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Routledge Contemporary Introductions to Philosophy
Philosophical Logic
Introductory logic is generally taught as a straightforward technical discipline. In this book, John MacFarlane helps the reader think about the limitations of, presuppositions of, and alternatives to classical first-order predicate logic, making this an ideal introduction to philosophical logic for any student who already has completed an introductory logic course.
The book explores the following questions. Are there quantificational idioms that cannot be expressed with the familiar universal and existential quantifiers? How can logic be extended to capture modal notions like necessity and obligation? Does the material conditional adequately capture the meaning of 'if'--and if not, what are the alternatives? Should logical consequence be understood in terms of models or in terms of proofs? Can one intelligibly question the validity of basic logical principles like Modus Ponens or Double Negation Elimination? Is the fact that classical logic validates the inference from a contradiction to anything a flaw, and if so, how can logic be modified to repair it? How, exactly, is logic related to reasoning? Must classical logic be revised in order to be applied to vague language, and if so how? Each chapter is organized around suggested readings and includes exercises designed to deepen the reader's understanding.
Key Features:
An integrated treatment of the technical and philosophical issues comprising philosophical logic Designed to serve students taking only one course in logic beyond the introductory level Provides tools and concepts necessary to understand work in many areas of analytic philosophy Includes exercises, suggested readings, and suggestions for further exploration in each chapter
The book explores the following questions. Are there quantificational idioms that cannot be expressed with the familiar universal and existential quantifiers? How can logic be extended to capture modal notions like necessity and obligation? Does the material conditional adequately capture the meaning of 'if'--and if not, what are the alternatives? Should logical consequence be understood in terms of models or in terms of proofs? Can one intelligibly question the validity of basic logical principles like Modus Ponens or Double Negation Elimination? Is the fact that classical logic validates the inference from a contradiction to anything a flaw, and if so, how can logic be modified to repair it? How, exactly, is logic related to reasoning? Must classical logic be revised in order to be applied to vague language, and if so how? Each chapter is organized around suggested readings and includes exercises designed to deepen the reader's understanding.
Key Features:
An integrated treatment of the technical and philosophical issues comprising philosophical logic Designed to serve students taking only one course in logic beyond the introductory level Provides tools and concepts necessary to understand work in many areas of analytic philosophy Includes exercises, suggested readings, and suggestions for further exploration in each chapter
238 pages, Paperback
First published January 1, 2020
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May 15, 2021
"Philosophical logic" in this book refers to philosophical investigations into the foundations of logic (rather than those into some logic distinctive of philosophical thought, or into findings of logic that may be applied to philosophy). It deals, specifically, (in chronological order across chapters) with the topics of quantification, modal logic, conditionals, logical consequence defined according to models, logical consequence defined according to proofs, relevance logic, and logical treatments of vagueness.
All of these chapters were fascinating; in each foundations of logic that I had believed were unquestionable or nature-given facts turned out to be either (1) contingent (and so there were alternatives that could replace them) or (2) conceptually problematic or contradictory with respect to either our intuitions about corresponding topics in philosophy or metaphysics, or principles found elsewhere in logic. An example of (2) is that it turns out our intuitive concepts of modality (e.g., possibility and necessity and their difference) could be modeled in distinct formal logics; there are more than 5 logical systems, each of which behave according to different rules. This helps us think through that in our ordinary language usage of modal terms, we use these ambiguously between different ways by which modality may operate; these ways are made explicit by these different logics. (This is dealt with in chapter 4).
Another example of (2) is the assumption in logic that if we have contradictory premises, we can conclude the proof with anything at all, and the inference is logically valid (this is called ex falso quolibet). In ordinary thinking, we do not commit to this; all humans believe in some contradictory beliefs, but we don't take this as warranting our concluding with anything we fancy. Relevance logic is a non-classical logic developed to get rid of this assumption of classical logic. It formalizes in logical terms the idea that the conclusion needs to be relevant to the premises. It does this by rejecting disjunctive syllogism: if we start with any proposition, proof rules allow us to add to it another proposition with the connective "or"; then if we have the negation of one of these propositions, we can conclude with the other proposition as true. Disjunctive syllogism is an essential step for ex falso quolibet, so rejecting it would reject the latter. A way to reject it is to commit to a four-valued logic, on which a proposition might be true, false, true and false, or neither; logical validity can then be defined in different ways, including that truth is preserved from the premises to the conclusion, non-falsity is preserved, or both. (This is dealt with in chapter 7).
An example of (1) is that the concept of logical truth or validity can be analyzed in very different ways. I had always thought it was an irreducible or primitive concept, but good heavens I was wrong! Intuitively, most of us implicitly define logical validity in modal terms, as that the premises necessarily entail the conclusion. But there are different interpretations of modality (as I gestured towards above), and each interpretation turns out to yield consequences that contradict our intuitions on which inferences are logically valid or not.
