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Chapter 5

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Chapter 5

Categorical Propositions

- 1. A deductive argument is one that claims to establish its conclusion conclusively.
- A valid deductive argument is one in which, if all the premises are true, the conclusion must be true.
- The theory of deduction aims to explain the relationship between premises and conclusion in valid arguments. It also aims to provide methods for evaluating deductive arguments.
- There are two major logical theories that have been developed to accomplish these aims: Aristotelian (or Classical) logic and Modern symbolic Logic.
- Chapters 5, 6, and 7 will cover Aristotelian logic. Chapters 8, 9, and 10 will cover Modern symbolic logic.

- Classical deductive (or Aristotelian) logic is based on the idea of categories, or classes.
- A categorical proposition affirms or denies, in whole or in part, that one class is included in another.
- Since categorical propositions deal with two states of two classes, there are four possible categorical propositions.
- (1) The universal affirmative (A) proposition states that every member of one class is also a member of the second class.
- (2) The universal negative (E) states that no member of one class is a member of the second.
- (3) In a particular affirmative (I), some members of one class are members of the second
- (4) And in a particular negative (O), some members of one class are not members of the second.

- The letter "S" stands for the class designated by the subject term of the proposition. The letter "P" stands for the class designated by the predicate term. Substituting any class-defining words for S and P generates actual categorical propositions.
- In classical theory, the four standard-form categorical propositions were thought to be the building blocks of all deductive arguments. Each of the four has a conventional designation: A for universal affirmative propositions; E for universal negative propositions; I for particular affirmative propositions; and O for particular negative propositions.

- All politicians are liars.
- All men are mortal.
- All good Web pages are written in html.
- All good men come to the aid of their party.
- All men have what it takes to become a successful salesman.
- All dogs go to heaven
- The universal affirmative as stated by Aristotle is not two-way. Consider, for example, example 1 above. It doesn’t mean that all liars are politicians. In example 2, we are not stating that all mortal things are men. (This “reverse” of the orignial statement is called the converse of the statement)

- No politician is intelligent.
- No man is immortal.
- No good Web pages contain Java or browser-specific tags.
- No good men will betray their principles.
- No men have what it takes to be a successful mother.
- The universal negative is effectively two-way, unlike the universal affirmative. In other words, a universal negative statement does imply its converse. For instance, in Example 1, we propose that no politicians are intelligent, and therefore imply that no intelligent people are politicians. In Example 2, we not only say that no men are immortal, but that no immortal beings are men. This is the most important distinction between the universal affirmative and the universal negative, functionally speaking.

- Some man is mortal.
- There is a woman who is a politician.
- At least one computer runs Microsoft products.
- There is a fun Web site.
- The particular affirmative states that there is at least one member of one class that is a member of a second.
- It doesn't imply that all members of one class are members of the second.These sentences sound strange: a more natural language might say that "This Web site is fun." or "Socrates is mortal." However, at this stage of the development of our logical language, we want to be able to distinguish between saying that there is at least one fun Web site and that a specific Web site is fun. While it is true that if this Web site is fun then there is at least one Web site that is fun, it doesn't necessarily follow that if there is at least one Web site that is fun, that this one is. You might think this Web site was lame and Yahoo was fun, for instance.
- This takes us naturally to the first thing to remember about the particular affirmative: It isn't exactly right to talk about the particular affirmative having a converse in the same way that it is to say that a universal term has one.
- However, there is an implication involved in certain natural language statements that forms the basis for most proofs of particular affirmative statements. It's simple: to prove that some A is B, all you need to do is find one example of when A is B, and bingo! You're done.

- Some fictional creatures are not mortal.
- Some Web sites are not fun.
- Some philosophers don't make sense.
- Some computers are not expensive.
- The connection between the particular affirmative and negative is easy to see. In fact, in our natural language, we often don't make much of a distinction between the two: modern logic doesn't either.
- For example, when you think of the negative particular statement "Some woman is not beautiful." it seems equivalent to the affirmative particular statement "Some woman is homely." On further examination, we see that this is only true if every woman is either beautiful or homely.
- Similarly to the particular affirmative, the particular negative can be proven by finding a single example. For instance, if we want to prove that some politician is corrupt, all we have to do is find one corrupt politician.

