# AIP Logic Chapter 4 Notes

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1 AIP Lgic Chapter 4 Ntes Sectin 4.1 Sectin 4.2 Sectin 4.3 Sectin 4.4 Sectin 4.5 Sectin 4.6 Sectin The Cmpnents f Categrical Prpsitins There are fur types f categrical prpsitins. Prpsitin Letter Name Prpsitin What is the assertin? A All S are P. The whle subject class [S] is included in the predicate class [P]. E N S are P. The whle subject class [S] is excluded frm the predicate class [P]. I Sme S are P. Part f the subject class [S] is included in the predicate class [P]. O Sme S are nt P. Part f the subject class [S] is excluded frm the predicate class [P]. 5 parts f a categrical prpsitin 1. Statement letter name (A, E, I r O). 2. Quantifier: "all", "n" r "sme" 3. Subject term 4. Cpula: either "are" r "are nt"; links/cuples the subject term with the predicate term. 5. Predicate term Things t remember: Standard-frm prpsitins have fur distinct cmpnents. Yu cannt cmbine the varius parts in lgic. The terms "subject" and "predicate" "d nte mean the same thing in lgic" that they mean in grammar. The frm "All S are nt P", is nt a standard frm. Tw ways f translating it are: 1. N S are P. r 2. Sme S are nt P. In this text there are nly three frms f quantifiers and tw types f cpulas (see abve). Other texts allw fr variatins, but fr simplicity s sake, we ll nly deal with this limited set.

2 2 Hurley Chapter 4 Lgic Ntes 4.2 Quality Quantity & Distributin Universal Prpsitins: assert smething abut every member f the S class A: All S are P. E: N S are P. Particular Prpsitins: assert smething abut ne r mre members f the S class I: Sme S are P O: Sme S are nt P. Affirmative Prpsitins: affirm class membership r put members int grups A: All S are P. I: Sme S are P. Negative Prpsitins: deny class membership r remve members frm grups. E: N S are P. O:: Sme S are nt P.

3 3 Hurley Chapter 4 Lgic Ntes 4.3 Venn Diagrams & the Mdern Square f Oppsitin 2 Ways t Interpret Categrical Prpsitins 1. Aristtelian: things actually exist in all prpsitins 2. Blean: n assumptins abut existence Aristtelian and Blean differ nly in regard t A and E prpsitins. Fr I and O prpsitins, there is a psitive claim abut existence (things actually exist). Jhn Venn (19th century): created Venn Diagram system. A E I O Cntradictries: A & O; E & I Review the mdern square f ppsitin. Cntradictry relatin = ppsite truth value

4 4 Hurley Chapter 4 Lgic Ntes Hw t test arguments fr validity using the Mdern Square f Oppsitin: a step-by-step guide. 1. Symblize the argument. 2. Draw a small square. 3. Plt the truth values given fr the premise and cnclusin nt the square. 4. Ask yurself if a) the statements are diagnally ppsed, and b) have ppsite truth values (i.e., ne is true and the ther is false). 5. If the answer t either questin is "n," then the argument is invalid. 6. If yu answer "yes" t bth questins, then the argument is valid. immediate inferences = arguments that have nly ne premise See prcess in actin Prcess fr testing arguments fr validity with Venn diagrams: a step-by-step guide. 1. Determine the letter names fr bth the premise and cnclusin statements. 2. Draw a Venn diagram fr the premise. 3. Draw a Venn diagram fr the cnclusin. 4. What t d when yu have a false statement: When yu have a statement that begins with the phrase "it is false that..." draw the diagram fr the cntradictry prpsitin f that statement. Fr example, if yu have an A statement, "Is is false that all S are P." [A statement false], draw the Venn diagram fr the statement, "Sme S are nt P." [O statement True] GIVEN STATEMENT False A False E False I False O DIAGRAM TO DRAW O I E A 5. If the tw diagrams express the same infrmatin (i.e., are identical), the argument is valid, therwise the argument is invalid.

