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<html> <!-- Mirrored from users.eniinternet.com/bradleym/RDMApp.html by HTTrack Website Copier/3.x [XR&CO'2014], Sun, 06 Nov 2022 06:59:39 GMT --> <head> <title> Reasoning And Decision Making - Milton N. Bradley</title> <link rel="stylesheet" href="style2.css"> </head> <body bgcolor="#E0FFFF"> <a name="top"></a> <center><br><br> <font class="booktitle"> Reasoning And Decision Making</font> </font> <br><br> <h1> <font class="chaptitle"> © Milton N. Bradley 2010</font> <hr></h1></center> <br><br> <center><a name="Appendix"></a> <font class="chaptitle"> <strong>Appendix - Introduction To Symbolic Logic</strong> </font><br> </center> <br><br><br> <font size= +1> This Appendix is included for completeness, because the time honored techniques it discusses form the logical (albeit theoretical) basis for all of the more practical problem solving techniques discussed in the following chapters that are our real interest. Because this material is very technologically dense and difficult, as noted in the Introduction, most readers will probably be well advised to either skim it lightly on first reading, or even skip it entirely. If you do either, the loss to your overall understanding will, in most cases, not be highly significant. Despite that, you will be well advised to invest the time and effort needed to master this material sooner rather than later, because the payoff to your overall understanding for so doing will be considerable. <br><br> Symbolic Logic is an ancient discipline dating at least back to the time of the famous Greek philosopher Aristotle (384-322 B.C.). Its purpose is to help make sense of human communication by providing a method which can enable us to distinguish between Arguments which can be relied upon and those which cannot.<br><br> This is a complex, well developed and mathematically oriented discipline which is often the subject of multi-semester courses, so it will not be possible for us to do much more here than to introduce its main ideas and briefly explore how they are applied. Our objective in doing this won’t be for you to master this important subject, but rather for you to obtain a broad grasp of how and when to use it. But it is important enough that those of you who really wish to perfect their Reasoning skills are strongly urged to invest the effort to further pursue this topic on your own.<br><br> Whatever your level of skill in utilizing Symbolic Logic, in order for this process to have any real chance of success it is first necessary that those communicating “be on the same page” - that is, to agree on the meaning of the words used in their communications. This may seem obvious and easy to accomplish, but, as we describe below, history clearly shows that it is all too often anything but! So let us digress briefly to address this important issue.<br><br> As a result there’s an important caveat that must be observed if the application of even the most powerful a set of techniques like REAP, Critical Thinking, or even Formal Symbolic Logic are to be really productive - the reliability of the proposer of the information must first be evaluated and then, if necessary, compensated for!<br><br> <strong>Categorizing</strong><br><br> Efficiently focusing upon the specifics of what’s important to any real world problem situation requires that there first exist an accurate description of that situation, and this in turn requires the ability to categorize. The verbal hierarchy used in categorizing is:<br><br> <OL type = DISC> <LI> <strong>Concepts</strong> = Ideas that represent general categories or types of things <LI> <strong>Genus</strong> = The broadest, most inclusive Concept class. <LI> <strong>Species</strong> = A less inclusive subclass of Genus. <LI> <strong>Referents</strong> = The individual items included in a subclass. </OL><br> Caution! Genus and Species are relative terms. (e.g. In one plausible formulation, Dog is the Genus for the Species Poodle. But in a different formulation, Dog may plausibly be considered a Species of the Genus Mammal, although perhaps calling it a sub-Genus might actually be more accurate. In either case, Aunt Mary’s pet dog Fido is a Referent.)<br><br> <strong>Defining Terms</strong><br><br> Preventing the considerable waste of time and effort often unnecessarily expended on mere verbal disputes, which can occur when people believe they disagree about some matter of substance when in reality they are just differently defining or using some key terminology, can frequently be achieved via the simple advance definition of key terms! And once they recognize what’s going on and settle on single definitions for the terms in question, what might otherwise be extremely antagonistic disputants may find that they don't disagree at all!<br><br> A classic case in point occurred at Supreme Headquarters of the Allied Expeditionary Forces (SHAEF) Europe during the key concluding days of WW II, when Supreme Commander Gen. Dwight D. Eisenhower was chairing a meeting with British, French and other Allied Generals. A proposal was made which the British requested be “tabled”, and the Americans objected. After hours of wrangling, it finally developed that in Britain “tabling” meant “address immediately” while in America it meant “defer”, so both sides had really been agreement all along, but simply didn’t know it because the supposedly “obvious” and common colloquial verb “table” had not been satisfactorily defined!<br><br> Dictionary definitions often provide an effective starting point, but aren’t always adequate because they sometimes consist almost entirely of a synonym for the term at issue, and are therefore only really useful if that synonym has just one possible meaning which both parties know. So the only feasible solution in cases in which a “ready made” definition satisfactory to all doesn’t already exist may be to conjure up one of your own!<br><br> <strong>Guidelines For Good Definitions</strong><br><br> <OL type = DISC> <LI> Include the Genus (general category) and Differentia (specifics which uniquely identify the particular item or concept of interest). <LI> Use the appropriate level of detail (neither too broad nor too narrow). <LI> Use only the essential (rather than trivial) attributes of the item being defined. <LI> Avoid circularity (= using something as its own “definition”). <LI> Avoid the negative, if possible . <LI> Avoid vagueness, obscure language, and metaphor. </OL><br> <strong>Some Key Symbolic Logic Definitions</strong><br><br> <strong>Statement = An assertion that can be determined to be true or false.