- Exercises P. 187

- Propositions are said to have quality—either affirmative or negative—and quantity—either universal or particular.
- Quality: A and I are affirmative (help AffIrmo – Latin for I affirm); E and O are negative (help nEgO- Latin for I deny)
- Quantity: A and E are universal; I and O are particular.
- In categorical propositions we use variations of the verb “to be” to connect the subject and predicate terms – called a copula
- Propositions may also be distributed or undistributed:
- A proposition is said to distribute a term if it refers to all members of the class designated by the term.
- All dogs go to heaven -
- In the A proposition above, for example, the subject term (dogs) is distributed, but the predicate term (things that go to heaven) is not.

- No men are immortal
- In the E proposition above, the subject term (men) is distributed because the whole class of men is excluded from the class of immortality; it also asserts that all immortals (predicate term) are excluded from being men.
- Some Mexicans come illegally
- In the I proposition, no assertion is made about Mexicans (subject) and no assertion is made about those who come illegally (predicate). NOT DISTRIBUTED
- Some male dancers are not strippers
- In this O proposition, nothing is said about all male dancers (subject), it says that only part of male dancers are excluded from the class of strippers. But these male dancers are excluded from the whole of the stripper class (predicate). Given the particular dancers being referred to, the proposition says that each and every member of the class of strippers is not one of these particular male dancers. Therefore, the whole class is referred to and the predicate class is DISTRIBUTED.
- Exer. P 193

- “Unprepared Students Never Pass”
Universals distribute Subjects.

Negatives distribute Predicates.

- “Any Student Earning B’s Is Not On Probation”
- A distributes Subject.
- E distributes Both.
- I distributes Neither.
- O distributes Predicate.

- The traditional square of opposition graphically displays the relationships that exist between the four different standard form categorical propositions: A,E,I, O.
- Propositions can be contradictories, contraries, subcontraries, subalterns, or superalterns.

- A and O propositions are contradictory, as are E and I propositions. Propositions are contradictory when the truth of one implies the falsity of the other, and conversely. Here we see that the truth of a proposition of the form All S are P implies the falsity of the corresponding proposition of the form Some S are not P. For example, if the proposition “all industrialists are capitalists” (A) is true, then the proposition “some industrialists are not capitalists” (O) must be false. Similarly, if “no mammals are aquatic” (E) is false, then the proposition “some mammals are aquatic” must be true.
- They cannot both be true and cannot both be false.

- A and E propositions are contrary. Propositions are contrary when they cannot both be true; if one is true, then other must be false. They can both be false. An A proposition, e.g., “all giraffes have long necks” cannot be true at the same time as the corresponding E proposition: “no giraffes have long necks.” Note, however, that corresponding A and E propositions, while contrary, are not contradictory. While they cannot both be true, they can both be false, as with the examples of “all planets are gas giants” and “no planets are gas giants.”
- By saying that ‘some’ rather than all or none, both statements would be false

- I and O propositions are subcontrary. Propositions are subcontrary when it is impossible for both to be false; if one is false the other must be true. They can both be true. Because “some lunches are free” is false, “some lunches are not free” must be true. Note, however, that it is possible for corresponding I and O propositions both to be true, as with “some nations are democracies,” and “some nations are not democracies.” Again, I and O propositions are subcontrary, but not contrary or contradictory.

- Two propositions are said to stand in the relation of Subalternation when the truth of the first (“the superaltern”) implies the truth of the second (“the subaltern”), but not conversely. A propositions stand in the Subalternation relation with the corresponding I propositions. The truth of the A proposition “all plastics are synthetic,” implies the truth of the proposition “some plastics are synthetic.” However, the truth of the O proposition “some cars are not American-made products” does not imply the truth of the E proposition “no cars are American-made products.” In traditional logic, the truth of an A or E proposition implies the truth of the corresponding I or O proposition, respectively. Consequently, the falsity of an I or O proposition implies the falsity of the corresponding A or E proposition, respectively. However, the truth of a particular proposition does not imply the truth of the corresponding universal proposition, nor does the falsity of an universal proposition carry downwards to the respective particular propositions.

