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# Logic midterm notes

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### Logic midterm notes

1. 1. ___________________________ CHAPTER II JUDGMENT and PROPOSITION Form is a revelation of essence. As the drop becomes the ocean, so the soul is deified, losing her Name and work, but not her essence. You must break the outside to let out the inside; to get at Kernel means breaking the shell. Even so to find nature herself all Her likeness has to be shattered. (Anonymous) Chapter Outline 1. Judgment Defined a) Elements of Judgment two known ideas comparison of these two ideas mental pronouncement of identity/non-identity between two ideas 2. Proposition Defined a) Parts of Proposition subject (S) predicate (P) copula (c) 3. Predicables species genus differentia property accident Suggested Learning Activities 1. Spotting the difference between a mere sentence and a judgment/proposition. 2. Imaginary ‘anatomy’ of the body. 3. Mind game: What’s more essential: a) character or fame? b) Beauty or brain? c) money or person? 4. ‘Measuring depth of knowledge’ game.The Proposition as Expression of Judgment Judgment is the second mental process or operation that essentially figures in theact of reasoning or thinking. A judgment is an act of the mind pronouncing an agreementor disagreement between two ideas. Three things are required in the making of judgment,namely: 1. two known ideas, their 2. comparison, and 3. the act of the intellectpronouncing their identity or non-identity. Here is how a judgment exemplified. “Acomputer is a machine.” This sentence is one written example of a judgment. The „computer‟is the subject term while „machine‟ is the predicate term. These two terms then represent thetwo ideas that the mind is comparing. The „is‟ in the same sentence represents the act ofpronouncement executed by the mind on the two ideas „computer‟ and „machine‟. In thisintellectual pronouncement, the mind is engaged in processing whether or not the two ideas are
2. 2. in agreement or disagreement and/or identical or non-identical with each other. The exampledemonstrates that the mind specifically makes a pronouncement of agreement since the verb„is‟ indicates that the mind recognizes the identity between the “computer‟ and „machine‟. Hadthe verb in the sentence been an “is not‟, it would have implied that the mind makes apronouncement of disagreement between the two ideas. Truth or, for that matter, falsity is then contained not in „idea‟ but in judgment. In otherwords, what is expressed by a judgment is either a truth or a falsity. If the judgment conforms tothe reality of the thing about which it is made, then it is true. If things are not as the judgmentasserts them to be, then the judgment is false. It is not therefore difficult to figure out that a judgment is always expressed in a sentence,and a declarative sentence at that. An interrogative sentence, or an exclamatory, or a request ora command, cannot constitute a judgment per se because in these kinds of sentences the minddoes not make a pronouncement of agreement or disagreement between two ideas or term.For instance, in this sentence, “Is Derrick Rose an NBA MVP?” no judgment is made by themind because it (the mind) is not engaged in making a pronouncement of agreement ordisagreement between the ideas „Derrick Rose‟ and „MVP‟. In asking question, the mind hasnot yet arrived to a definitive judgment whether one idea is affirmed or denied of another idea.Hence, no determination of truth or falsity is reached yet by the mind in an interrogativesentence. The same can also be said in exclamatory and imperative sentences. For examples:“Oh my goodness!” and “Please, hurry up.” In both sentences, the mind does not make anycomparison between two ideas. Thus, no judgments per se have been made by the mind inthese two sentences. Just as idea is externally expressed in either verbal or written word technically called term,a judgment is also expressed in an oral or written sentence technically called proposition. Aproposition is a judgment expressed in a sentence. It can then be readily stated here thatall propositions are sentences but not all sentences are propositions. Since there are sentencesthat express neither truth nor falsity (thus they don‟t contain judgment), then not all sentencesare propositions. For instance, this interrogative sentence, “What day is today?”, although asentence, is not however a proposition because it does not contain a judgment. It expressesneither truth nor falsity because it does not imply a mental act of pronouncing agreement ordisagreement between two ideas. Every proposition consists of a subject (S), a predicate (P),and the copula (c); subject and predicate are the matter, and the copula is the form of theproposition. It is the copula which indicates whether one term is denied or affirmed of thesubject. For example, in the proposition, “Her car is expensive,” the copula „is‟ expressesaffirmation; in the proposition, “Meriam Defensor Santiago is not media-friendly,” the copula„is not‟ expresses denial or negation. Since the copula expresses the present act of the mind inits judgment, it is always expressed in the present tense of the indicative mood of the verb „tobe‟ even if the sentence refers to past or future events. When a proposition is structured or constructed grammatically as “S c P,” it is said to be inits logical form. Examples: “Bin Laden is a Muslim extremist” and “Angel Locsin is notarrogant.” These propositions are in their logical form. However, not all sentences employ „tobe‟ always as their verbs. Examples: “The communist countries disintegrated,” and “Brazil willwin hopefully again in the next Football World Cup.” The verbs used in these sentences are notfrom the infinitive „to be,‟ and neither are they in the present tense, but past and future tensesrespectively. In cases like the above, what is needed is simply to reduce the propositions totheir logical form. So, “The communist countries disintegrated,‟ can be reduced to its logicalform which is, ”The communist countries are the counties that disintegrated.”; “Brazil will winhopefully again in the next Football World Cup,” becomes “Brazil is the team that will winhopefully again in the next Football World Cup.” Basically, no truth or falsity value is altered inthe propositions reduced to their logical forms.