</strong><br><br> Statements may be either simple or compound.</strong> 2 + 3 = 5 is an example of a simple Statement which is true. 2 + 3 = 6 is a simple Statement which is false. “The Moon is made of green cheese” is another simple statement which is false.<br><br> <strong>Compound Statement = Two or more simple Statements joined by one or more Symbolic Logic Operators</strong> <br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="100%" colspan="3" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">Symbolic Logic Operators</font></strong></td> </tr> <tr> <td width="33%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong><font size="4">Connective</strong></td> <td width="20%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> <font size="4">Symbol</strong></td> <td width="34%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> <font size="4">Formal Name</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31">Not</td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="5"> ~~</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1">Negation</td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1">And</td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="5"> &n</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1">Conjunction</td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1">Or</td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="5"> Ú?</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1">Disjunction</td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1">If ...then</td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="5"> É?</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1">Conditional</td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1">...if and only if...</td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="5"> º?</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1">Biconditional</td> </tr> </table> </div> <br><br> <strong>Explanation = A collection of related Statements in which the Conclusion is already accepted as fact</strong>, and therefore need not be proved. In this case, the only function of the Premises is to improve understanding. (Example: The Sun appears to rise and set every day because the Earth rotates on its axis once every 24 hours.)<br><br> <strong>Proposition = A Statement in which something is affirmed or denied, so that it can be properly characterized as either True (T) or False (F).<br><br> Categorical Proposition = A Proposition that asserts a relationship between two categories of (logical or physical) things.<br><br> A Categorical Proposition consists of 4 elements: <OL type = DISC> <LI> 2 elements (Subject and Predicate) which determine the content of the Proposition <LI> 2 elements (Quantifier and Copula) which determine its kind (= “quality”).</strong><br><br> <strong>These elements are defined as follows:</strong> <OL type = DISC> <LI> <strong>Subject</strong><br> The class, category, or concept that is the concern of the Proposition. <LI> <strong>Predicate</strong><br> The class, category or concept which is related by the Proposition to the Subject. <LI> <strong>Quantifier</strong><br> Sets the proportion of the Subject about which the Proposition makes a claim. Only two proportions matter in Categorical Logic: “all”, and “less than all”. <strong>If the whole Subject class is referred to, the Statement is called Universal; if less than the whole is referred to, it is called Particular.</strong> <LI> <strong>Copula (= linking term) Determines inclusion or exclusion. When the relation is inclusion, the Proposition is called Affirmative. When the relation is exclusion the Proposition is called Negative.</strong><br><br> The copula is always some form of the verb "to be".(e.g. "is", "are", "was", "were", "will be")<br><br> example: <OL type = DISC> <LI> Categorical Proposition = “All humans are mortal.” <OL type = DISC> <LI> Quantifier = “All” <LI> Subject = “humans” <LI> Copula = “are” <LI> Predicate = “mortal” </OL></OL><br> The thing to be wary of in practice is that the Statement being analyzed will often not be in the “standard form” shown, and/or will contain other verbiage which may make it difficult to precisely determine its meaning. There is no simple solution to this problem that will always suffice, so all of the REAP technique, the analysis of seller’s gimmicks, your own experience and good judgment, plus Symbolic Logic must be employed in solving it. <br><br> <strong>The Four Types (Forms) of Categorical Proposition:</strong> <OL type = DISC> <LI> <strong>A = Universal Affirmative => All S are P</strong> (e.g. All men are mortal.) <LI> <strong>E = Universal Negative => No S are P</strong> (e.g. No pigs can fly.) <LI> <strong>I = Particular Affirmative => Some S are P</strong> (e.g. Some students are female.) <LI> <strong>O = Particular Negative => Some S are not P</strong> (e.g. Some large people aren’t fat.) </OL><br><br> <OL type = DISC> <LI> <strong>The Predicate of an Affirmative Proposition is regarded as having Particular Quantification. <LI> The Predicate of a Negative Proposition is regarded as having Universal Quantification.</strong><br><br> The following considerations are useful in translating “ordinary language” (= non-standard) Propositions into standard form: <OL type = DISC> <LI> When the grammatical Predicate of a sentence does not explicitly include a class or Concept but instead does so by implication, the sentence may be rewritten to create one. (E.g. "Deers run fast." does not actually have a Predicate Term. But this can be translated into "Deers are fast runners.", in which the Predicate Term is clearly seen to be “the class of fast runners”). <LI> Although the Copula is always some form of the verb "to be", its tense may be ignored because it’s not important to the logic of the Proposition. <LI> Although the Subject Term usually occurs first in a Proposition, even if the positions of the Subject and Predicate are switched the Proposition’s logical structure is unaffected.. (e.g. The poetic "Soft is the wind." Here it should be apparent that the Subject of the Proposition is really “the wind” and not “soft” so "The wind is soft" would be closer to standard form.) <LI> When the Subject Term is explicitly singular, as in the case of proper names and definite descriptions, it should be treated it as a class of one. But that means the whole class is being referred to, so Singular Propositions are then treated as Universals. <LI> Nonstandard Quantifiers or no Quantifiers at all are provided. <OL type = DISC> <LI> When no Quantifier is given, one must judge from the context (as best one can) whether a Universal or Particular Proposition is being