5 5 Hurley Chapter 4 Lgic Ntes 4.4 Cnversin, Obversin & Cntrapsitin (three mves t alter subjects and predicates) General Ntes: 1. We are adding a new truth value fr prpsitins in Sectin 4.4. It is called undetermined. 2. Each mve belw can nly be perfrmed n the statements indicated within the explanatin. When a mve is applied t a statement accrding t the rules belw, it is called a legal mve. 3. When yu make a legal mve, statements keep/retain their riginally truth values. Thus, if a statement is true and yu make a legal mve, the statement stays true. When a statement is false and yu make a legal mve, the statement stays false. 4. When a mve is applied t a statement and vilates ne f the rules belw, it is called an illegal/illicit mve and the truth value fr the resulting statement will be undetermined. 5. When yu make a legal mve, the beginning statement and the statement generated after making the mve are said t be lgically equivalent (i.e., this indicates bth statements mean the same thing). Cnversin: switch subject and predicate Review the diagrams t be sure yu understand the relatin being expressed. What is it? Cnversin is a mve that allws us t switch subjects and predicates fr E & I statements nly. When is it a legal mve? Cnversin can nly be used n E & I statements. Hw d I cnvert E & I statements? Switch the subject and predicate terms. Examples: T cnvert the statement: N cats are fish, we switch the subject and predicate terms t N fish are cats. Since these tw statements mean the same thing, we state that they are lgically equivalent. Cntrapsitin: tw steps Review the diagrams t be sure yu understand the relatin being expressed. What is it? Cntrapsitin is a mve that allws us t switch the subject and predicate terms fr A & O statements while als changing each term t its cmplement. When is it a legal mve? Cntrapsitin can be used n A & O statements nly. Hw d I cntrapse statements? this is a tw-step prcess 1. Switch subject and predicate terms. 2. Change bth the subject and predicate term t its cmplement (i.e., ppsite). Examples: T cntrapse the statement: All cats are fish, we first switch the subject & predicate terms t the A statement yielding "All fish are cats. Next we change each term t its cmplement and the final statement is All nn-fish are nn-cats. Obversin: tw steps Yu d nt have t shw each step, just the riginal and final statements. Since these tw statements mean the same thing, we state that they are lgically equivalent. Review the diagrams t be sure yu understand the relatin being expressed. What is it? Obversin is a mve that allws us t switch the quality and predicate terms fr all fur statements: A, E, I & O. When is it a legal mve? Obversin can be used n A, E I & O statements.

6 6 Hurley Chapter 4 Lgic Ntes Hw d I bvert statements? this is a tw-step prcess 1. Change the quality f the statement (i.e., mve hrizntally acrss the square frm the current statement letter) and, 2. Change the predicate term t its cmplement (i.e., ppsite). Examples: T bvert the statement: N cats are fish, we first mve hrizntally t the A statement yielding "All cats are. Next we change the predicate term t its cmplement and the final statement is All cats are nn-fish.. Yu d nt have t shw each step, just the riginal and final statements. Since these tw statements mean the same thing, we state that they are lgically equivalent. Testing arguments fr validity. Three Steps: 1. Symblize the argument. 2. Determine which f the three new mves: cnversin, cntrapsitin r bversin has taken place between the premise and cnclusin f the argument. 3. If the mve is a legal mve, then the argument is valid. If the mve is illegal/illicit, the argument is invalid. Nte: it is critical that yu memrize legal versus illegal mves (r learn which statements are nt lgically equivalent) t d well n the quiz and subsequent tests n this material. 4.5 The Traditinal Square f Oppsitin (mves t determine truth relatinships fr statements with the same subject & predicate) This square is ften called the Aristtelian Square f Oppsitin. 1. All f the general rules intrduced at the beginning f sectin 4.4 still apply here. 2. Thus when we add the three new mves belw, there will be legal versus illegal mves. 3. It is pssible t have an illegal cntrary statement, illegal subcntrary statement and illegal subalternatin. The truth value f the resulting statements (after an illegal mve) is undetermined. 4. T understand the cncepts belw, yu must understand the cncept f a minimum cnditin. The minimum cnditin is met when the criteria fr meeting a rule are fulfilled: e.g. each credit card has a minimum mnthly payment fr peple wh carry balances mnth t mnth. When yu make the minimum payment, yu are fulfilling the minimum cnditin t keep the accunt in gd standing. Anther example is that requirement that yu have a minimum amunt f credits t graduate alng with a passing prtfli. When bth requirements are fulfilled, yu have met the minimum cnditins fr graduatin. Remember: 1. Aristtelian: things actually exist in all prpsitins 2. Blean: n assumptins abut existence Here are the three new mves (i.e., relatinships) we are adding t the square:

