{"id":4,"date":"2025-04-08T16:28:57","date_gmt":"2025-04-08T16:28:57","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/?p=4"},"modified":"2025-10-13T16:57:49","modified_gmt":"2025-10-13T16:57:49","slug":"introduction","status":"publish","type":"front-matter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/front-matter\/introduction\/","title":{"raw":"Introduction","rendered":"Introduction"},"content":{"raw":"<h2>0.1 \u00a0Why study logic?<\/h2>\r\nLogic is one of the most important topics you will ever study.\r\n\r\n\u201cHow could you say such a thing?\u201d you might well protest. \u00a0And yet, consider: \u00a0logic teaches us many things, and one of these is how to recognize good and bad arguments. \u00a0Not just arguments about logic\u2014<span class=\"em\">any<\/span>\u00a0argument.\r\n\r\nNearly every undertaking in life will ultimately require that you evaluate an argument, perhaps several.\u00a0You are confronted with a question: \u00a0Should I buy this car or that car? \u00a0Should I go to this college or that college?\u00a0Did that scientific experiment show what the scientist claims it did? \u00a0Should I vote for the candidate who\u00a0promises\u00a0to lower taxes, or for the one who says she might raise them?\u00a0And so on.\u00a0Our lives are a long parade of choices.\u00a0 When we try to answer such questions, in order to make the best choices, we often have only one tool:\u00a0an argument. We listen to the reasons for and against various options, and must choose between them.\u00a0And so, the ability to evaluate arguments is an ability that is very useful in everything that you will do\u2014in your work, your personal life, your deepest reflections.\r\n\r\nIf you are a student, note that nearly every discipline, be it a science, one of the humanities, or a study like business, relies upon arguments. \u00a0Evaluating arguments is the most fundamental skill common to math, physics, psychology, literary studies, and any other intellectual endeavor. \u00a0Logic alone tells you how to evaluate the arguments of any discipline.\r\n\r\nThe alternative to developing these logical skills is to be always at the mercy of bad reasoning and, as a result, you will make bad choices. \u00a0Worse, you will always be manipulated by deceivers. \u00a0Speaking in Canandaigua, New York, on August 3, 1857, the escaped slave and abolitionist leader Frederick Douglass observed that:\r\n<blockquote>\r\n<div>Power concedes nothing without a demand. It never did and it never will. Find out just what any people will quietly submit to and you have found out the exact measure of injustice and wrong which will be imposed upon them, and these will continue till they are resisted with either words or blows, or with both. The limits of tyrants are prescribed by the endurance of those whom they oppress.<sup class=\"super\"><a id=\"ftnt_ref1\" href=\"https:\/\/milnepublishing.geneseo.edu\/concise-introduction-to-logic\/front-matter\/0-introduction\/#ftnt1\">[1]<\/a><\/sup><\/div><\/blockquote>\r\nWe can add to Frederick Douglass\u2019s words that: find out just how much a person can be deceived, and that is just how far she\u00a0will be deceived. \u00a0The limits of tyrants are also prescribed by the reasoning abilities of those they aim to oppress. \u00a0And what logic teaches you is how to demand and recognize good reasoning, and so how to avoid deceit. \u00a0You are only as free as your powers of reasoning enable.\r\n<h2>0.2 \u00a0What is logic?<\/h2>\r\nSome philosophers have argued that one cannot define \u201clogic\u201d. Instead, one can only show logic, by doing it and teaching others how to do it. \u00a0I am inclined to agree. \u00a0But it is easy to describe the benefits of logic. \u00a0For example, in this book, you will learn how to:\r\n<ul class=\"lst-kix_list_2-0 start\">\r\n \t<li class=\"marg-left\">Identify when an argument is good, and when it is bad;<\/li>\r\n \t<li class=\"marg-left\">Construct good arguments;<\/li>\r\n \t<li class=\"marg-left\">Evaluate reasons, and know when they should, and should not, be convincing;<\/li>\r\n \t<li class=\"marg-left\">Describe things with a precision that avoids misunderstanding;<\/li>\r\n \t<li class=\"marg-left\">Get a sense of how one can construct the foundations of arithmetic;<\/li>\r\n \t<li class=\"marg-left\">Begin to describe the meaning of \u201cpossibility\u201d and \u201cnecessity\u201d.