claimed. For example, "Lions are carnivores." would be a Universal Proposition about all lions, while "Lions are circus animals." would be a Particular Proposition making a claim about some lions. <LI> Any Proposition of the form "All S are not P" (where S = Subject Term, and P = Predicate Term) is ambiguous. It may reasonably be translated into either an E form or an O form Proposition with no certain way to decide between them, so once again judgement is required. <LI> Because "some" = "at least one" and Logic cares only about whether the claim is about (the whole class) or (less than the whole class), words such as "few", "several", "many" and "most" must all be translated as “some” (i.e. Particular rather than Universal). </OL></OL></OL><br><br> <strong>Distribution<br><br> A term is called Distributed if its Proposition makes a claim about each and every member of that category. <OL type = DISC> <LI> A Propositions => The Subject term is Distributed and the Predicate term is not. <LI> E Propositions => Both Subject and Predicate terms are Distributed. <LI> I Propositions => Neither Subject nor Predicated term is Distributed. <LI> O Propositions => The Subject term is not Distributed but the Predicate term is.</OL><br><br> Rules For Distribution<br><br> <OL type = DISC> <LI> The Subject of any Universal Proposition is Distributed, but the Subject of any Particular Proposition is not. <LI> The Predicate of any Negative Proposition is Distributed, but the Predicate of any Affirmative proposition is not.</strong></OL><br><br> <left><strong>Arguments</strong><br><br> Because most of the communications of interest are those intended to persuade us in some fashion or other, our main concern will be with what are technically termed “Arguments” (not to be confused with disputes) in Formal Symbolic Logic, so it will be to our advantage to briefly define and explain that crucial concept now, as well as to show how it differs from an Explanation. <br><br></OL></OL> <strong>ARGUMENT = A collection of related Statements (= Propositions) which purports to prove something:</strong><br><br> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>ARGUMENT = STATEMENT SUPPORTED + SUPPORTING EVIDENCE</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <center> <strong>Or</strong> </center> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>ARGUMENT = CONCLUSION + PREMISES</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center><br> <center> <strong>And</strong> </center> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <center><strong>Every ARGUMENT contends that:<br> The CONCLUSION is true <br> because the PREMISES are true.</strong></center> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center><br> <strong>How To Detect CONCLUSIONS</strong><br><br> <OL type = DISC> <LI> In most essays, any Conclusions will appear in either or both the introductory and concluding paragraphs. <LI> Certain indicator words signal Conclusions: “therefore, thus, hence, so, in conclusion, as a result, in short, the point is, Q.E.D.”. </OL><br><br> <strong>Real life Arguments almost always contain one or more of the following additional types of Statement:</strong><br><br> <OL type = DISC> <LI> <strong>Supporting Materials</strong><br><br> <OL type = DISC> <LI> Data <LI> Examples <LI> Illustrations <LI> Analogies <LI> Literature citations <LI> Comments</OL><br><br> <LI> <strong>Status Assessments</strong><br><br> <OL type = DISC> <LI> Convince that the Argument is important. <LI> Define the range of application of the Argument's Conclusions. <LI> Identify the source of and/or validate the "truth" of the Premises and/or Conclusion(s).</OL><br><br> <LI> <strong>Explanations</strong><br><br> <OL type = DISC> <LI> Key concepts or terms <LI> Data <LI> Context</OL><br><br> <LI><strong>Summaries/Restatements/References to Related Arguments:</strong><br><br> <OL type = DISC> <LI> The stated position. <LI> The opposition view.</OL><br><br> <LI><strong>Unsupported and/or Irrelevant Statements</strong> (= Claims which either do not fit the argument, are controversial and/or undefended, or are actually unrelated to the subject at issue.) These are most often included with the deliberate intent to distract, obfuscate, or confuse, but sometimes may just reflect the sloppy thinking processes of the Argument’s creator.<br><br> <LI> <strong>Summaries/ Restatements of the Premise(s) or Conclusion(s).</strong><br><br> These Supporting Statements may or may not be illuminating or confusing, convincing (i.e. provide Inductive support) or not, but none of them can in any way influence the Deductive logic underlying our analysis.<br><br> <strong>Two Key Definitions</strong><br><br> <strong>EMPIRICAL PREMISE</strong> = A Premise that claims that some state of affairs either exists or doesn’t exist.<br><br> <strong>CONCEPTUAL PREMISE</strong> = A Premise whose truth depends upon the meaning of certain key words in its statement.<br><br> <strong>The consequence of these definitions is that:</strong> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Your First Step In Evaluating An Argument Must Be To Examine Its Premises <br> To Determine Whether They Are Empirical Or Conceptual.</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <strong>And because</strong> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong><center>Every ARGUMENT contends that<br> The CONCLUSION is (presumed) true because the PREMISES are true.</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <strong>This means that</strong><br><br> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Determining the truth or falsity of its Premises<br> is the essential step at the heart of the process<br> of determining the validity and soundness of Arguments.</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <strong>Methods For Deteremining The Truth Of Premises</strong><br><br> <OL type = DISC> <LI> Consistency with established fact. <LI> Reference to established authority. <LI> Personal experience. <LI> Internal consistency.</OL><br><br> The first three of these methods are all concerned with fact on some level, and are therefore applicable to Empirical Premises. The fourth is concerned with the structure of the Statement itself, and therefore applies only to Conceptual Premises.<br><br> The caution to be observed with the second method is that the qualifications and credibility of the “authority” cited must be above question, else the “proof” of the statement’s truth falls apart. The well known example of competing “experts” in jury trials demonstrates the danger here.