- Inferences from square of opposition:
- A number of very useful immediate inferences may be readily drawn from the information embedded in the traditional square of opposition. Given in the truth, or the falsehood, of anyone of the four standard form categorical propositions, it will be seen that the truth or falsehood of some or all of the others can be inferred immediately.
- A being given as True: E is false; I is true; O is false.
- E being given as True: A is false; I is false; O is true.
- I being given as True: E is false; A and O are undetermined.
- O being given as True: A is false; E and I are undetermined.
- A being given as False : O is true , E and I are undetermined.
- E being given as False: I is true; A and O are undetermined.
- I being given as False: A is false; E is true; O is true.
- O being given as False: A is true; E is false; I is true.

If true:AAll men are wicked creatures. If false:

falseENo men are wicked creatures undetermined

trueISome men are wicked creatures. undetermined

falseOSome men are not wicked creatures. true

- If true:ENo men are wicked creatures. If false:
falseAAll men are wicked creatures undetermined

falseISome men are wicked creatures. true

trueOSome men are not wicked creatures. undetermined

- If true:ISome men are wicked creatures. If false:
- Undetermined AAll men are wicked creatures false
- FalseENo men are wicked creatures. true
- undetermined O Some men are not wicked creatures. True
- If true:OSome men are not wicked creatures. If false:
- falseAAll men are wicked creatures true
- undeterminedENo men are wicked creatures. false
- undeterminedISome men are wicked creatures. true

- The square of opposition has three important kinds of immediate inference that are not directly associated with the square of opposition:
- (1) Conversion
- (2) Obversion
- (3) Contraposition

- Conversion
- An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid.
- Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false.
- Steps for Conversion: Reversing the subject and the predicate terms in the premise.
- Valid Conversions
- ConvertendConverse

- Example:
- All bags are mangoes.-A
- Some mangoes are bags.-I
- No men are intelligent.-E
- No intelligent are men.-E
- Some cows are tables.-I
- Some tables are cows.-I
- Some students are not cats.
- (not valid)

- Conversion is valid in the case of E and I propositions. “No women are American Presidents,” can be validly converted to “No American Presidents are women.”
- An example of an I conversion: “Some politicians are liars,” and “Some liars are politicians” are logically equivalent, so by conversion either can be validly inferred from the other.
- Note that the converse of an A proposition is not generally valid form that A proposition.
- For example: “All bananas are fruit,” does not imply the converse, “All fruit are bananas.”
- A combination of subalternation and conversion does, however, yield a valid immediate inference for A propositions. If we know that "All S is P," then by subalternation we can conclude that the corresponding I proposition, "Some S is P," is true, and by conversion (valid for I propositions) that some P is S. This process is called conversion by limitation.

- ConvertendA proposition: All IBM computers are things that use electricity. Converse A proposition: All things that use electricity are IBM computers.
- ConvertendA proposition: All IBM computers are things that use electricity. Corresponding particular: I proposition: Some IBM computers are things that use electricity. Converse (by limitation)I proposition: Some things that use electricity are IBM computers.
- The first part of this example indicates why conversion applied directly to A propositions does not yield valid immediate inferences. It is certainly true that all IBM computers use electricity, but it is certainly false that all things that use electricity are IBM computers.
- Conversion by limitation, however, does yield a valid immediate inference for A propositions according to Aristotelian logic. From "All IBM computers are things that use electricity" we get, by subalternation, the I proposition "Some IBM computers are things that use electricity." And because conversion is valid for I propositions, we can conclude, finally, that "Some things that use electricity are IBM computers."

- The converse of“Some S is not P,” does not yield an valid immediate inference.
- ConvertendO proposition: Some dogs are not cocker spaniels.Converse O proposition: Some cocker spaniels are not dogs.
- This example indicates why conversion of O prepositions does not yield a valid immediate inference. The first proposition is true, but its converse is false.

- Obversion
- An inference formed by changing the quality of a proposition and replacing the predicate term by its complement. Obversion is valid for any standard form Categorical proposition.
- Obversion is the only immediate inference that is valid for categorical propositions of every form. In each of the instances, the original proposition and its obverse must have exactly the same truth-value, whether it turns out to be true or false.
- Steps for Obversion:
- Replace the quality of the given statements. That is, if affirmative, change it into negative, and if negative, change it into affirmative.
- Replace the predicate term by its complementary term.
- Valid Obversions
- Obverted Obverse
- A: All S is P. E: No S is non-P.
- E: No S is P A: All S is non-P.
- I : Some S is P O : Some S is not non-P
- O: Some S is not P I: Some S is non-P

- Obversion - A valid form of immediate inference for every standard-form categorical proposition. To obvert a proposition we change its quality (from affirmative to negative, or from negative to affirmative) and replace the predicate term with its complement. Thus, applied to the proposition "All cocker spaniels are dogs," obversion yields "No cockerspaniels are nondogs," which is called its "obverse." The proposition obverted is called the "obvertend."