3. 3. Putting the proposition in its logical form makes its meaning clear and minimizes a lot ofmisinterpretation, misunderstanding or confusion arising from the vague formulation of theproposition.The Predicables Predicables (or logical universals) are the different modes or ways in which a universal(used as a predicate term in a proposition) can be predicated of its subject. Understanding thefunctionality of predicables in the thinking process enhances depth of knowledge about realities.In fact, predicables serve in determining that there can be different levels of depth inunderstanding things. Thus they anchor the thinker‟s capacity to define or describe realities. That there are only five predicables can be seen by considering the essential and accidentalconnections that a predicate of a proposition has with its subject. There is an essentialconnection between a subject and predicate, if the latter expresses something that is intrinsic tothe nature or essence of the subject; the connection is accidental or non-essential when thepredicate expresses something that is extrinsic to the essence or nature of the subject. Thus,for instance, „rational‟, if predicated of „man‟ in the proposition “Man is rational,” indicatesessential connection between the two precisely because rationality constitutes a part of man‟sessence or nature; „religious fundamentalist,‟ if predicated of the same as in this proposition“Man is a religious fundamentalist,” indicates only an accidental connection between them sincebeing a religious fundamentalist is not a necessary part of man‟s nature or essence. The following are the five predicables: 1. Species refers to the manner in which a universal idea when used as a predicate of aproposition expresses the whole essence of its subject. Example: „Man is a rational animal.‟Precisely, „rational animal,‟ as predicated of „man‟ expresses the whole essence of what man isin the real order. 2. Genus refers to the manner in which a universal idea when used as a predicate of aproposition expresses a part of the essence of its subject, that part which the subject has incommon with other species in the same class. Example: „Man is an animal‟; „animal‟ aspredicated of „man‟ in the proposition expresses only a part of the essence of what man is.Aside from man, however, there are also many other species of which „animal‟ can bepredicated, such as: „Dog is an animal.‟ „Dolphin is an animal.‟ 3. Differentia refers to the manner in which a universal idea when used as a predicate of aproposition expresses a part of the essence of its subject, that part which distinguishes onespecies from another under the same genus. So: „Man is rational‟ and „Brute is irrational” aretwo propositions whose predicates „rational‟ and irrational‟ express a part of the essence of theirrespective subjects thus making the latter different (differentia) from each other, albeit theybelong to the same genus of „animal.‟ 4. Property refers to the manner in which a universal idea when used as a predicate of aproposition expresses something that flows necessarily from the essence, though not of theessence itself. Example: „Man is capable of writing.‟ The predicate of this proposition „capableof writing‟ is not of and even a part of the essence of what man is because man would stillremain to be a man even if he does not write. Yet, from man‟s rational nature, the capacity forwriting flows necessarily. Otherwise, writing would not be possible. 5. Accident refers to the manner in which a universal idea when used as a predicate of aproposition expresses something of its subject which is neither of its essence nor necessarilyconnected with its essence, but is merely contingently connected with the essence. Example:„Man lives in a condominium.‟ In this proposition, the predicate „lives in a condominium‟ is neverof and part of the essence of what man is; neither is it necessarily flowing from man‟s essence,but it is only an accidental circumstance or condition of man. It can entirely be otherwise.
4. 4. _______________________ CHAPTER III TYPES OF PROPOSITIONS For thousand of years man has lost touch with his original intelligence......He starts interfering in the course of nature with a mind that is centered and one-pointed, and analyzes everything, and breaks it down into bits... The moment you do that you lost contact with your original know-how... ...getting back to being able to trust our original intelligence, ...(is drawing) an entirely new course for the development of civilization. Alan Ginsberg Chapter Outline I. The General Types of Propositions A. Propositions according to their Quality 1. Affirmative 2. Negative B. Propositions according to their Quantity 1. Universal 2. Singular 3. Particular 4. Collective C. General Propositions expressed in Logical Shorthand Symbols 1. A proposition or Universal affirmative 2. E proposition or Universal negative 3. I proposition or Particular affirmative 4. O proposition or Particular negative D. Propositions according to Relation between subject(s) and predicate(s) 1. Analytic (essential, necessary, ‘a priori’) 2. Synthetic (accidental, contingent, ‘a posteriori’) II. The Special Types of Propositions A. Single Categorical Propositions 1. Simple 2. Composite o Complex o Modal Necessary Impossible Possible Contingent B. Multiple Categorical Propositions C. Hypothetical Propositions 1. Overtly Multiple Categorical a) Conditional a) Copulative b) Disjunctive b) Adversative c) Conjunctive c) Relative d) Causal e) Comparative 2. Covertly Multiple Categorical a) Exclusive b) Exceptive c) Reduplicative d) Specificative
5. 5. Suggested Learning Activities 1. Group power point presentation of ‘standardization’ models of consumer products. Elaboration of the practical benefits of this ‘standardization’ model by each group. 2. Random samples of mnemonics to enhance the mental processes of memory and imagination. 3. Board contest: ANALYSIS AND SYNTHESIS Knowing considerably well all possible appearances of propositions in their grammaticalconstructions contributes to the clarity and keenness of thinking and reasoning. More so, themind will easily penetrate to the truth and falsity found in the judgment and proposition if it cansee through the intricacies of linguistic structures. Hence, the knowledge of the various types ofpropositions can come in handy here. There are general and special types of propositions asdelineated below.The General Types of Propositions The general types are based on the quality, quantity, and relation of subject and predicatefound in proposition. Quality of Proposition. From the standpoint of quality a proposition is either affirmativeor negative. The quality of a proposition affects the copula and makes the proposition eitheraffirmative or negative. In other words, the predicate is either affirmed or denied of the subject.Examples: „Bioinformatics is an academic course taught in medical school,‟ and „Thearchitectural design of the CICCT building is modern.