<br><br> The caution to be observed with the third method is that of our own limitations as observers, which may easily lead us to reach erroneous conclusions about the content and meaning of our experiences. The well known unreliability of eyewitness testimony demonstrates the danger here.<br><br> When the Premises and Conclusion of a presentation are clearly spelled out, understanding and evaluating the author’s position is a fairly routine (if often somewhat complex) logical exercise, as we shall soon see. Unfortunately the real world is anything but neat and clean, so the Arguments we encounter in the press, on TV, and especially in political discourse are not always clearly spelled out, and in all too many cases are even deliberately obscured to the extent that they must actually be reconstructed before their Premises and Conclusions become apparent. So in order to be able to correctly identify those key elements in the blizzard of obfuscation the author may have thrown up, it is frequently necessary is that we seek indicators of their presence:<br><br> <strong>Premise indicators are phrases like:<br><br> <OL type = DISC> <LI> After all, ... <LI> Given that... <LI> Inasmuch as... <LI> Because... <LI> Since... <LI> In view of the fact that... <LI> Whereas... </OL><br><br> Conclusion indicators are phrases like:<br><br> <OL type = DISC> <LI> So... <LI> Thus... <LI> Therefore... <LI> Accordingly... <LI> It follows that... <LI> Hence... <LI> Consequently... <LI> In short,... <LI> We can conclude that...</strong></OL><br><br> Things become even more difficult when there are no apparent indicators at all! Then the best strategy is to try to find a statement which can serve as the Conclusion of the presentation's Argument - assuming that one exists. If it doesn’t, then there is no choice but to try to construct one of your own, and this can be a tricky business because it will require your (obviously subjective) interpretation of the author’s intent. And if you get that interpretation wrong, of course, everything else that follows therefrom will necessarily be largely useless.<br><br> <strong>Types of Argument:<br><br> <OL type = DISC> <LI> Deductive<br><br> An argument is Deductive if its major premise is based on a rule, law, principle, or generalization.</strong><br><br> In this form of Argument, if the Premises are true then the Conclusion is necessarily also true! So <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Deductive Arguments are always either completely Valid or Invalid, <br>with no middle ground.</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Deduction Begins With The General (The Rule) <br> And Ends With The Specific.</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <LI> <strong>Inductive<br><br> An Argument is Inductive if its major premise is based on observation or experience.</strong><br><br> Inductive arguments are all comparisons between two sets of events, ideas, or things. In this form of Argument, even if the Premises are true the best that can be said about the Conclusion is that it is likely to be true. So <strong>Inductive Arguments are classified as Strong or Weak, depending on how we assess the probability that their conclusions are true.</strong><br><br> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Induction Proceeds From The Specific (The Observation)<br> To The General. </strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> Despite this, the difference between Inductive and Deductive reasoning is primarily in the way the arguments are expressed, and any Inductive Argument can alternatively be expressed Deductively, and conversely. Which of these approaches is best in any given situation is a matter of judgment, influenced in no small part by the degree of confidence desired (and attainable) in the Argument’s Conclusion(s).<br><br> The result is that the Conclusion of a valid Deduction never contains more information than was contained the Premises; while the conclusion of even the strongest Induction always does. And that’s why <br><br> </OL> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Deductions Are Always Certain!</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Inductions Are Always Uncertain,<br> to Greater Or Lesser Degree!</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <strong>Argument Reliability Assessment</strong><br><br> <OL type = DISC> <LI> <strong>Valid Argument</strong><br><br> A Deductive Argument in which if the Premises are true then the Conclusion must also be true! (Whether or not the Premises are actually true is another matter entirely!)<br><br> <LI> <strong>Invalid Argument</strong><br><br> A Deductive Argument in which even if the Premises are true the Conclusion may still be false. An Invalid Argument is always also Unsound.<br><br> <LI> <strong>Strong Argument</strong><br><br> An Inductive Argument in which the Premises provide probable support for the Conclusion.<br><br> <LI> <strong>Weak Argument</strong><br><br> An Inductive Argument in which even if the Premises are true they don’t provide much support for the Conclusion. Weak Arguments are always Uncogent.<br><br> <LI> <strong>Sound Argument</strong><br><br> A Valid Deductive Argument in which all of the Premises are true.<br><br> <LI> <strong>Unsound Argument</strong><br><br> A Valid Deductive Argument in which either the Conclusion doesn’t follow from the Premises or some of the Premises are untrue.<br><br> <LI> <strong>Cogent Argument</strong><br><br> A Strong Inductive Argument with true Premises.<br><br> <LI> <strong>Uncogent Argument</strong><br><br> A Strong Inductive Argument some of whose Premises are untrue.<br><br> </OL> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Statements Can Be True Or False, <br> But They Can’t Be Valid Or Invalid <br> In The Sense These Terms Are Used In Logic.</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <center> <br><table class="txtbox"> <tr> <td width="30"> </td> <td align="middle" valign="top"> <br> <font class="txtboxsmall"> <strong>Arguments Can Be Valid Or Invalid, <br> But They Can’t Be True Or False.</strong> </font> <br><br> </td> <td width="30"> </td> </tr> </table><br> </center> <br> <strong>Evaluating Arguments</strong><br><br> The use of techniques like Truth Tables and Venn Diagrams can expose logical flaws in Arguments, and are therefore extremely valuable in distinguishing fact from fiction.