- Example:
- All females are perfect beings.-A
- No females are non-perfect beings.-E
- No female are perfect beings.-E
- All female are non-perfect beings.-A
- Some female are perfect beings.-I
- Some females are not non-perfect beings.-O
- Some female are not perfect beings.-O
- Some female are non-perfect beings.-I

- The obverse is logically equivalent to the obvertend. Obversion is thus a valid immediate inference when applied to any standard-form categorical proposition.
- The obverse of the A proposition "All S is P" is the E proposition "No S is non-P."
- The obverse of the E proposition "No S is P" is the A proposition "All S is non-P."

- The obverse of the I proposition "Some S is P" is the O proposition "Some S is not non-P."
- The obverse of the O proposition "Some S is not P" is the I proposition "Some S is non-P."
- ObvertendA-proposition: All cartoon characters are fictional characters. ObverseE-proposition: No cartoon characters are non-fictional characters.
- ObvertendE-proposition: No current sitcoms are funny shows. ObverseA-proposition: All current sitcoms are non-funny shows.

- ObvertendI-proposition: Some rap songs are lullabies. ObverseO-proposition: Some rap songs are not non-lullabies.
- ObvertendO-proposition: Some movie stars are not geniuses. ObverseI-proposition: Some movie stars are non-geniuses.

- As these examples indicate, obversion always yields a valid immediate inference.
- If every cartoon character is a fictional character, then it must be true that no cartoon character is a non-fictional character.
- If no current sitcoms are funny, then all of them must be something other than funny.
- If some rap songs are lullabies, then those particular rap songs at least must not be things that aren't lullabies.
- If some movie stars are not geniuses, than they must be something other than geniuses.

- An inference formed by replacing the subject term of a proposition with the complement of its predicate term, and replacing the predicate term by the complement of its subject term. Not all contrapositions are valid.
- Contraposition is a reliable immediate inference for both A and O propositions; that is, the contrapositive of any A or O proposition is true if and only if the original proposition was true. Thus, in each of the pairs, both propositions have exactly the same truth-value.
- Note: In contraposition the subject of the conclusion is contradictory of the predicate of the premise and predicate of the conclusion is contradictory of the subject of the premise.
- Steps for Contraposition:
- a. Convert the statement: reverse the subject and the predicate terms.
- b. Replace both terms by their complementary terms.
- Valid Contrapositions
- Premises Contrapositive
- A: All S is P. A: All non-P is non-S.
- E: No S is P O: Some non-P is not non-S.
- (By limitation)
- I : Some S is P (Contraposition not valid)
- O: Some S is not P O: Some non-P is not non-S.

- Contraposition is a process that involves replacing the subject term of a categorical proposition with the complement of its predicate term and its predicate term with the complement of its subject term.
- Contraposition yields a valid immediate inference for A propositions and O propositions. That is, if the proposition
- All S is P is true, then its contrapositive
All non-P is non-S is also true.

- For example:
- Premise
- A proposition: All logic books are interesting things to read.
- Contrapositive
- A proposition: All non interesting things to read are non logic books.

- The contrapositive of an A proposition is a valid immediate inference from its premise. If the first proposition is true it places every logic book in the class of interesting things to read. The contrapositive claims that any non-interesting things to read are also non-logic books—something other than a logic book—and surely this must be correct.

- Premise:
- I-proposition: Some humans are non-logic teachers.
- Contrapositive
- I-proposition: Some logic teachers are not human.
- As this example suggests, contraposition does not yield valid immediate inferences for I propositions. The first proposition is true, but the second is clearly false.

- E premise:
- No dentists are non-graduates.
- The contrapositive is: No graduates are non-dentists.
- Obviously this is not true.