‟ In these instances, the copula „is‟ impliesthat each of the predicates of both propositions is affirmed of the subject. In these otherpropositions „The consultant‟s grasp of the human psychology of the entire institution is notimpressive,‟ and „The feasibility study of the graduating accountancy students is not that viable,‟the copula implies that each of the predicates of both propositions is denied of the subject. As a rule, in an affirmative proposition, the predicate is always used according to the wholeof its comprehension and a part of its extension; it is, therefore, always a particular term. In anegative proposition the predicate is always used according to a part of its comprehension andthe whole of its extension; it is, therefore, always a universal term. This logical rule should beborne in mind because it bears important relation to the other laws and methods of correctthinking that will be treated later in the other chapters. Quantity of Proposition. The quantity of a proposition affects the extension of the wholejudgment as a judgment. It expresses the number of individuals to whom the judgment orproposition applies. Since the predicate is applied to the subject, the proposition will be true ofall the individuals contained in the extension of the subject. From the standpoint of quantity,then, propositions will be 1. universal, 2. particular, 3. singular, and 4. collective, whicheverway the extension of the subject is taken. 1. A proposition is universal if the subject is a universal term applied distributively toeach and all of the class. Examples: „All normal cows are four-legged.‟ „Every voter is acitizen.‟ The use of „every‟ to prefix a subject denoting universality can never be ambiguous.However, there are cases that when „all‟ is prefixed to a subject, it would not necessarilyindicate universality of a proposition. Take this for instance: „All players filled up the lockerroom.‟ This can be taken only collectively not distributively because to say, „Every (single)player filled up a locker room,‟ does not make any real sense. So, here, we must look to themeaning of the judgment or proposition. 2. A proposition is particular when the subject is a universal term used partly andindeterminately. „Particularity‟ of a term is indicated by some words like „some‟ or „not all,‟attached to the subject. Examples: „Some artists are very eccentric,‟ „Not all nursing studentsare females.‟ But sometimes, a statement with the word „all‟ attached to the subject makes thatstatement appear like a universal proposition at first glance. Yet, a closer scrutiny would reveal
6. 6. it to be otherwise. Example: „All Boholanos are not handsome.‟ This statement seems on theface of it to be universal. But this is certainly not the sense intended, because obviously thereare Boholanos who are evidently (according to ultimate aesthetic standard) handsome.Therefore, “All Boholanos are not handsome,” in its real sense, implies, „Not all Boholanos arehandsome,‟ which conversely is the same as saying „Some Boholanos are not handsome.”Hence, it makes it a particular proposition. 3. A proposition is singular when the subject applies to a single individual only. Such arethe statements: „Carlos Polestico Garcia was a true-blooded Talibon, Bohol native.‟ „Thatengineering student is not attentive.‟ „The present Pope is a German.‟ Singular propositionshave the same value as universal propositions and are treated the same way because thesubject is taken according to the whole of its extension, which in this case, is one. 4. A proposition is collective when the subject is a collective term, applying to all takentogether as a class, but not to the individuals composing the class. Examples: „TeamPhilippines emerged as the champion in the 23rd South East Asian Games (SEA Games).‟ „Alltroops of the 7th Infantry Brigade of the Philippine Army surrounded the swampy territory ofthe insurgents.‟ „The entire gallery of spectators witnesses the buffoonery of the senators.‟Since a collective term represents many considered as one, it is taken according to the whole ofits extension and it, too, is treated as a universal. Addendum: There is one kind of proposition which has no definite sign of quantityattached to the subject. This is called indefinite proposition. Example: „Athletes are physicallygifted.‟ „Mga Cebuanos hawod ug kamao mopili kinsa ang angayan nga mamuno sa nasud.‟„Teenagers are restless.‟ Such and similar propositions indicate no definite quantity. Theyevidently mean „some‟ or „all‟ and are either particular or universal propositions. The sense ofthe statement or the context in which they are used must give us the exact quantity. All propositions, considered simultaneously according to their quality and quantity result tofour general propositions, namely: the universal affirmative, the universal negative, theparticular affirmative and the particular negative. They are deemed general propositions inorder to point out the fact that practically all propositions can be reduced to this generalclassification. This classification will in turn provide a kind of standardization of proposition thusenhancing the mind‟s capacity for better and orderly comprehension. Why then there are only four general types of propositions? As far as quantity is concerned,all judgments have the value of either universal or particular propositions because singular andcollective propositions are equivalent to universal propositions as elucidated beforehand. And,by reason of quality, all propositions will either be affirmative or negative. Logicians expressthem in a sort of logical shorthand symbols with the letter A standing for universal affirmative; Efor universal negative; I for particular affirmative; and O for particular negative. The letters Aand I are the letters A and I in the Latin word „AffIrmo‟ which means „I affirm.‟ Letters E and Oare derived from „nEgO,‟ meaning „I deny.‟ This therefore gives us the following scheme. Universal affirmative (A) ---------- ex: All men are mortal. or Every man is mortal. Universal negative (E) ---------- ex: No man is a mechanical invention. or All men are not mechanical inventions. Particular affirmative (I) ---------- ex: Some men are altruistic. Particular negative (O) ---------- ex: Some men are not narcissistic. Relation between Subject and Predicate of Proposition. From the standpoint of therelation between subject and predicate, propositions will either be necessary or contingent.When the relation is a necessary matter, the essence or nature of the subject is alwayscontained in the comprehension of the predicate or vice versa; hence, an analysis of one will