<br><br> <strong>Truth Tables<br><br> For each type of Compound Statement, a 'truth table” can be constructed which spells out: <OL type = DISC> <LI> All of the possible combinations of truth and falsity for the included Simple Statements, plus The resulting truth or falsity of the Compound Statement of which they are a part:</OL></strong><br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="30%" colspan="3" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">And</font></strong></td> </tr> <tr> <td width="33%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong><font size="4">Statement A</strong></td> <td width="20%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> <font size="4">Statement B</strong></td> <td width="34%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> <font size="4">A & B</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong>T </strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>F</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> T</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>F</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F</strrong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>F</strong></td> </tr> </table> </div> <br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="30%" colspan="3" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">Or</font></strong></td> </tr> <tr> <td width="33%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong>Statement A</strong></td> <td width="20%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> Statement B</strong></td> <td width="34%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> A Ú B</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> T</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> T</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>F</strong></td> </tr> </table> </div> <br><br> <div align="center"> <center> <table border="6" width="20%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="30%" colspan="2" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">Not</font></strong></td> </tr> <tr> <td width="33%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong><font size="4">A</strong></td> <td width="33%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> <font size="4">~ A</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>T</strong></td> <td width="33%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F</strong></td> <td width="33%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> T</font></strong></td> </tr> </table> </div> <br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="20%" colspan="3" align="center" bordercolor="#FFFFFF" height="30"> <strong><font size="5">If - Then</font></strong></td> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> T</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>F</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F<strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> T</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F<strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> </table> </div> <br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="30%" colspan="3" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">If And Only If (= Iff)</font></strong></td> </tr> <tr> <td width="33%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong><font size="4">Statement A</strong></td> <td width="20%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> <font size="4">Statement B</strong></td> <td width="34%" align="center" bordercolor="#000000" height="20" BORDER="1"><strong> <font size="4">A iff B</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong>T </strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>T</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>F</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> T</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>F</strrong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> F</font></strong></td> <td bordercolor="#000000" align="center" height="31" BORDER="1"><strong>T</strong></td> </tr> </table> </div> <br><br> These elementary truth tables can be extended to determine the truth value of far more complex compound statements containing many logical operators, such as are routinely encountered in the real world. This is a complicated process which is the subject of entire courses and is therefore infeasible for us to go into further here, but it is well worth your while to explore on your own.<nr><br> To apply these truth tables, all that is necessary is to replace the symbols (A and B) by their respective statements, and the answer regarding the truth or falsity of the consequent compound statement will then be instantly known.<br><br> Example:<br><br> A = The weather is hot.<br> B = It’s September<br><br> If A = T, B = F: A and B = F, A or B = T, If A then B = F, A iff B = F<br> if A = F, B = T: A and B = F, A or B = T, If A then B = T, A iff B = F<br><br> That is, to cite just one of these results for A and B: If the weather is hot but it’s not September, then the compound statement “The weather is hot if and only if it is September” is (obviously) false.<br><br> For simple compound statements with only a single logical operator like those in these tables, such a formal process hardly seems necessary. But for complex compound statements with many logical operators functioning in combination, correctly finding one’s way through the logical maze can be daunting, and in such situations the discipline provided by the table may become the only feasible way to assure the obtaining correct answer regarding the compound statement’s truth or falsity.<br><br> <strong>Inductive Inferences</strong><br><br> Because we have time here for only a brief survey of Symbolic Logic, our emphasis will be on Deductive Reasoning, but the subject of (Inductive) Reasoning By Analogy is so important and pervasive that it’s essential it at least receive mention.<br><br> This process of inductive inference is at the center of both “brand loyalty” in the commercial sphere and many of our key relationships in the personal sphere. Because we have had good past experiences with products of brand “X”, we are induced to expect similar good future experience with new products from the same manufacturer, and this forms the basis for our product selection/brand loyalty. And it also forms the basis for our dismay if and when said new product turns out to be a “lemon” that fails to live up to our expectations. In the personal sphere, because person “X” is a leader of our church or an executive of our company, we confidently expect that they will exhibit exemplary behavior in all related activities. And this accounts for the vast dismay we feel if the Church funds have been embezzled, or as happened in the Enron scandal, when the company was bankrupted and the employees and stockholders were defrauded, while the executives walked off with untold $ millions!