7 7 Hurley Chapter 4 Lgic Ntes Pints t nte: Cntrary relatin (A & E statements nly): at least ne is false S, if we already knw that either the A r the E prpsitin is false, the truth value f the remaining prpsitin is undetermined. Subcntrary relatin (I & O statements nly): at least ne is true S, if we already knw that either the I r the O prpsitin is true, the truth value f the remaining prpsitin is undetermined. Subalternatin relatin (relatin between A & I statements and E & O statements): truth flws dwnward and falsity flws upward If we are given an A r an E prpsitin that is false, the truth value f the crrespnding I r O prpsitin is undetermined. Als, if we are given an I r an O prpsitin that is true, the truth value f the crrespnding A r E prpsitin is undetermined. Testing Immediate Inferences There are tw kinds f immediate inferences: thse yu can use the square f ppsitin t test, (i.e., subjects and predicates remain the same) and thse in which yu will have t reduce the number f terms thrugh cnversin, bversin and cntrapsitin befre using the square t test. Testing Immediate Inferences using the traditinal square f ppsitin nly Three Steps: 1. Symblize the argument 2. Assume the premise and cnclusin are true unless yu see the phrase: "It is false that..." 3. Determine the type f relatin (i.e., the mve that has ccurred) that exists between the premise and cnclusin.

8 8 Hurley Chapter 4 Lgic Ntes 4. Using the basic relatins frm the traditinal square f ppsitin, deduce the truth value f the cnclusin. 5. If the mve is a legal mve, then the argument is valid. If the mve is illegal/illicit (i.e., the cnclusin is undetermined), the argument is invalid. Three Fallacies:these fallacies ccur when arguments ask us t vilate the relatinal rules in the traditinal square. Any time the prblem generates an undetermined truth value fr the cnclusin, the argument is invalid and ne f the fallacies belw has been cmmitted. 1. Illicit cntrary (An A r E premise is false and the cnclusin is undetermined): argument tries t use an invalid applicatin f the cntrary relatin. 2. Illicit subcntrary (An I r O premise is true and the cnclusin is undetermined): argument tries t use an invalid applicatin f the subcntrary relatin. 3. Illicit subalternatin (Truth is flwing upwards r falsity is flwing dwnwards between A & I statements r E & O statements): argument tries t use an invalid applicatin f the subalternatin relatin. The Prfs: Sectin 4.5 Part V (Testing Immediate Inferences using cnversin, bversin and cntrapsitin plus the traditinal square f ppsitin) Please understand that yu will nt be able t cmplete these prblems unless yu thrughly understand the mves presented in sectins 4.4 & 4.5. If yu d nt learn thse mves, yu will nt understand hw t chse yur mves in step 4 belw. Hence, if yu have prblems with the prfs, my advice is always the same, g back and learn the mves again. A Step-by-step prcess fr ding these prblems: 1. Symblize the argument. 2. The bject f the game is t transfrm the premise statement int the cnclusin statement using nly legal mves. 3. Determine which terms (subject and predicate) may have t be switched and/r transfrmed int their cmplements. This determinatin will help yu chse the mves that yu will use. 4. Sme hints fr chsing yur mves: 1. If nly ne term is t be changed t its cmplement, then it is likely that yu will use bversin alng the way. 2. If the terms have t be switched, it is likely yu will need t use cnversin. 3. If bth terms must be switched and changed t their cmplements, then cntrapsitin will be used. 4. Mving arund the square des nt change subjects and predicates, it changes letter names and truth values nly. 5. Keep track f letter names and truth values fr each statement because they will influence yur next mve in the prf.

9 9 Hurley Chapter 4 Lgic Ntes Make a chart fr each prblem that lks like this: (This example is frm the exercises: Part V, #9.) Letter Name Truth Value Prpsitin Inference Name O F Sme nn-l are nt S. given E F N nn-l are S. subalternatin E F N S are nn-l. cnversin A F All S are L. bversin O T Sme S are nt L. cntradictry 4.6 Venn Diagrams and the Traditinal Standpint 1. Nte hw the cncept f existence changes the diagramming f relatinships. 2. The diagrams have been mdified t shw that the subalternatin mve can prduce valid arguments. 3. Tw diagrams change fr the purpse f prving arguments valid r invalid by subalternatin: The A diagram:yu will use this diagram fr premise statements in tw cases: 1) a true A statement r 2) a false O statement The E diagram: yu will use this diagram fr premise statements in tw cases: 1) a true E statement r 2) a false I statement 4. Nte the changes in prving validity when we add the existence symbl t the A and E prpsitins. There are nw direct inferences that can be made frm the universal prpsitins