7. 7. reveal the other and thereby reveal the truth of the proposition. Examples: „Man is a rationalanimal.‟ „A square is a quadrangle.‟ „Two and two are four.‟ „A fish is capable of breathingunderwater.‟‟ But when this relation is a contingent matter, neither the subject nor the predicateis contained in the comprehension of the other, and an analysis of one will not reveal the other;since this relation is based on a contingent fact, we can prove the truth of the proposition onlyby experience. Examples: „Carbon Market is a stone‟s throw away from the universitycampus.‟ „The Mt. Pinatubo eruption in 1991 effected a change in global temperature.‟ „Thesuper typhoon left a huge area of destruction.‟ The necessary proposition is, therefore,analytic, while the contingent proposition is synthetic; the former is also called „a priori‟ andthe latter „a posteriori.‟The Special Types of Propositions One function of language is to convey thought and the truth or falsehood of this thought fromone mind to another. But the complexity of language wrought by myriad of words tends tomuddle the meaning of a judgment or proposition. The knowledge of these special types ofpropositions will definitely facilitate in understanding better the thought-contents or judgmentsimplied in many different grammatical or propositional structures. Before exposing all these special propositions, it will be helpful to understand first andforemost the individual characteristic feature(s) of a single, a multiple, a categorical, and anhypothetical proposition. A single proposition consists of one subject and one predicate.Examples: „A cab is a vehicle.‟ „The football field is deserted.‟ „An Oversea-Filipino-Worker(OFW) is a modern day martyr.‟ A multiple proposition consists of two or more propositionsunited into one sentence. Examples: „Ringo and Paul were members of the Beatles.‟ „RobertJaworski was a professional basketball player and a senator.‟ „If an earthquake hits Cebu, manydilapidated buildings will collapse.‟ A categorical proposition attributes (affirms or denies) thepredicate of its subject outright. In other words, the predicate is directly asserted to (eitheraffirmed or denied of) its subject in a categorical proposition. Example: „Corruption isimmoral.‟ „Sex per se is not evil.‟ A hypothetical proposition asserts the dependence of onejudgment on another. Example: “If Manny Pacquiao does not focus on his training, he cannotremain a pound-for-pound boxing king for long.‟ Single Categorical Proposition. The categorical proposition, as has been stated, makes adirect assertion (hence it is also called assertoric) of agreement or disagreement betweensubject and predicate. The single categorical proposition contains but a single sentence in itsconstruction – one subject, one predicate, and the copula. If the subject, the predicate and thecopula are not modified in any way, i.e., no qualification whatsoever enters into each and all ofthem (the subject, the predicate and copula), then it is simple categorical. In other words, thesubject and predicate of a single categorical proposition are simple terms (consisting only of oneword) and the copula is at the same time not modified. Example: „Love is madness.‟In this example, the subject „Love‟ is unmodified or it is a simple term; the predicate „madness‟ isalso a simple term; and the copula „is‟ is not modified. But if a qualification enters into thesubject or predicate or copula, what results is a composite single categorical proposition.Examples: „This man is a passionate revolutionary.‟ „That lady wearing signature jeans is anex-nun.‟ „The intramural games may not be held this semester.‟ In the first example, thepredicate „passionate revolutionary‟ is not a simple but a compound term; in the second, thesubject „lady wearing a signature jeans‟ is also a compound term; and in the third, the verb „maynot be held‟ indicates a modified copula. Hence, all three propositions are compositepropositions. There are two kinds of these composite propositions, namely, the complex and the modal.The complex proposition is a composite single sentence in which both the subject and thepredicate or either one is a complex term. That‟s why „Colonel Gringo Honasan is a soldier,‟ „He
8. 8. is a renegade lawmaker too,‟ and „The Armed Forces of the Philippines is a graft-riddengovernment establishment,‟ are all examples of complex (composite single categorical)propositions. The modal proposition is a composite single sentence in which the copula is somodified as to express the manner (mode) in which the predicate belongs to the subject.The qualification affects the copula, that is, it states whether the objective connection betweensubject and predicate expressed by the copula is necessary, impossible, possible, orcontingent. The necessary modal proposition states that the predicate belongs to the subject andmust belong to it. Examples: „God is necessarily powerful.‟ „A tsunami must be extremelyhuge.‟ In these examples, „is necessarily‟ and „must be‟ are modified copulas. The impossibleproposition states that the predicate does not and cannot belong to the subject. Examples:„Man cannot live forever.‟ „It is impossible that the globe is flat.‟ The possible propositionenunciates the fact that the predicate is not actually found in the subject, but it might be.Such are: One Filipino basketball player will be able to play in an NBA team.‟ „The mountainclimbers could have asphyxiated.‟ The underlined parts of both propositions given above alsoindicate modified copulas. The contingent proposition states that the predicate actuallybelongs to the subject, but it need not. Examples: „It is not necessary that the audiencekeeps on standing.‟ „Blood need not be shed to attain peace.‟ Again, both examples showmodified copulas. To facilitate familiarization, here is a chart of the single categorical propositions: Single Categorical Propositions Simple Composite Complex Modal Necessary Possible Contingent Impossible Multiple Categorical Propositions. Multiple categorical propositions contain two ormore sentences in their very construction. Some of these are overtly, while the others arecovertly multiple; the latter are called „exponibles.‟