An alternative way to go is to define logical validity in terms of provability; an inference is logically valid if it can be proven in a system of logic. But the problem is that we could invent just any logical system with its idiosyncratic rules (as long as its systematic and obeys other general criteria). So if we take this approach, we need to find ways to rule out proof rules that look bad and could be used to justify inferences that are obviously not logically valid. Dag Prawitz does just this: he offers a method for testing an inference rule, for whether it is a good one. The basic idea is that the elimination rule of a certain logical connective necessary for getting premises to yield a certain conclusion must be definable in terms of the corresponding introduction rule. The elimination rule for a certain connective can be mechanically derived from the corresponding introduction rule (whose justification ultimately comes down to our primitive commitment to certain "canonical arguments," but expanding on this in full is something I cannot do here). So logical validity may be understood on the basis of proof rules, and the goodness of a proof rule is self-justifying. This amounts to a certain constructivist picture of logic: logical validity isn't a fundamental, god-given matter, but it depends on whichever somewhat relative ways we do proofs (while there are constraints on which logical connectives are legitimate, there is still a plurality of them, and it just happens to be that classical logic latched onto the set that it did). (This is dealt with in chapter 7).
The book is full of fascinating questions and topics. I arbitrarily chose the ones I've named above to expand on a bit. I highly recommend anyone who is interested in how logic really works or where it comes from!
All of these chapters were fascinating; in each foundations of logic that I had believed were unquestionable or nature-given facts turned out to be either (1) contingent (and so there were alternatives that could replace them) or (2) conceptually problematic or contradictory with respect to either our intuitions about corresponding topics in philosophy or metaphysics, or principles found elsewhere in logic. An example of (2) is that it turns out our intuitive concepts of modality (e.g., possibility and necessity and their difference) could be modeled in distinct formal logics; there are more than 5 logical systems, each of which behave according to different rules. This helps us think through that in our ordinary language usage of modal terms, we use these ambiguously between different ways by which modality may operate; these ways are made explicit by these different logics. (This is dealt with in chapter 4).
Another example of (2) is the assumption in logic that if we have contradictory premises, we can conclude the proof with anything at all, and the inference is logically valid (this is called ex falso quolibet). In ordinary thinking, we do not commit to this; all humans believe in some contradictory beliefs, but we don't take this as warranting our concluding with anything we fancy. Relevance logic is a non-classical logic developed to get rid of this assumption of classical logic. It formalizes in logical terms the idea that the conclusion needs to be relevant to the premises. It does this by rejecting disjunctive syllogism: if we start with any proposition, proof rules allow us to add to it another proposition with the connective "or"; then if we have the negation of one of these propositions, we can conclude with the other proposition as true. Disjunctive syllogism is an essential step for ex falso quolibet, so rejecting it would reject the latter. A way to reject it is to commit to a four-valued logic, on which a proposition might be true, false, true and false, or neither; logical validity can then be defined in different ways, including that truth is preserved from the premises to the conclusion, non-falsity is preserved, or both. (This is dealt with in chapter 7).
An example of (1) is that the concept of logical truth or validity can be analyzed in very different ways. I had always thought it was an irreducible or primitive concept, but good heavens I was wrong! Intuitively, most of us implicitly define logical validity in modal terms, as that the premises necessarily entail the conclusion. But there are different interpretations of modality (as I gestured towards above), and each interpretation turns out to yield consequences that contradict our intuitions on which inferences are logically valid or not.
An alternative way to go is to define logical validity in terms of provability; an inference is logically valid if it can be proven in a system of logic. But the problem is that we could invent just any logical system with its idiosyncratic rules (as long as its systematic and obeys other general criteria). So if we take this approach, we need to find ways to rule out proof rules that look bad and could be used to justify inferences that are obviously not logically valid. Dag Prawitz does just this: he offers a method for testing an inference rule, for whether it is a good one. The basic idea is that the elimination rule of a certain logical connective necessary for getting premises to yield a certain conclusion must be definable in terms of the corresponding introduction rule. The elimination rule for a certain connective can be mechanically derived from the corresponding introduction rule (whose justification ultimately comes down to our primitive commitment to certain "canonical arguments," but expanding on this in full is something I cannot do here). So logical validity may be understood on the basis of proof rules, and the goodness of a proof rule is self-justifying. This amounts to a certain constructivist picture of logic: logical validity isn't a fundamental, god-given matter, but it depends on whichever somewhat relative ways we do proofs (while there are constraints on which logical connectives are legitimate, there is still a plurality of them, and it just happens to be that classical logic latched onto the set that it did). (This is dealt with in chapter 7).
The book is full of fascinating questions and topics. I arbitrarily chose the ones I've named above to expand on a bit. I highly recommend anyone who is interested in how logic really works or where it comes from!
December 6, 2022
Very well written.
Displaying 1 - 2 of 2 reviews