<\/li>\r\n<\/ul>\r\nThat is by no means a complete list of the many useful things that logic can provide. \u00a0Some of us believe that logic and mathematics are ultimately the same thing, two endeavors with the same underlying structure distinguished only by different starting assumptions. \u00a0On such a view, we can also think of logic as the study of the ultimate foundations of mathematics. \u00a0(This is a reasonable characterization of logic, but those afraid of mathematics need not fear: \u00a0logic must become quite advanced before its relation to mathematics becomes evident.)\r\n\r\nUltimately, the only way to reveal the beauty and utility of logic is to get busy and do some logic. \u00a0In this book, we will approach the study of logic by building several precise logical languages and seeing how we can best reason with these. \u00a0The first of these languages is called \u201cthe propositional logic\u201d.\r\n<h2>0.3 \u00a0A note to students<\/h2>\r\nLogic is a skill. \u00a0The only way to get good at understanding logic and at using logic is to practice. \u00a0It is easy to watch someone explain a principle of logic, and easier yet to watch someone do a proof. \u00a0But you must understand a principle well enough to be able to apply it to new cases, and you must be able to do new proofs on your own. \u00a0Practice alone enables this.\r\n\r\nThe good news is that logic is easy. \u00a0The very goal of logic is to take baby steps, small and simple and obvious, and after we do this for a long while we find ourselves in a surprising and unexpected new place. \u00a0Each step on the way will be easy to take. \u00a0Logic is a long distance walk, not a sprint. \u00a0Study each small step we take, be sure you know how to apply the related skills, practice them, and then move on. \u00a0Anyone who follows this advice can master logic.\r\n<h2>0.4 \u00a0A note to instructors<\/h2>\r\nThis book incorporates a number of features that come from many years of experience teaching both introductory and advanced logic.\r\n\r\nFirst, the book moves directly to symbolic logic. \u00a0I don\u2019t believe that informal logic is worth the effort that it requires. \u00a0Informal logic largely consists of memorization (memorizing seemingly disconnected rules, memorizing fallacies, and so on). \u00a0Not only is this sure to be the kind of thing that students will promptly forget, but it completely obscures the simple beauty of why the various rules work, and why the fallacies are examples of bad reasoning. \u00a0A student who learns symbolic logic, however, is learning a skill. \u00a0Skills are retained longer; they encourage higher forms of reasoning; and they have far more power than a memorized list of facts. \u00a0Once one can recognize what makes an argument good, one can recognize the fallacies, regardless of whether one has memorized their names.\r\n\r\nSecond, this book focuses on some of the deeper features of logic, right at the beginning. \u00a0The notions of semantics and syntax are introduced in the first chapter. \u00a0Ideas like <span class=\"em\">theorem<\/span>, and a <span class=\"em\">model<\/span>, are discussed early on. \u00a0My experience has shown that students can grasp these concepts, and they ultimately pay off well by greatly expanding their own understanding.\r\n\r\nThird, this book uses examples, and constructs problems, from our intellectual history in order to illustrate key principles of logic. \u00a0The author is a philosopher, and understands logic to be both the method of philosophy and also one of the four fundamental sub-disciplines of philosophy. \u00a0But more importantly, these examples can do two things. They make it clear that arguments matter. \u00a0Weighty concerns are discussed in these arguments, and whether we accept their conclusions will have significant effects on our society. \u00a0Seeing this helps one to see the importance of logic. \u00a0These examples can also make this book suitable for a logic course that aims to fulfill a requirement for an introduction to the history of thought, an overview of Western civilization, or the knowledge foundations of a related discipline.\r\n\r\nFourth, I follow a no-shortcuts principle. \u00a0Most logic textbooks introduce a host of shortcuts. \u00a0They drop outer parentheses, they teach methods for shrinking truth tables, and so on. \u00a0These moves often confuse students, and for no good reason: \u00a0they have no conceptual value. \u00a0I suspect they only exist to spare the impatience of instructors, who would like to write expressions and truth tables more quickly. \u00a0In this book, except in the last chapter that looks to advanced logic, we will not introduce exceptions to our syntax, nor will we spend time on abridged methods. \u00a0The only exception is writing \u201c<span class=\"em strong\">T<\/span>\u201d for <span class=\"em\">true<\/span>\u00a0and \u201c<span class=\"em strong\">F<\/span>\u201d for <span class=\"em\">false<\/span>\u00a0in truth tables.