9. 9. The overtly multiple categorical propositions are plainly composed of two or morepropositions. They are five in number: 1. copulative 2. adversative 3. relative 4. causal and5. comparative. The copulative proposition is a multiple categorical proposition which has two or moresubjects, or two or more predicates, or two or more subjects and predicates, combined intoone sentence. Examples: „SM and Gaisano Country Malls are supermarkets.‟ „The passengerbus skidded and plunged into a ravine.‟ „Tony Parker and Boris Diaw are non-American NBAplayers and Spurs team mates.‟ Each of these sentences can be resolved into as many singlepropositions as there are different subjects and predicates. The truth of copulative categoricalproposition depends upon the truth of all the single sentences which compose it. The adversative (discretive) proposition is a multiple categorical proposition which consistsof two propositions united in opposition to each other by conjunctions like, „although,‟„yet,‟ „even if,‟ „but,‟ etc. Examples: „Bill Gates became so successful, even if he dropped outof college.‟ „A winner stumbles countless of times, yet he never quits.‟ „The torturers might havebroken him physically, but not his spirit.‟ The relative proposition is a multiple categorical proposition which expresses arelationship of time or place between two sentences. Examples: „The incidence of bombingin Iraq diminished a bit after they have installed a government.‟ „Before the cock crowed threetimes, Peter chickened out and disowned his master.‟ „The demagogues or the traditionalpoliticians squirmed in their seats during the entire period that the witnesses divulged theirshenanigans.‟ The causal proposition is a multiple categorical proposition which combines twostatements in such a way that the one is given as the reason or cause of the other. Thewords „for‟ or „because‟ or „due to‟ or „since‟ are some words used to connect the statements.Examples: „There is massive poverty and hunger in the Philippines due to the absence ofmoral leadership.‟ „Because of intolerance and bigotry, terrorism of all kinds persists in theworld today.‟ The comparative proposition is a multiple categorical proposition which compares therelation between a subject and a predicate with the same relation between another subjectand predicate, and expresses the degree of this relationship as being either less or equal orgreater. Examples: „Rafael Nadal is a better lawn tennis player than Roger Federer in claycourt.‟ „As you live, so you shall die.‟ „An Apple laptop is flashier in design than an Acer.‟ The covertly multiple categorical propositions have the appearance of single propositions,although they are really multiple. Their composition lies concealed in some words and needs anexposition to show their multiple characters; hence, they are styled „exponibles,‟ and thesentences into which the general proposition can be resolved are called the „exponents.‟ The„exponibles‟ are four in number: 1. exclusive, 2. exceptive, 3. reduplicative, and 4.specificative. The exclusive proposition is a multiple categorical proposition which contains someparticles of speech like „only,‟ „alone,‟ „solely,‟ „none but,‟ etc., indicating the exclusion of anyother predicate from this subject or any subject from this predicate. Examples: „Only you canmake my dreams come true.‟ „Not only is the Philippines not economically progressive.‟ Theaffirmative exclusive exponible will be resolved by means of a copulative proposition, in whichone sentence is affirmative and the other negative; thus, „Only you can make my dreams cometrue,‟ will become, „You can make my dreams come true and no one or nobody else can.‟ Thenegative exclusive exponible will be resolved into two negative exponents; the sentence „Notonly is the Philippines not economically progressive,‟ will become „The Philippines is noteconomically progressive and some other countries are not also economically progressive.‟ The exceptive exponible is a multiple categorical proposition which contains a particle ofspeech like „except‟ or „save‟ to indicate that a portion of the extension of the predicate doesnot apply to the subject, or vice versa. Examples: „All hostages except an old mother were not
10. 10. released by the kidnappers.‟ „The entire class passed save those who dropped out.‟ Theresolution of such exponible should go like this: „An old mother was released by the kidnappers,and all others were not.‟ The reduplicative and specificative propositions are multiple categorical propositions whichcontain an expression which duplicates the subject or predicate, giving them specialemphasis. The expressions „as such‟ „as a or such an‟ and „such a‟ are some expressionsemployed in reduplicative or specificative propositions. Example of reduplicative proposition: „Ahovercraft, as an amphibious vehicle, can operate both on land and on sea.‟ In thereduplicative proposition, the duplicated subject or predicate, can be taken as the reason itselfor cause for the connection between the subject and predicate. Example of a specificativeproposition: „Niña, as a student, is a freshman.‟ The duplicated subject or predicate in thespecificative proposition merely implies a circumstantial condition of the connection betweensubject and predicate. For facility, here is a schematic outline of all multiple categoricalpropositions: Multiple Categorical Overt Covert Copulative Adversative Relative Causal Comparative Exclusive Exceptive Reduplicative Specificative The Hypothetical Propositions. The hypothetical proposition differs from the categorical(assertoric) proposition, because it does not declare an unqualified affirmation or denial, butexpresses the dependence of one affirmation or denial on another affirmation or denial. Thisproposition has three distinct kinds: the 1. conditional, the 2. disjunctive, and the 3.conjunctive. The conditional proposition is a hypothetical proposition which expresses a relation invirtue of which one proposition necessarily flows from the other because a definite condition isverified or not verified. Example: „If the international benchmark of the price of oil continues torise, social unrest will occur in many third world countries.‟ The statement containing „if‟ iscalled „antecedent’ proposition; and the other is the „consequent.‟ The truth of thishypothetical proposition does not depend on the truth of the two statements taken individuallyand separately, but on the relation between them. The disjunctive proposition is a hypothetical proposition which contains an„either – or‟ statement, indicating that the implied judgment cannot be true together nor falsetogether, but one must be true and the other (or others) false. Example: “A student can eitherbe a scholar or a truant.‟ There are two variations of disjunctive proposition, namely: perfect orproper disjunction which implies that the parts should be such that neither can be true or falsetogether; example: „A body is either in motion or at rest.‟ An imperfect or improper disjunction
11. 11. is taken in wider sense and implies then „at least one, but possibly some or all of the parts. . .‟Example: Either St. Peter or St. Paul or St. John died in Rome.‟ In this wider sense, all are notallowed to be false, but all may be true. A conjunctive proposition is a hypothetical proposition which expresses a judgment thattwo alternative assumptions are not and cannot be true simultaneously. Examples: „An aceathlete cannot be a disciplined person and voracious eater at the same time.‟ „A person cannotbe a hero and a villain at the same time.‟ A working knowledge of the various types of categorical and hypothetical propositions willenable us to unravel the maze of words and detect the truth of the judgment implied in them.
