30 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
31 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
32 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.
35 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
38 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
39 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
40 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>
42 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,
43 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!
46 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
47 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>
51 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
65 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
66 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>
70 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
71 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
72 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>
74 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
75 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,
76 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
79 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
80 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>
102 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.
163 <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
164 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>
166 <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>
187 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
191 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>
209 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
210 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,
235 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
236 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”).
240 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
241 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.)
243 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,
249 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
252 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
255 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"
281 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”
282 (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
396 <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.)
397 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>
401 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
481 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
484 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
487 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.
490 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
491 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
492 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
493 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>
533 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
534 - 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)
535 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>
585 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.
586 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>
608 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
609 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)
612 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.
653 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>
662 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>
714 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>
947 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
948 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>
950 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>
960 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>
962 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,
963 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
968 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>
970 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
971 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”
972 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
973 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>
978 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
981 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
982 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>
1010 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,
1013 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,
1064 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>
1087 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>
1089 <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>
1106 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>
1176 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>
1307 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.
1308 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>
1310 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
1311 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
1312 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>
1314 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
1315 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
1398 * 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>
1399 * 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>
1420 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.
1433 <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. Bradley</font></a>