<br><br> ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------<br> <strong>IMPORTANT NOTE!</strong><br><br> The material which follows is somewhat more complex, detailed and difficult than the preceding, so it will probably be difficult for any normal person to absorb it completely from the very limited treatment provided here. For that reason all that is expected is that you get a general understanding of what it’s all about and why/how it works. <br><br> But because this material describes a long established and very powerful technique for making sense of and assessing the validity of many of the Arguments with which we are all inundated daily, it is important that you at least be aware of its existence so that you can learn more about it on your own should you be so inclined. And, as earlier noted, any investment of time and energy required to master this material will certainly ulttimately payoff to your advantage.<br><br> -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------<br><br> <strong>Deductive Inferences<br><br> A Deductive Inference may be of two kinds:<br><br> <OL type = DISC> <LI> Immediate Inference<br> An Argument consisting of exactly 2 Categorical Propositions: <br> 1 Premise and 1 Conclusion <LI> Mediated Inference (= A Categorical Syllogism)<br> An Argument consisting of: <OL type = DISC> <LI> exactly 3 Categorical Propositions (2 Premises and 1 Conclusion) <LI> containing 3 categorical terms, <LI> each of which is used twice.</strong></OL></OL><br><br> The classical example:<br> All men are mortal.<br> Socrates is a man.<br> Therefore Socrates is mortal.<br><br> It is important to recognize that an Argument containing more than 3 terms might still qualify as a Categorical Syllogism if it can be translated into an equivalent Argument containing the required exactly 3 terms. The danger here, of course, is that the “translation” must be valid, and not do substantive damage to the Argument being presented.<br><br> A second key danger to be avoided is that each of the 3 terms in the Syllogism must be used in exactly the same sense throughout. (e.g. If the term “men” is used to mean “human beings” in one Statement but “male humans” in another, the Argument would not qualify as a Syllogism.)<br><br> <OL type = DISC> <LI> <strong> The Major Premise of the Syllogism is a Categorical Proposition that contains the Predicate of the Conclusion and the Middle Term. <LI> The Minor Premise of the Syllogism contains the Subject of the Conclusion and the Middle Term. <LI> The Major Term of the Syllogism is whatever is employed as the Predicate Term of its Conclusion. <LI> The Minor Term of the Syllogism is whatever is employed as the Subject Term of the Conclusion. <LI> The Middle Term of the Syllogism doesn't occur in the Conclusion at all, but must be employed in somewhere in each of its Premises. </OL></strong><br><br> <strong>Syllogism Rules:<br><br> <OL type = DISC> <LI> There are only three terms in a Syllogism (by definition). <LI> The Middle Term is not in the Conclusion (by definition). <LI> The Quantity of a term cannot become greater in the Conclusion. <LI> The Middle Term must be Universally Quantified in at least one Premise. <LI> At least one Premise must be Affirmative. <LI> If one Premise is Negative, the Conclusion is Negative. <LI> If both Premises are Affirmative, the Conclusion is Affirmative. <LI> At least one Premise must be Universal. <LI> If one Premise is Particular, the Conclusion is Particular. <LI> If both Premises are Universal, the Conclusion is Universal.</strong></OL><br><br> <strong>Standard Form<br><br> A Categorical Syllogism in Standard Form always has a:<br><br> <OL type = DISC> <LI> Major Premise <LI> Minor Premise <LI> Conclusion<br> in that order.</strong><br><br> </OL> Although arguments in ordinary language are frequently stated in a different arrangement, it is always possible to restate them in Standard Form for analysis. The simple procedure is:<br><br> <OL type = DISC> <LI> Identify the conclusion, and place it in the final position <LI> whichever premise contains the Conclusion’s Predicate term must be the Major Premise that should be stated first. <LI> the remaining Premise is the Minor Premise, and appears in the middle.</OL><br><br> <strong>The Standard Form of a Syllogism is described in terms of 2 factors: Mood, and Figure.</strong><br><br> <OL type = DISC> <LI> <strong>Mood = A list of which categorical propositions (A, E, I, or O) it comprises, in the order in which they appear.</strong><br><br> <OL type = DISC> <LI> <strong>A = Universal Affirmative => All S are P</strong> (e.g. All men are mortal.) <LI> <strong>E = Universal Negative => No S are P</strong> (e.g. No pigs can fly.) <LI> <strong>I = Particular Affirmative => Some S are P</strong> (e.g. Some students are female.) <LI> <strong>O = Particular Negative => Some S are not P</strong> (e.g. Some large people aren’t fat.)</OL><br><br> Example: A Syllogism with a Mood of OAO has an O Proposition as its Major Premise, an A Proposition as its Minor Premise, and another O Proposition as its Conclusion.<br><br> <strong>There are 4 kinds of Categorical Proposition but a Syllogism uses only 3 of them at a time, so the total number of possible Moods (arrangements) = 64.<br><br> S: Subject of the Conclusion.<br> P: Predicate of the Conclusion.<br> M: Middle Term. of the Conclusion</strong><br><br> <strong>Each Syllogistic Mood can appear in four distinct versions called Figures.</strong> <br><br> <LI><strong>Figure = The position in which the Middle Term appears in the two Premises.</strong><br><br> <OL type = DISC> <LI> First-Figure Syllogism = The Middle Term is the Subject Term of the Major Premise and the Predicate Term of the Minor Premise <LI> Second-Figure Syllogism = The Middle Term is the Predicate Term of both Premises <LI> Third-Figure Syllogism = The Middle Term is the Subject Term of both Premises <LI> Fourth-Figure Syllogism = The Middle Term appears as the Predicate Term of the Major Premise and the Subject Term of the Minor Premise. </strong></OL></OL><br><br> These four Figures may be depicted as follows:<br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="20%" colspan="5" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">The Four Syllogism Figures</font></strong></td> </tr> <tr> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong><font size="4"></strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong><font size="4">1st</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">2nd</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">3rd</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">4th</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>Major</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>MP</strong></td> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>PM </strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>MP</strong></td> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>PM </strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>Minor</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>SM</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>SM</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>MS</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>MS</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>Conclusion</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>SP</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>SP</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>SP</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>SP</strong></td> </tr> </table> </div> <br><br> Since each of the 64 possible Moods can have 4 Figures, the total number of possible Categorical Syllogisms is 64 x 4 = 256.<br><br> The power of a Syllogism is that after it has been put into Standard Form and checked for informal fallacies, its validity or invalidity may be determined directly via mere inspection of the form!!<br><br> </strong> <br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="20%" colspan="4" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">Unconditionally Valid Syllogisms</font></strong></td> </tr> <tr> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong><font size="4">AAA</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong><font size="4">EAE</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">IAI</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">AEE</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>EAE</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AEE</strong></td> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>AII </strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>IAI</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AII</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>EIO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>OAO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>EIO</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>EIO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AOO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>EIO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>-</strong></td> </tr> </table> </div> <br><br> <div align="center"> <center> <table border="6" width="60%" cellspacing="0" cellpadding="11" bordercolordark="#000000" bordercolorlight="#C0C0C0" bordercolor="#000000" height="373"> <tr> <td align="center" width="20%" colspan="5" align="center" bordercolor="#FFFFFF" height="30"> <strong> <font size="5">Conditionally Valid Syllogisms</font></strong></td> </tr> <tr> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong><font size="4">Figure 1</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong><font size="4">Figure 2</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">Figure 3</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">Figure 4</strong></td> <td width="20%" align="center" bordercolor="#000000" height="31" BORDER="1"><strong> <font size="4">Required<br>Conditions</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AAI<br>EAO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AEO<br>EAO</strong></td> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>-</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AEO</strong></td> <td align="center" bordercolor="#000000" height="31" BORDER="1"><strong>S Exists</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>-</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>-</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AII<br>EAO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>EAO</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>M Exists</strong></td> </tr> <tr> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>-</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>-</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>-</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>AAI</strong></td> <td align="center" bordercolor="#000000" BORDER="1" height="31"><strong>P Exists</strong></td> </tr> </table> </div> <br><br> What do these tables tell us??<br><br> In the first table, these syllogisms are Valid whether or not their terms denote actually existing things! In the second table, the syllogisms listed are valid only if the designated term denotes an actually existing thing. So all that we have to do is to put an unknown Syllogism into Standard Form, determine its mood and figure, and see if it appears in our tables to instantly know whether or not it is valid! Surely a quick path to the truth. <br><br> Arguments in the real world frequently incorporate what can (often only with considerable effort) really be parsed into multiple Syllogisms, and there are methods that have been developed for handling such situations and then converting those arguments into standard Syllogistic Form so that these techniques can be applied. But all of that is beyond our present intent of introducing these logical concepts as a preferred way to determine the validity of the blizzard of verbiage with which we are all confronted every day. As noted earlier, you will be well advised to explore this important topic further on your own.<br><br> Finally, before we leave this subject it is useful to recognize that life is not fair, so that there are a host of ways, either deliberately or through inadvertence, in which what appear to be solid logical arguments can go wrong, but only one way to get them right. This situation was cleverly characterized many years ago by Chessmaster Dr. Savielly G. Tartakower when he wryly observed that “All the little errors are there, waiting to be made”. Although he was talking about Chess, his astute observation applies equally well to Logical Fallacies, which have been neatly catalogued as follows:<br><br> <strong>Guide To The Logical Fallacies © 1995-2000 Dr. Stephen Downes, Research Officer, National Research Council Canada</strong> (Included here with permission.)