12. 12. ____________________________ CHAPTER IV OPPOSITION OF PROPOSITIONS ____________________________ “Not to speak with a man who can be spoken with is to lose a man. To speak to a man who cannot be spoken with is to waste words. He who is truly wise never loses a man; he too, never wastes his words.” Chapter Outline I. An introductory note on Chapters IV and V II. The Logical Oppositions of Propositions A, E, I, and O 1. Subalternation 2. Contradiction 3. Contrariety 4. Subcontrariety III. The Laws of Logical Opposition 1. Laws of Subalternation 2. Laws of Contradiction 3. Laws of Contrariety 4. Laws of Subcontrariety Suggested Learning Activities 1. Connecting the dots. 2. Mind’s Power of “finishing other’s sentence.” 3. Figuring out “things that are better left unsaid” 4. Pick up lines games Note: For easy understanding of chapters IV and V, it is beneficial to situate these lessons properly. Both the logicaloppositions of propositions (chap. IV) and eductions (chap. V) are two intellectual processes that are inferential by nature.Inference is a process whereby the mind, in its act of thinking, draws out or derives a certain judgment from one judgment thatit (mind) already comprehends or considers. Specifically, logical oppositions of propositions and eductions are immediateinferences in contrast to reasoning (proper) which can be called more fittingly as a mediate inference. The distinction betweenimmediate and mediate inferences will be made clearer in the treatment of both in their respective chapters. The logical opposition of proposition refers to the relation which exists betweenpropositions having the same subject and the same predicate, but differing in quality, orin quantity, or in both. There are four possible ways for a proposition to have the same subjectand the same predicate, but differing in quality, or in quantity or in both: a universal affirmative(A), a universal negative (E), a particular affirmative (I), a particular negative (O). Examples:(A) – „All men are learned.‟ (E) – „No men are learned.‟ (I) – „Some men are learned.‟ and (O) –„Some men are not learned.‟ The four oppositional relations that can possibly exist amongthese propositions are exemplified in the subjoined Square of Opposition.
13. 13. All men are learned. No men are learned. A contrariety E subalternation contradiction subalternation I subcontrariety O Some men are learned. Some men are not learned. This diagram illustrates the four relations resulting from four oppositions, namely: 1.subalternation, 2. contradiction, 3. contrariety, and 4. subcontrariety. Note on the Square of Opposition Seen against the square above, Subaltenation can be traced with line starting from the topmost of both the right and left corners of the square and downward to their lowest counterparts. It shows then the relations between A to I and E to O or vice versa. Contrariety is the line connecting A and E or vice versa. Subconrariety is the line between I and O or vice versa. Contradiction is the diagonal line inside the square connecting A to O and E to I or vice versa. Subalternation is the opposition between a universal and particular affirmative (A and I),and between a universal and particular negative (E and O). Both propositions, the universaland particular, are called „subalterns‟; the universal (A and E) being the „subalternant,‟ whilethe particular (I and O) is the „subalternate.‟ Contradiction is the opposition existing between the universal affirmative (A) and aparticular negative (O), and between universal negative (E) and a particular affirmative (I). Contrariety is the opposition existing between a universal affirmative (A) and a universalnegative (E). Subcontrariety is the opposition existing between a particular affirmative (I) and a particularnegative (O). Laws of Logical Opposition. Our mind, in its natural state, that is, even without yet aformal training or orientation in scientific logic, actually follows already patterns in reasoning orthinking. These various thinking patterns and structures are explicated as mental laws ormethods in a formal study we now familiarly know as logic. Some of these very basic lawsgovern the various relations existing among the four general propositions. The most practicalbenefit we can derive from the knowledge of these laws is that it makes us figure out easilywhether or not judgment or proposition and its possible implications can be true or false. Theseare the following laws. 1. Law of Subalternation: A-I and E-O There are two rules for this relation: the first rulestates that the truth of the universal involves the truth of the particular, but the truth of theparticular does not involve the truth of the universal. In other words, if A is true, I must also be
14. 14. true; and if E is true, O must also be true. If I is true, A need not be true, but is doubtful(doubtful means that the proposition is either true or false); if O is true, E need not be true, but isdoubtful. There are thus two sections to this first rule. The first section implies that it is always logicalto conclude from the truth of the universal to the truth of the particular. What is true of „all‟individuals of a class must also be true of „some‟ of these individuals, because what is true ofthe „whole‟ must be true of every „part‟ of the whole. If „All men are mortal,‟ then surely „Somemen are mortal.‟ And if „No men are pigs,‟ then „Some men are not pigs.‟ either. On the otherhand, if I is true, we cannot conclude, in virtue of the proposition as such, that A is also true; andif O is true, we cannot validly conclude that E is also true; What is true of „some‟ need not betrue of „all‟, because what is true of a „part‟ of a class need not be true of the „whole‟ of the class.Examples: „Because some men are left-handed,‟ it does not follow that „All men are left-handed.‟ and because „Some men are not polyglot,‟ it does not follow that „No men are polyglot.‟It might happen, of course, that what is true of „some‟ is also true of „all‟ and what is true of a„part‟ is also true of the „whole.‟ Examples: Both „Some men are mortal‟ (I), and „Some men arenot pigs‟ (O) are true. So: „All men are mortal‟ (A), and „No men are pigs‟ (E) are also true. We are, therefore, never warranted to conclude from the truth of the particular to the truth ofthe universal. It may be so, but it need not be so. In virtue of this phase of subalternation law,we cannot validly argue from „some‟ to „all‟ and from the „part‟ to the „whole.