20(3)pp55-86TeachingSupplement3.pdf TS 74 Testing the Validity ofDisjuctive Arguments Using Physical Models RON LEONARD University of Nevada, Department of Phi losophy, 4505 Maryland Parkway, Box 455028, Las Vegas, Nevada, 89154-5028 U.S.A E-mail: pleroma@mindspring.com Introduction. Methods for testing the validity of arguments often require formal techniques Ihat cannot be app lied without knowledge of symbolic logic. Simple physica l models, such as Venn diagrams, that visua ll y represent logical relationships can help stu- dents to test validity whi le they are still in the process of learning the formal tools. However. Venn Diagrams are rest ricted to categorical arguments.Th is motivated me to develop a similar method for arguments employing propos itional logic. In the previous issue of this journal, I introduced a method for using the physi- cal model of a thermometer to test the validity of conditional arguments.The pro- posal advanced here uses diagrams of a sc ratch-and -w in ticket for arguments based on disjunctions ("or" statements). Its guiding principle is the Counterexample Method, according to which an argument is invalid when it is poss ible for all of its premises to be true and its conclusion false. Analogously, if it is possible to dia- gram all and only the information expressed in the premises while exc luding the information expressed in the conc lusion, the diagram visually shows that the argu- ment is invalid. Scratch-and -win tickets are sma ll rectangu lar cards conta ining a numbe r of circles coated with paint. Under each may be written notice of a pri ze, which may be revealed when the paint is scratched off. Fo r the purpose of our model we must assume that there is a prize somewhere on each ticket. According to the rules, a specific number of circles may contain a prize, but the number of painted ci rcles that the player may scratch is specified. This model provides a relatively straight- forward method for testing the validity of disjunctive arguments. Before formulating the genera l method, it will be helpful to apply it to a few examples, including diagrams of each argument. The first thing we must deter- mine is whether the "or" of a disjunction is exclusive or inclusive. A disjunction is exclusive whenever the truth of one disjunct excludes the truth of the other disjunct(s) (such as getting heads when flipping a coin makes it impossible that it is also tails). In such cases a scratch-and-win diagram that tests the validity of a disjunctive argument constructed from an exclusive "or" has only one prize . A © in/ormal Logic, 20.3 (2000) Teaching Supplement #3: pp. TS 74-TS 83. Testing the Validity of Disjunctive Arguments TS 75 disjunction is inclusive when more than one disjunct may be true at the same time (such as tomorrow being windy or rainy). In such cases a scratch-and-win dia- gram may have more than one prize. When an argument denies disjuncts, rather than affirming them, the distinction between the inclusive and exclusive "o r" is irrelevant. In contrast, as I will illustrate later, the distinction is relevant when an argument affirms the disj unct. Examples Example 1: Denying a disjunct of an inclusive "or· J statement: Either natural disaster or misuse of human technology will calise massive loss of life on Earth during the next millennium. No natural catastrophe will occur during the next millennium. Therefore, misuse of human technology will cause massive loss of life on Earth during next millennium. Figure 1 Massive loss of life Massive loss of life on earth in 3M due on earth in 3M due to natural disaster to human technology ~2 VALID The rectangle represents a scratch-and -w in ticket; its two circles represent the two possibilities given in the disjunctive prem ise, labeled accordingly. It is under- stood that the circles in the diagram are originally painted over. Because neither possibility that is stated in the disjunctive prem ise excludes the other, they are inclusive. Consequently, the ticket is labeled to show that it contains at least one pnze. TS 76 Ron Leollal'd A premi se that either affirm s or denies an alte rnati ve expressed in a disj unct is re presented by sc ratching the pa int off the corresponding circle. The seco nd premi se in Examp le I denies that the first di sj unct is true , which means that there is no prize there . This is shown by the pain t scratched off to revea l the word "NOT " in that circle. Because the ticket must contain a pri ze so mewhere, it is obvious that it must lie under the other circle. ( In sofar as no premise affirms or denies that possibil ity, the circle remains painted in the diagram.) According to the diag ram , all and only the in formation in the premises contains (ex presses) the information in the conclusion, so the argument is val id. Example 2: Denying two disjuncts of an inclusive "o r " statement; The government will fund Medicare by borrowing morc money or signifi- cantly increasing premium s or drastical ly reducing benefits. The government wi ll not signific antly increase premiums (because sen- iors wi ll neve r accept hi gher prem iums). The governme nt w ill not drast ica ll y reduce benefits (because sen iors w ill never accept reduced benefits). Therefore. the government wi ll fund M edicare by borrowing more money. U. S. borrows more money to fund Medicare Figure 2 U. S. raises Medicare premiums VALID U. S. cuts Medicare benefits drastically Testing the Validity of Dijjunclive Arguments TS 77 The above example illustrates how the principle used in the previous case may apply to more than two possibilities (disjuncts), and that their order on the ticket is logically irrelevant. In addition , it shows how