\r\n\r\nFifth, this book includes a final chapter introducing some advanced topics in logic. \u00a0The purpose of this chapter is to provide students with some understanding of the exciting things that they can study if they continue with logic. \u00a0In my experience, students imagine that advanced logic will be just more proofs in first order logic. \u00a0Giving them a taste of what can come next is valuable. \u00a0My hope is that this chapter will motivate students to want to study more logic, and also that it can serve as a bridge between their studies in basic logic and the study of advanced logic.\r\n\r\nFinally, about typesetting: \u00a0quotation is an important logical principle, and so I adopted the precise but comparatively rare practice of putting punctuation outside of quotes. \u00a0This way, what appears in the quotations is alone what is being defined or otherwise mentioned. \u00a0I use italics only to indicate the meaning of a concept, or to distinguish symbolic terms of the object language from functions of the object language. \u00a0Bold is used to set aside elements of our metalanguage or object language.\r\n<h2>0.5 \u00a0Contact<\/h2>\r\nThe editors would appreciate any comments, advice, or discoveries of errata.\u00a0 They can be contacted at: allyson.turner@ccaurora.edu\r\n<h2>0.6 \u00a0Acknowledgements<\/h2>\r\nThe typesetting of proofs used the lplfitch LaTex package developed by John Etchemendy, Dave Barker-Plummer, and Richard Zach.\r\n\r\n<hr \/>\r\n\r\n<div>\r\n\r\n<a id=\"ftnt1\" href=\"https:\/\/milnepublishing.geneseo.edu\/concise-introduction-to-logic\/front-matter\/0-introduction\/#ftnt_ref1\">[1]<\/a>\u00a0From Blassingame (1985: 204), in a speech titled \u201cThe Significance of Emancipation in the West Indies.\u201d\r\n\r\n<\/div>","rendered":"<h2>0.1 \u00a0Why study logic?<\/h2>\n<p>Logic is one of the most important topics you will ever study.<\/p>\n<p>\u201cHow could you say such a thing?\u201d you might well protest. \u00a0And yet, consider: \u00a0logic teaches us many things, and one of these is how to recognize good and bad arguments. \u00a0Not just arguments about logic\u2014<span class=\"em\">any<\/span>\u00a0argument.<\/p>\n<p>Nearly every undertaking in life will ultimately require that you evaluate an argument, perhaps several.\u00a0You are confronted with a question: \u00a0Should I buy this car or that car? \u00a0Should I go to this college or that college?\u00a0Did that scientific experiment show what the scientist claims it did? \u00a0Should I vote for the candidate who\u00a0promises\u00a0to lower taxes, or for the one who says she might raise them?\u00a0And so on.\u00a0Our lives are a long parade of choices.\u00a0 When we try to answer such questions, in order to make the best choices, we often have only one tool:\u00a0an argument. We listen to the reasons for and against various options, and must choose between them.\u00a0And so, the ability to evaluate arguments is an ability that is very useful in everything that you will do\u2014in your work, your personal life, your deepest reflections.<\/p>\n<p>If you are a student, note that nearly every discipline, be it a science, one of the humanities, or a study like business, relies upon arguments. \u00a0Evaluating arguments is the most fundamental skill common to math, physics, psychology, literary studies, and any other intellectual endeavor. \u00a0Logic alone tells you how to evaluate the arguments of any discipline.<\/p>\n<p>The alternative to developing these logical skills is to be always at the mercy of bad reasoning and, as a result, you will make bad choices. \u00a0Worse, you will always be manipulated by deceivers. \u00a0Speaking in Canandaigua, New York, on August 3, 1857, the escaped slave and abolitionist leader Frederick Douglass observed that:<\/p>\n<blockquote>\n<div>Power concedes nothing without a demand. It never did and it never will. Find out just what any people will quietly submit to and you have found out the exact measure of injustice and wrong which will be imposed upon them, and these will continue till they are resisted with either words or blows, or with both. The limits of tyrants are prescribed by the endurance of those whom they oppress.