‟ Hence, the secondsection of the first rule is established: The truth of the particular does not involve the truth of theuniversal; the truth of the universal will always be doubtful. The second rule of the law of Subalternation states: The falsity of the particular involves thefalsity of the universal; but the falsity of the universal does not involve the falsity of theparticular. Here we begin with the falsity of one of the subaltern propositions (I to A, O to E),and the rule states: if I is false, A is also false; and if O is false, E is also false. But if A is false,I need not be false; and if E is false, O need not be false. There are thus two sections to this second rule. The first section implies that it is valid for themind to draw conclusion from the falsity of the particular proposition to the falsity of theuniversal. The reason is clear. In order that something be true of „all‟, it must be true of everyindividual that belongs to the „all‟; that something be true of the „whole,‟ it must be true of every„part‟ contained in the „whole‟. How, then, can something be true of „all‟ if it is false of „some‟?That would mean that „all‟ are, although „some‟ of the „all‟ are not; and it would follow that every„part‟ of the „whole‟ is, although a „part‟ of the „whole‟ is not; the same „some‟ and the same „part‟would then both „be‟ and „not be‟ something at the same time. Hence, if it is false that „Somemen are pigs,‟ it is all the more false to state that „All men are pigs‟; and if it is false that „Somemen are not mortal,‟ it is also false to say that „No men are mortal.‟ From the falsity, therefore,of the propositions, I or O, we must conclude to the falsity of the respective universalproposition, A or E. The second section of this second rule reads: If A is false, I need not be false; and if E isfalse, O need not be false. In order that a universal be true, every individual of the class andevery „part‟ of the „whole‟ must be included in the truth of the universal; hence, the universal willbe false, if not every individual of the universal and not every „part‟ of the „whole‟ is included inthe truth of the universal statement. This means that if a universal proposition is false, some ofits individuals must also be false, but some (of the others) may be true. But if „some‟ may betrue, even if the universal („all‟) is false, it is obvious that we cannot legitimately conclude fromthe falsity of the universal to the falsity of the particular. Thus, if „All men are learned‟ (A), and„No men are learned‟ (E) are both false, respectively, „Some men are learned‟ (I) and “Somemen are not learned‟ (O) are true. This proves definitely that that the falsity of the universaldoes not involve the falsity of the particular: the particular may be true, even if the universal isfalse. However, the particular may also be false. Examples: Both „All men are pigs‟ (A) and„No men are mortal‟ (E) are false; so, „Some men are pigs‟ (I) and, „Some men are not mortal‟
15. 15. (O) are also false. We thus see that, when the universal proposition is false, its respectiveopposite (subaltern) particular proposition may be either true or false. The falsity of theuniversal only entitles us to conclude that some of the individuals are false, leaving the matterundecided whether the others are true of false. Hence, the falsity of the universal propositions(A and E) does not involve the falsity of their respective opposite (subaltern) particularpropositions (I and O): the particular proposition may or may not be false together with theuniversal; the falsity of the particular will always be in doubt, when the universal proposition isfalse. We thus see the truth of the Law of Subalternation: the truth of the universal involves thetruth of the particular, but the truth of the particular does not involve the truth of the universal;the falsity of the particular involves the falsity of the universal, but the falsity of the universaldoes not involve the falsity of the particular. The Law of Contradiction. A-O and E-I. This law also has double phases, of which thefirst rule is: contradictories cannot be true together. If A is true, O is false; if O is true, A isfalse; if E is true, I is false; if I is true, E is false. In an affirmative universal (A) proposition, it isasserted that the predicate is affirmed of each and every individual belonging to the subject: „Allmen are mortal.‟ If this is true, then it must be false to deny this statement of „some‟; hence, thestatement that „Some men are not mortal‟ (O) cannot be true. In a negative universal (E)proposition, it is asserted that the predicate must be denied of each and every individualbelonging to the subject: „No men are pigs.‟ If this statement is true, then it must be false tosay that „Some men are pigs‟ (I). What is true of all must be true of every one of the class; tostate at the same time that „all are‟ and „some are not,‟ and that „none more‟ and „some are,‟would violate the Principle of Contradiction. Hence, if the universal affirmative (A) is true, theparticular negative (O) must be false; and if the universal negative (E) is true, the particularaffirmative (I) must be false. The law works also the opposite way: if O is true, A is false; and if I is true, E is false. If it istrue that „Some men are not intelligent‟, it is certainly false that „All men are intelligent‟; and ifit‟s true that „Some men are mortal,‟ it must be false to assert that „No men are mortal.‟ Andthus the rule is established: Contradictories cannot be true together. The second rule reads: Contradictories cannot be false together. If A is false, O is true;if E is false, I is true; if O is false, A is true; if I is false, E is true. If it is false that „All men areintelligent,‟ it must be true that „Some men are not intelligent.‟ (A-O). If it is false that „No menare learned,‟ it must be true that „Some men are learned.‟ (E-I). If it is false that „Some menare not mortal,‟ it must be true that „All men are mortal.‟ If it is false that „Some men are pigs,‟ itmust be true that „No men are pigs.‟ From the above, it will be clear that the contradictory pairs form a perfect opposition amongthemselves: they can be neither true nor false together; one must be true and the other must befalse. There is no neutral middle ground between contradictories: a thing either is or is not; ifthe one is true, the other must be false. Law of Contrariety: A-E. The rules are: Contraries cannot be true together; contrariesmay be false together. If A is true, E is false; if E is true, A is false. If A is false, E may be trueor false; if E is false, A may be true or false. The first rule states that contraries cannot be true together; if one of the contraries is true,the other contrary must be false. The correctness of this rule is easily demonstrated with thehelp of the Law of Subalternation and the Law of Contradiction. Suppose the universalaffirmative proposition (A) is true: „All men are mortal.‟ According to the Law of Contradiction, ifa universal proposition is true, its contradictory proposition must be false. Therefore, it is false tosay that „Some men are not mortal.‟ (I). Now, according to the Law of Subalternation, „thefalsity of the particular involves the falsity of the universal.‟ So, from the falsity of „Some menare not mortal,‟ (O) the statement, „No men are mortal‟ (E) is also false. Hence, if A is true, E isfalse, i.e., confirming that contraries cannot be true together.