practical arguments in ordinary lan- guage often rely on common assumptions concerning attitudes and values. In this case, assuming that the government will conform to the wishes of seniors, the two premises that imply the denial of two of the alternatives reveal no prize under the two corresponding circles in the diagram. So, the prize must be located under the remaining circle, which corresponds to what the conclusion asserts . This argument too is valid because the diagram that represents all and only the informa- tion in the premises cannot be constructed without also representing the informa- tion in the conclusion. Er:ample 3: Denying a disjunct of an exclusive "or" statement: In roulette, the result of each play is red, black or (rarely) green. My spin was not red. Therefore, my spin was black. Figure 3 INVALI D This example differs from Example 2 in that the premises have only eliminated one of tile three possible locations for the prize, and that these possibilities are exclu- sive because only one of them can be true. The prize might be under either of the two remaining circles. Since it is still possible (however unlikely) that it might be under the 'jgreen" circle. the conclusion could be false. The diagram, constructed from all and only the information in the premises does not contain the information in the conclusion, so the argument is invalid. TS 78 Ron Leonard Example 4: Denying a disjunct of an inclusive "or " s tatement, with Q disjunctive conclusion : Cr iminals sentenced to death sho uld be exec uted by poison gas, e lectro- cuti on, or lethal injection. Crim inals shou ld not be electrocuted. Therefore , criminals should be ex- ecuted either by poison gas or lethal injection. Figur e 4 Poison gas Lethal injection VALI D This differs from the other examples in that there is more than one possibility that would make the conclusion true. Here the "or" is inclusive because there may be more than one method of executio n that shou ld be used, even though it woul d be bizarre to use more than one of them for a particular crimi nal. The possibility that the "electrocution" circ le contains a prize has been denied, and the d iagram illus- trates that there must be a prize under at least one of the two rema ining, unscratched ci rcl es. Since this is what the conclusion cla ims, the premises conta in the infor- mation stated in the conclusion . Consequent ly, the argument is valid . Example 5: Affirming a disjunct of an exclusive "or " statement: Ei ther the third millennium begins on January 1,200 I or the third millen- nium begins on January I, 2000. The third millennium begins on January I 200 I (because exactly 2000 years will have elapsed from 0) the beginning of the Common Era) until January I, 2001). Therefore, the new mi llennium did not begin on Janua ry 1, 2000 . Testing the Validity of Disjunctive Argumel1ts TS 79 Fi gur e 5 PRIZE millenium begins 1-1-2001 millenium begins 1-1-2000 VALID The difference from the previous example is that one of the disjuncts is affirmed, rather than denied. So, there is a prize located in the circle representing the af- firmed disjunct. Hence, the word "PRIZE" appears in that circle. Next, we must determine whether the argument uses an exclusive or inclusive "or," Here, if either disjunct is true, the other must be false, so the disjuncts are exclusive. Conse- quently, the ticket is labeled to show that there is one, and only one, prize on the ticket. Since it has been determined that the tirst circle contains the prize, the other circle cannot. Because the conclusion denies that the unscratched circle contains a prize , all and only the information in the premises also contains the information in the conclusion. Thus, the argument is valid . Example 6: Affirming a disjunct o/an inclusive "or" statement: Either we ban CFC's or the ozone layer will become dangerously de- pleted. We have banned CFC ' s. Therefore, the ozone layer will not become dangerously depleted. Example 6 differs from the previous one in that both possibilities could be true, for we could ban CFC's and the ozone layer could still become dangerously depleted (from the effect of some other chemical, for instance). Hence, we are dealing with an inclusive "or" statement, so we label the ticket to show that it contains at least one prize. The second premise asserts that we have banned CFC's, which means that there is a prize under the corresponding circle , so we show the word " PRIZE " revealed there. The conclusion claims that there is no prize under the TS 80 ROil Leonard PRIZE Fi gure 6 Ozone layer will be dangerously depleted INVALID other circle, but because the argument uses an inclusive "or," it is possible that another prize cou ld be found under the other circle. Hence, a ll and only the infor- mation in the premises does not contain the information in the conc lusio n: it is possible for the premises to be true and the conclusion false. Thus the argument is in val id. Example 7. Affirming a dlsjunct of an exclusive "or" s tatement: Prese ntl y, the largest country in the world is Russia or Ch ina or the U.S. or A ustra lia or Brazil or Canada, Canada has been sho wn to have the greatest land mass. Therefore, Russia is not the largest country in the wor ld. Example 7 (see Figure 7 on page TS 80) is a complex examp le where severa l possibilities are mutually exclusive, and where one of them is asserted in orde r to conclude th at one of the others is false. When a scratch -and-w in ticket must represent these many possibilities, it is more conven ient to use two rows of cir- cles, just as do many actual ti ckets. Again , asserting that Canada has the largest land mass is equivalent to finding the