<sup class=\"super\"><a id=\"ftnt_ref1\" href=\"https:\/\/milnepublishing.geneseo.edu\/concise-introduction-to-logic\/front-matter\/0-introduction\/#ftnt1\">[1]<\/a><\/sup><\/div>\n<\/blockquote>\n<p>We can add to Frederick Douglass\u2019s words that: find out just how much a person can be deceived, and that is just how far she\u00a0will be deceived. \u00a0The limits of tyrants are also prescribed by the reasoning abilities of those they aim to oppress. \u00a0And what logic teaches you is how to demand and recognize good reasoning, and so how to avoid deceit. \u00a0You are only as free as your powers of reasoning enable.<\/p>\n<h2>0.2 \u00a0What is logic?<\/h2>\n<p>Some philosophers have argued that one cannot define \u201clogic\u201d. Instead, one can only show logic, by doing it and teaching others how to do it. \u00a0I am inclined to agree. \u00a0But it is easy to describe the benefits of logic. \u00a0For example, in this book, you will learn how to:<\/p>\n<ul class=\"lst-kix_list_2-0 start\">\n<li class=\"marg-left\">Identify when an argument is good, and when it is bad;<\/li>\n<li class=\"marg-left\">Construct good arguments;<\/li>\n<li class=\"marg-left\">Evaluate reasons, and know when they should, and should not, be convincing;<\/li>\n<li class=\"marg-left\">Describe things with a precision that avoids misunderstanding;<\/li>\n<li class=\"marg-left\">Get a sense of how one can construct the foundations of arithmetic;<\/li>\n<li class=\"marg-left\">Begin to describe the meaning of \u201cpossibility\u201d and \u201cnecessity\u201d.<\/li>\n<\/ul>\n<p>That is by no means a complete list of the many useful things that logic can provide. \u00a0Some of us believe that logic and mathematics are ultimately the same thing, two endeavors with the same underlying structure distinguished only by different starting assumptions. \u00a0On such a view, we can also think of logic as the study of the ultimate foundations of mathematics. \u00a0(This is a reasonable characterization of logic, but those afraid of mathematics need not fear: \u00a0logic must become quite advanced before its relation to mathematics becomes evident.)<\/p>\n<p>Ultimately, the only way to reveal the beauty and utility of logic is to get busy and do some logic. \u00a0In this book, we will approach the study of logic by building several precise logical languages and seeing how we can best reason with these. \u00a0The first of these languages is called \u201cthe propositional logic\u201d.<\/p>\n<h2>0.3 \u00a0A note to students<\/h2>\n<p>Logic is a skill. \u00a0The only way to get good at understanding logic and at using logic is to practice. \u00a0It is easy to watch someone explain a principle of logic, and easier yet to watch someone do a proof. \u00a0But you must understand a principle well enough to be able to apply it to new cases, and you must be able to do new proofs on your own. \u00a0Practice alone enables this.<\/p>\n<p>The good news is that logic is easy. \u00a0The very goal of logic is to take baby steps, small and simple and obvious, and after we do this for a long while we find ourselves in a surprising and unexpected new place. \u00a0Each step on the way will be easy to take. \u00a0Logic is a long distance walk, not a sprint. \u00a0Study each small step we take, be sure you know how to apply the related skills, practice them, and then move on. \u00a0Anyone who follows this advice can master logic.<\/p>\n<h2>0.4 \u00a0A note to instructors<\/h2>\n<p>This book incorporates a number of features that come from many years of experience teaching both introductory and advanced logic.<\/p>\n<p>First, the book moves directly to symbolic logic. \u00a0I don\u2019t believe that informal logic is worth the effort that it requires. \u00a0Informal logic largely consists of memorization (memorizing seemingly disconnected rules, memorizing fallacies, and so on). \u00a0Not only is this sure to be the kind of thing that students will promptly forget, but it completely obscures the simple beauty of why the various rules work, and why the fallacies are examples of bad reasoning. \u00a0A student who learns symbolic logic, however, is learning a skill. \u00a0Skills are retained longer; they encourage higher forms of reasoning; and they have far more power than a memorized list of facts. \u00a0Once one can recognize what makes an argument good, one can recognize the fallacies, regardless of whether one has memorized their names.<\/p>\n<p>Second, this book focuses on some of the deeper features of logic, right at the beginning. \u00a0The notions of semantics and syntax are introduced in the first chapter. \u00a0Ideas like <span class=\"em\">theorem<\/span>, and a <span class=\"em\">model<\/span>, are discussed early on. \u00a0My experience has shown that students can grasp these concepts, and they ultimately pay off well by greatly expanding their own understanding.