tell you (while we’re still friends) that a normal person can actually come to enjoy logic. It looks like logic is about formulas in some esoteric language, but really it’s about people. But you won’t believe us until we’ve made significant progress in our journey. So let’s get going, if you have the courage. Perhaps we’ll meet again in Chapter 1. v Note to instructors The present text differs from most other logic books we know in the follow- ing ways. (a) Only the sentential calculus is treated. (b) Sentential semantics are built around the concept of meaning (sets of truth-assignments). (c) The derivation system is particularly simple in two respects. Assump- tions are cancelled by filling in open circles that flag live hypotheses; also, there is only one rule that cancels assumptions. (d) It is shown how probabilities can be attached to formulas. (e) Sentential Logic is evaluated as a theory of “secure inference” in English. (f) Having noted deficiencies in Logic’s treatment of English conditionals, several alternatives to standard logic are explored in detail. There are some exercises, but not enough. We will gratefully acknowledge any assistance in this matter (contact us about format). vi Acknowledgements Whatever is original in our discussion is little more than reassembly of ideas already developed by other scholars. The earlier work we’ve relied upon is acknowledged along the way. The book has benefitted from perspicacious feedback from Michael McDer- mott, and from eagle-eye proofing by Roger Moseley (a surgeon!). These gen- tlepeople should not be held responsible for errors and confusions that remain. The pictures that grace the chapters were composed by Anne Osherson. We acknowledge support from NSF grant IIS-9978135 to Osherson. Chapter 1 Introduction 1 2 CHAPTER 1. INTRODUCTION 1.1 Reasoning Suppose you had to choose one feature of mental life that sets our species apart from all others. It should be a capacity exercised virtually every day that affects the human condition in countless ways. Although present in attenuated form in other mammals, it should reach its most perfected state in people. What feature of mental life would you choose? It seems obvious that the only contender for such a special human capacity is love. What could be more remarkable about our species than the tendency of its members to form stable and deeply felt attachments to each other, tran- scending generations and gender, often extending to entire communities of het- erogeneous individuals? Love is surely what separates us from the grim world of beasts, and renders us worthy of notice and affection. (For discussion, see [49].) Alas, this book is about something else. It concerns reason, which is also pretty interesting (although not as interesting as love). The capacity for rea- soning may be considered the second most distinguishing characteristic of our species. We’re not bad at it (better, at any rate, than the brutes), and like love it seems necessary to keep the human species in business. To make it clearer what our subject matter is about, let us consider an ex- ample of reasoning. Suppose that c1 and c2 are cannonballs dropped simulta- neously from the top story of the Tower of Pisa. They have identical volumes, but c1 weighs one pound whereas c2 weighs two. Could c2 hit the ground be- fore c1? If it does, this is almost surely due to the fact that 2-pound objects fall faster than 1-pound objects. Now, c2 can be conceived as the attachment of two, 1-pound cannonballs, so if it fell faster than c1 this would show that two, 1-pound objects fall faster attached than separately. This seems so unlikely that we are led to conclude that c2 and c1 will land at the same time. Just this line of thought went through the mind of Galileo Galilei (1564 - 1642), and he probably never got around to dropping cannonballs for verification (see Cooper [21]). Galileo’s thinking, as we have presented it, has some gaps. For one thing, there is unclarity about the shapes of the two 1-pound cannonballs that com- 1.1. REASONING 3 pose c2. Still, the reasoning has evident virtue, and we are led to wonder about the biological conditions necessary for an organism to dream up such clever arguments, or even to follow them when presented by someone else. Related questions arise when thought goes awry. In an often cited passage from a prestigious medical journal, the author infers the probability of breast cancer given a negative mammogram from nothing more than the probability of a negative mammogram given breast cancer, taking them to be equal. That no such equivalence holds in general is seen by comparing the high probabil- ity of having swum in the ocean given you live in the Bahamas with the low probability of living in the Bahamas given you’ve swum in the ocean (or com- paring the probability of an animal speaking English given it’s a mammal with the probability of it being a mammal given it speaks English, etc.). What is it about our brains that allow such errors to arise so easily? The causes and consequences of good and bad thinking have been on the minds of reflective people for quite a while. The Old Testament devotes space to King Solomon’s son and successor Rehoboam. Faced with popular unrest stemming from his father’s reliance on forced labor, Rehoboam had the stun- ning idea that he could restore order with the declaration: “Whereas my father laid upon you a heavy yoke, I will add to your yoke. Whereas my father chas- tised you with whips, I shall chastise you with scorpions.” (Kings I.12.11) Social disaster ensued.1 Much Greek philosophical training focussed on distinguish- ing reliable forms of inference from such gems as: This mutt is your dog, and he is the father of those puppies. So, he is yours and a father, hence your father (and the puppies are your siblings).2 The authors of the 17th century textbook Logic or the Art of Thinking lament: “Everywhere we encounter nothing but faulty minds, who have practically no ability to discern the truth.
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