prize under the corresponding circ le. The word "PRlZE" thus appears there. Since these possibilities are exclusive, the ticket contains only one prize. The conclusion denies that Russia is the largest country. which m·eans that there is no prize under that circl e. Since a ll and only the informa- ti on in the premises inescapab ly contai ns the in formation asserted in the conclu- sion, the argument is valid . Testing {he Validity of Disju.nctive Arguments TS 81 Fi gure 7 PRIZE VALID Method of Diagramming Let us now consider the general rules for using the physical model of a scratch- and-win ticket for conducting validity tests. I. Draw a large rectangle (ticket) to include all, and only , the information in the premises of the argument. It is crucial that we never use information from the conclusion to construct the diagram. The test is to determine whether the infor- mation contained in the premises includes that contained in the conclusion. No doubt, if we have already included information from the conclusion, we will find it in the diagram , but that will make the test worthless. By analogy, when testing a water sample for lead, if we added lead to the sample before doing the test, the res ults would be useless. 2. Draw and label as many circles as there are possibilities (disjuncts) in the dis- junctive ( . . . or. .. ) premise. Each circle is understood to have either the word "PRIZE" (for a circle containing a prize) or the word "NOT" (for a circle that does not contain a prize) under the surface. TS 82 Ron Leonard 3. Determ ine whether, in reality, the poss ibili ti es given are inclusive or exclusive. Whenever Ihe possibililies are exclus ive, the trulh of one possibililyexcludes Ihe Irulh of lhe other possibililY, on ly o ne of them can be Iru e. This means Ihal only one of Ihe c ircles co nlains a prize . Whenever Ihey are inclusive, thaI is, where more than one possibility may be true at the same time, then any number o f circles may contain a pri ze. If exclusive, label Ihe licket "ONL Y ONE PRI ZE" al Ihe lOp; if inclusive, label Ihe licket "A T LEAST ONE PRIZ E" al Ihe top. 4. Premises that ei ther affirm or deny a given possibility are interpreted as analo- gous 10 sc ratching th e su rface of Ihe correspond ing c ircl e to reveal Ihe app ro- priate word. For each premise that affirms a disjunct, write the word "PRIZE " in the corresponding circle; for each premise that denies a disjunct, write th e word "NOT" in Ih at circle . Method of Determining Validity Given that the po ss ibili ti es are either exclusive or inclusive, determine whether any of the ci rcle s th aI have not been sc ratched cannot/may/musl co ntain a prize. J. For "O nl y One Prize" tickets (exc lusive "or" statements), whenever the word " PRlZE" has already appeared (because a disjuncI has been affinned), Ihe re- maining circles cannot conlain a prize. (The si ng le prize already has been won.) 2. For "A t Least One Prize" tickets (inclusive "or" state ments), whenever the word " PR1ZE" has already appeared (a disj uncI has bee n affirmed), any of Ihe remaining circles may also contain a prize. 3. For e ither soJ1 oflickel , wheneverthe word " PRIZE" has nOI ap peared (disj uncts have only been denied- nol affirmed): ( I) If a s ing le unscralched circle re- mains, it must con lain th e prize, but (2) if more Ihan one unscralched ci rc le remains, any of them may contain a prize (for exclus ive "o r," exac tly one circle wi ll). We tesl for validity by delermining whether the diagram (constructed by using all and only Ihe information from Ih e premises) corresponds 10 what Ihe conc lus ion claims (conta ins the information found in the conclusion). If so, Ihen the argu- ment is valid; oth erwi se, it is invalid. 4. If the conclusion affi rms that there is a prize within a si ngle c ircle, the argument is va lid whenever Ihe diagram in d icates thaI Ihere must be a prize there (I here must be a prize somewhere; it has not yet a ppeared; there is only one unscralched circl e where it could be-Examples 1,2). Olherwise it is invalid (Example 3). 5. I f the conclusio n affirms that there is a prize under one of two or more unscratched circles, the argument is val id whenever a prize must be under at least one o f Ihem (I here musl be a prize somewhe re; il has not yel appeared ; the concl usion allows for it to be under any of the remaining circles-Example 4). Olherwise it is invalid. 6. If the conclusion denies Ihal th ere is a prize under a single ci rcle, the argu ment is valid whenever th e circle cannot contain a prize (all of the prizes contained on Testing the Validity of Disjunctive Arguments TS 83 the ticket have already been won; the conclusion denies that there is a prize under a remaining unscratched circle-Examples 5,7). Otherwise it is invalid (Example 6). Conclusions The method for testing validity by using physical models has several advantages: I. It provides a reliable procedure for determining the validity of disjunctive argu- ments. 2. The diagrams are easy to draw. 3. The method's simplicity makes it easy to use. In particular, it applies to an disjunction, regardless of whether its disjuncts are positive or negative. 4. Students need not comprehend the subject matter of the argument. Lack of background knowledge sometimes limits the test for validity that uses counterexamples. Where subject matter is technical or obscure, students may have little idea what it would mean tor the premises or conclusion to be true or false. 5. Insofar as these physical models are analogous to logical forms , working with them will especially aid the visual learner to grasp the general logical forms underlying arguments having diverse content.