<br><br> <strong>Fallacies of Distraction</strong><br><br> * False Dilemma: two choices are given when in fact there are three options<br> * From Ignorance: because something is not known to be true, it is assumed to be false<br> * Slippery Slope: a series of increasingly unacceptable consequences is drawn<br> * Complex Question: two unrelated points are conjoined as a single proposition<br><br> <strong>Appeals to Motives in Place of Support</strong><br><br> * Appeal to Force: the reader is persuaded to agree by force<br> * Appeal to Pity: the reader is persuaded to agree by sympathy<br> * Consequences: the reader is warned of unacceptable consequences<br> * Prejudicial Language: value or moral goodness is attached to believing the author<br> * Popularity: a proposition is argued to be true because it is widely held to be true<br><br> <strong>Changing the Subject</strong><br><br> <strong>* Attacking the Person:</strong><br> * (1) the person's character is attacked<br> * (2) the person's circumstances are noted<br> * (3) the person does not practice what is preached<br><br> <strong>* Appeal to Authority:</strong><br> * (1) the authority is not an expert in the field<br> * (2) experts in the field disagree<br> * (3) the authority was joking, drunk, or in some other way not being serious<br> * Anonymous Authority: the authority in question is not named<br> * Style Over Substance: the manner in which an argument (or arguer) is presented is felt to affect the truth of the conclusion<br><br> <strong>Inductive Fallacies</strong><br><br> * Hasty Generalization: the sample is too small to support an inductive generalization about a population<br> * Unrepresentative Sample: the sample is unrepresentative of the population as a whole<br> * False Analogy: the two objects or events being compared are relevantly dissimilar<br> * Slothful Induction: the conclusion of a strong inductive argument is denied despite the evidence to the contrary<br> * Fallacy of Exclusion: evidence which would change the outcome of an inductive argument is excluded from consideration<br><br> <strong>Fallacies Involving Statistical Syllogisms</strong><br><br> * Accident: a generalization is applied when circumstances suggest that there should be an exception<br> * Converse Accident : an exception is applied in circumstances where a generalization should apply<br><br> <strong>Causal Fallacies</strong><br><br> * Post Hoc: because one thing follows another, it is held to cause the other<br> * Joint effect: one thing is held to cause another when in fact they are both the joint effects of an underlying cause<br> * Insignificant: one thing is held to cause another, and it does, but it is insignificant compared to other causes of the effect<br> * Wrong Direction: the direction between cause and effect is reversed<br> * Complex Cause: the cause identified is only a part of the entire cause of the effect<br><br> <strong>Missing the Point</strong><br><br> * Begging the Question: the truth of the conclusion is assumed by the premises<br> * Irrelevant Conclusion: an argument in defense of one conclusion instead proves a different conclusion<br> * Straw Man: the author attacks an argument different from (and weaker than) the opposition's best argument<br><br> <strong>Fallacies of Ambiguity</strong><br><br> * Equivocation: the same term is used with two different meanings<br> * Amphiboly: the structure of a sentence allows two different interpretations<br> * Accent: the emphasis on a word or phrase suggests a meaning contrary to what the sentence actually says<br><br> <strong>Category Errors</strong><br><br> * Composition : because the attributes of the parts of a whole have a certain property, it is argued that the whole has that property<br> * Division: because the whole has a certain property, it is argued that the parts have that property<br><br> <strong>Non Sequitur</strong><br><br> * Affirming the Consequent: any argument of the form: If A then B, B, therefore A<br> * Denying the Antecedent: any argument of the form: If A then B, Not A, thus Not B<br> * Inconsistency: asserting that contrary or contradictory statements are both true<br><br> <strong>Syllogistic Errors</strong><br><br> * Fallacy of Four Terms: a syllogism has four terms<br> * Undistributed Middle: two separate categories are said to be connected because they share a common property<br> * Illicit Major: the predicate of the conclusion talks about all of something, but the premises only mention some cases of the term in the predicate<br> * Illicit Minor: the subject of the conclusion talks about all of something, but the premises only mention some cases of the term in the subject<br> * Fallacy of Exclusive Premises: a syllogism has two negative premises<br> * Fallacy of Drawing an Affirmative Conclusion From a Negative Premise: as the name implies<br> * Existential Fallacy: a particular conclusion is drawn from universal premises<br><br> <strong>Fallacies of Explanation</strong><br><br> * Subverted Support :The phenomenon being explained doesn't exist<br> * Non-support:: Evidence for the phenomenon being explained is biased<br> * Untestability: The theory which explains cannot be tested<br> * Limited Scope: The theory which explains can only explain one thing<br> * Limited Depth: The theory which explains does not appeal to underlying causes<br><br> <strong>Fallacies of Definition</strong><br><br> * Too Broad : The definition includes items which should not be included<br> * Too Narrow: The definition does not include all the items which should be included<br> * Failure to Elucidate: The definition is more difficult to understand than the word or concept being defined<br> * Circular Definition: The definition includes the term being defined as a part of the definition<br> * Conflicting Conditions: The definition is self-contradictory<br><br> Please note the close relationship between this list of Logical Fallacies and both the “Seller’s Gimmicks” and the Impediments to Logical Thought presented earlier, all of which are really only slightly different ways of expressing the same key idea. In the next chapter we shall begin our exposition of specific techniques which can serve to improve your Decision Making/Problem Solving.<br><br> <strong>Click Here To Go To</strong><a href="RDMCh3.html"><font size=+1><font Color="#0033FF"><strong> Chapter 3</strong></font></a> <br><br> <strong>Click Here To Return To</strong><a href="RDMCh2.html"><font size=+1><font Color="#0033FF"><strong> Chapter 2 </strong></font></a> <br><br> <strong>Click Here To Return To</strong><a href="RDMTOC.html"><font size=+1><font Color="#0033FF"><strong> Table Of Contents </strong></font></a> <br><br> <strong>Click Here To Return To</strong><a href="index.html"><font size=+1><font Color="#0033FF"><strong> Milt's Go Page</strong></font></a> <br> <br><br> <strong>Click Here To Email Your Comments/Suggestions To</strong><font size=+2><font color="#0033FF"><a href="mailto:bradleym@eniinternet.com?subject=Reasoning And Decision Making Comments/Suggestions"> Milton N. 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