<\/p>\n<p>Third, this book uses examples, and constructs problems, from our intellectual history in order to illustrate key principles of logic. \u00a0The author is a philosopher, and understands logic to be both the method of philosophy and also one of the four fundamental sub-disciplines of philosophy. \u00a0But more importantly, these examples can do two things. They make it clear that arguments matter. \u00a0Weighty concerns are discussed in these arguments, and whether we accept their conclusions will have significant effects on our society. \u00a0Seeing this helps one to see the importance of logic. \u00a0These examples can also make this book suitable for a logic course that aims to fulfill a requirement for an introduction to the history of thought, an overview of Western civilization, or the knowledge foundations of a related discipline.<\/p>\n<p>Fourth, I follow a no-shortcuts principle. \u00a0Most logic textbooks introduce a host of shortcuts. \u00a0They drop outer parentheses, they teach methods for shrinking truth tables, and so on. \u00a0These moves often confuse students, and for no good reason: \u00a0they have no conceptual value. \u00a0I suspect they only exist to spare the impatience of instructors, who would like to write expressions and truth tables more quickly. \u00a0In this book, except in the last chapter that looks to advanced logic, we will not introduce exceptions to our syntax, nor will we spend time on abridged methods. \u00a0The only exception is writing \u201c<span class=\"em strong\">T<\/span>\u201d for <span class=\"em\">true<\/span>\u00a0and \u201c<span class=\"em strong\">F<\/span>\u201d for <span class=\"em\">false<\/span>\u00a0in truth tables.<\/p>\n<p>Fifth, this book includes a final chapter introducing some advanced topics in logic. \u00a0The purpose of this chapter is to provide students with some understanding of the exciting things that they can study if they continue with logic. \u00a0In my experience, students imagine that advanced logic will be just more proofs in first order logic. \u00a0Giving them a taste of what can come next is valuable. \u00a0My hope is that this chapter will motivate students to want to study more logic, and also that it can serve as a bridge between their studies in basic logic and the study of advanced logic.<\/p>\n<p>Finally, about typesetting: \u00a0quotation is an important logical principle, and so I adopted the precise but comparatively rare practice of putting punctuation outside of quotes. \u00a0This way, what appears in the quotations is alone what is being defined or otherwise mentioned. \u00a0I use italics only to indicate the meaning of a concept, or to distinguish symbolic terms of the object language from functions of the object language. \u00a0Bold is used to set aside elements of our metalanguage or object language.<\/p>\n<h2>0.5 \u00a0Contact<\/h2>\n<p>The editors would appreciate any comments, advice, or discoveries of errata.\u00a0 They can be contacted at: allyson.turner@ccaurora.edu<\/p>\n<h2>0.6 \u00a0Acknowledgements<\/h2>\n<p>The typesetting of proofs used the lplfitch LaTex package developed by John Etchemendy, Dave Barker-Plummer, and Richard Zach.<\/p>\n<hr \/>\n<div>\n<p><a id=\"ftnt1\" href=\"https:\/\/milnepublishing.geneseo.edu\/concise-introduction-to-logic\/front-matter\/0-introduction\/#ftnt_ref1\">[1]<\/a>\u00a0From Blassingame (1985: 204), in a speech titled \u201cThe Significance of Emancipation in the West Indies.\u201d<\/p>\n<\/div>\n","protected":false},"author":158,"menu_order":1,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"front-matter-type":[12],"contributor":[],"license":[],"class_list":["post-4","front-matter","type-front-matter","status-publish","hentry","front-matter-type-introduction"],"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/front-matter\/4","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/front-matter"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/types\/front-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":3,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/front-matter\/4\/revisions"}],"predecessor-version":[{"id":330,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/front-matter\/4\/revisions\/330"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/front-matter\/4\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=4"}],"wp:term":[{"taxonomy":"front-matter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/front-matter-type?post=4"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=4"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=4"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}