10 10 Hurley Chapter 4 Lgic Ntes t their respective particular prpsitins (I and O) via subalternatin and vice versa ging upward frm the particulars t the universals. 5. When d I use the new diagrams? Use the new diagrams when yu have t draw an A r E diagram fr a premise statement nly. When yu are drawing an A r E diagram fr a cnclusin statement, cntinue t draw the ld diagram. 6. Hw des this change ur view f an argument's validity? By using this technique yu are shwing that arguments can als be valid under the rule f subalternatin: when A is true, I is als true and when E is true, O is als true. 4.7 Translating Ordinary Language Statements int Categrical Frm Tw Benefits 1. Can manipulate using square f ppsitin and new argument evaluatin techniques learned in this chapter. 2. Renders statements "cmpletely clear and unambiguus." Types f Transfrmatins: 1. Terms Withut Nuns Review the sentence beginning "Nuns and prnuns." 2. Nn-standard Verbs We are wrking with the frm f the verb "t be." Varius tenses (i.e., will, will nt, has, has nt). This invlves translating all ther cpulas int statements that cntain the phrases "are" r "are nt." 3. Singular Prpsitins Watch fr plural frms f nuns as they shuld nt be translated in this matter. 4. Adverbs and Prnuns Wrds t lk ut fr - the ht list: Spatial Adverbs where wherever anywhere Tempral Adverbs when whenever anytime

11 11 Hurley Chapter 4 Lgic Ntes Spatial Adverbs everywhere nwhere Tempral Adverbs always never NOTE: There is a "switching the rder" trick that must ccur if ne f the abve wrds ccurs in the middle f a statement. 5. Unexpressed Quantifiers Here, "quantifiers are implied but nt expressed." The trick is figuring ut if we are talking abut "all" r "sme" f the nun in questin. Nte the last tw examples in this sectin use tw different uses f the wrd children. 6. Nnstandard Quantifiers The frm "All S are nt P" is nt standard frm. Translatin determines meaning: e.g., All athletes are nt superstars. The previus statement is nt a universal prpsitin, but rather a particular claim. Read "At least ne athlete is nt a superstar." 7. Cnditinal Statements Cnditinal statements are always rendered as universals! NOTE: When a cnditinal statement appears in the middle f a sentence, "the statement must be restructured s that it the beginning." Transpsitin: applies t cnditinal statements where bth terms are negated. Nte/reread the middle paragraphs n p.246 and g ver the examples again. The wrd "unless" means "if nt"again, the transpsitin rule applies here. Carefully g ver the examples. 8. Exclusive Prpsitins Wrds t lk fr: nly, nne but, nne except Tw step Prcess t render statement int standard frm: 1. First phrase as a cnditinal statement. 2. Transfrm int a categrical statement. (Remember all cnditinal statements are translated as universals as nted abve in #7.) Nte als hw individual references wrk. Basically, these references generate tw categrical prpsitins. Our curse ignres these special cases. When nly and nne but are in the middle f a sentence, they are transpsed. Only can be rendered in many ways. Thus, it is ambiguus.

12 12 Hurley Chapter 4 Lgic Ntes 9. "The Only" If the wrds "the nly" appear at the beginning f a phrase, they can be replaced by the wrd "all" an n transpsitin is necessary. But, if these wrds appear in the middle f a phrase, then the statement must be transpsed befre putting it int standard frm. "The nly" is like nly in that is ambiguus and has t be rendered using tw statements fr clarity. 10. Exceptive Prpsitins Tw frms: 1. All except S are P. 2. All but S are P. These statements generate tw standard frm prpsitins. Key Wrd Translatin Hint whever wherever always anyne never, etc. use all tgether with persns, places and times a few sme if.. then use "all" r "n" unless "if nt" nly nne but, nne except, and n.except. use "all" switch rder f terms the nly "all" all but, all except, few tw statements required

13 13 Hurley Chapter 4 Lgic Ntes "nt every" and nt all Key Wrd Translatin Hint "sme are nt" there is "sme"

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