16. 16. The same line of thinking applies if we begin with E, the universal negative, as true: „Nomen are pigs‟ (E). Since this is true, its contradictory (I) must be false, namely, „Some men arepigs.‟ And since this particular affirmative (I) is false, it also involves (according to the Law ofSubalternation) the falsity of the universal (A) “All men are pigs.‟ Therefore if E is true, A mustbe false; or the rule: Contraries cannot be true together. The second rule states that contraries may be false together; if one contrary is false, theother contrary may also be false (though it need not be false, but may be true). Examples: If„All men are intelligent‟ (A) is false, its contradictory, „Some men are not intelligent‟ (O), must betrue. But the Law of Subalternation states that „the truth of the particular proposition does notinvolve the truth of the universal.‟ Hence, although it is true that „Some men are not intelligent‟(O), we cannot conclude legitimately that „No men are intelligent‟ (E) is also true; E may be trueor false. Hence, both contraries may be false. Similarly, granting that „No professors arelearned‟ (E) is false, its contradictory, „Some professors are learned‟ (I) must be true. Again,„the truth of the particular proposition does not involve the truth of the universal.‟ Therefore, thetruth of A is not established; it may be true to say „All professors are learned‟; but the statementmay also be false. Contraries then may be false together. Law of Subcontrariety: I-O. The twofold rule here states: Both subcontraries cannot be falsetogether; but both subcontraries may be true together. The first rule demands: If I is false, O is true; if O is false, I is true. Examples: If „Some menare pigs‟ (I) is false, according to the Law of Contradiction, “No men are pigs‟ (E) must be true.Now, if a universal proposition is true, its particular proposition (according to the Law ofSubalternation) is also true. So, it must be true to state that „Some men are not pigs‟ (O). Wecan arrive at the same result in a different way. If (I) „Some men are pigs‟ is false, (A) „All menare pigs‟ (according to the Law of Subalternation) is also false; but if A is false, (O) „Some menare not pigs‟ (according to the Law of Contradiction) must be true. Hence if I is true, O must betrue. We arrive at the same conclusion, if we begin with the falsity of O. If O is false (‟Some menare not mortal.‟), its contradictory A („All men are mortal.‟) must, according to the Law ofContradiction, be true. But if A („All men are mortal.‟) is true, I („Some men are mortal.‟) must,according to the Law of Subalternation, also be true. Hence, if O is false, I must be true. Wethus see the first rule governing subcontrary propositions (I and O): Subcontraries cannot befalse together; at least one of the two must be true. The second rule of the subcontraries (I and O) states that both may be true together: if I istrue, O may be true; if O is true, I may be true. Suppose I is true: „Some Filipinos areCebuanos.‟ The contradictory of this proposition, namely E, must be false, and we cannotassert that „No Filipinos are Cebuanos.‟ But we know from the Law of Subalternation that „thefalsity of the universal does not involve the falsity of the particular.‟ Hence, even though E isfalse, we cannot conclude to the falsity of O: O may be true, that is, „Some Filipinos are notCebuanos.‟ We can also begin with O as true. If O is true, that is, “Some men are not learned‟,its contradictory A is false, and that is, „All men are learned.‟ Since, however, we cannotconclude from the falsity of the universal to the falsity of its particular (Law of Subalternation), itdoes not follow that I is also false: the I statement, that is, „Some men are learned‟ may be true. We have thus established the twofold rule governing the subcontrary propositions I and O:Both subcontraries cannot be false together; but both subcontraries may be true together. The following are legitimate conclusions from the study of the different oppositionalrelations of propositions (subalternation, contradiction, contrariety, and subcontrariety): If A is true: then I is true, E is false, O is false. If A is false: then O is true, E is doubtful, I is doubtful. If E is true: then O is true, A is false, I is false. If E is false: then I is true, A is doubtful, O is doubtful. If I is true: then E is false, A is doubtful, O is doubtful.
17. 17. If I is false: then O is true, A is false, E is true. If O is true: then A is false, E is doubtful, I is doubtful. If O is false: then I is true, E is false, A is true. Logical Opposition of Modals. The treatment of modal propositions is similar to theordinary categorical propositions, but the logical opposition affects the mode itself. The„necessary‟ mode resembles the A proposition; the „impossible mode, the E proposition; the„possible‟ mode, the I proposition; the „contingent‟ mode, the O proposition. The logicalopposition of the modal propositions may affect only the mode, or it may affect both the modeand the quantity of the propositions. Conclusion. The method of concluding from the truth or falsity of one statement to the truthor the falsity of another is called immediate inference. It is called „immediate‟, because we canpass directly from the one to the other, without the necessity of adducing any other idea orjudgment as proof. The Square of Opposition, therefore, with its relations of subalternation,contradiction, contrariety, and subcontrariety, will act as a powerful aid toward correct thinking.