- The contrapositive of an E proposition does not yield a valid immediate inference. This is because the propositions "No S is P" and "Some non-P is non-S" can both be true. But in that case "No non-P is non-S," the contrapositive of "No S is P," would have to be false.
- A combination of subalternation and contraposition does, however, yield a valid immediate inference for E propositions. If we know that "No S is P" is true, then by subalternation we can conclude that the corresponding O proposition, "Some S is not P," is true, and by contraposition (valid for O propositions) that "Some non-P is not non-S" is also true. This process is called contraposition by limitation.

- Premise:
- E-proposition: No Game Show Hosts are Brain Surgeons.
- Contrapositive
- E proposition: No non-Brain Surgeons are non-Game show hosts.
- Premise:
- E proposition: No game show hosts are brain surgeons.
- Corresponding particular O proposition: Some game show hosts are not brain surgeons.
- Contrapositive
- O proposition: Some non-brain surgeons are not non-game show hosts.

- The first part of this example indicates why contraposition applied directly to E propositions does not yield valid immediate inferences. Even if the first proposition is true then the second can still be false. This may be hard to see at first, but if we take it apart slowly we can understand why. The first proposition, if true, clearly separates the class of game show hosts from the class of brain surgeons, allowing no overlap between them. It does not, however, tell us anything specific about what is outside those classes. But the second proposition does refer to the areas outside the classes and what it says might be false. It claims that there is not even one thing outside the class of brain surgeons that is, at the same time, a non-game show host. But wait a minute. Most of us are neither brain surgeons nor game show hosts. Clearly the contrapositive is false.

- Contraposition by limitation, however, does yield a valid immediate inference for E propositions according to Aristotelian logic. By subalternation from the first proposition we get the O proposition "Some game show hosts are not brain surgeons." And then by contraposition, which is valid for O propositions, we get the valid, if tongue-twisting O proposition, "Some non-brain surgeons are not non-game show hosts."

- O proposition.
- Premise:
- Some flowers are not roses.
- Some non-roses are not non-flowers.
- This is valid. Thus we can see that contraposition is a valid form of inference only when applied to A and O propositions. Contraposition is not valid at all for I propositions and is valid for E propositions only by limitation.

- Example:
- All citizens are voter.-A
- All non-voters are non-citizens.-A
- No politicians are honest.-E
- Some-non-honest are not non-politicians.-O
- (by limitation)
- Some applicants are graduate. -I
- (cannot be contraposited)
- Some students are not scholarship holders.-O
- Some non-scholarship holders are not non-students.-O
- Exer. P. 205-207

- The problem of existential import presents some problems for the relationships suggested by the traditional square of opposition.
- As a result, most modern logicians adopt a different interpretation of the square, called Boolean.
- Under this interpretation, particular propositions (I and O) have existential import; but universal propositions (A and E) do not.

- Aristotelian logic suffers from a dilemma that undermines the validity of many relationships in the traditional Square of Opposition. Mathematician and logician George Boole proposed a resolution to this dilemma in the late nineteenth century. This Boolean interpretation of categorical propositions has displaced the Aristotelian interpretation in modern logic.

- The source of the dilemma is the problem of existential import. A proposition is said to have existential import if it asserts the existence of objects of some kind. I and O propositions have existential import; they assert that the classes designated by their subject terms are not empty. But in Aristotelian logic, I and O propositions follow validly from A and E propositions by subalternation. As a result, Aristotelian logic requires A and E propositions to have existential import, because a proposition with existential import cannot be derived from a proposition without existential import.

- A and O propositions with the same subject and predicate terms are contradictories, and so cannot both be false at the same time. But if A propositions have existential import, then an A proposition and its contradictory O proposition would both be false when their subject class was empty.
- For example:
- Unicorns have horns. If there are no unicorns, then it is false that all unicorns have horns and it is also false that some unicorns have horns.

- The Boolean interpretation of categorical propositions solves this dilemma by denying that universal propositions have existential import. This has the following consequences:
- I propositions and O propositions have existential import.
- A-O and E-I pairs with the same subject and predicate terms retain their relationship as contradictories.
- Because A and E propositions have no existential import, subalternation is generally not valid.
- Contraries are eliminated because A and E propositions can now both be true when the subject class is empty. Similarly, subcontraries are eliminated because I and O propositions can now both be false when the subject class is empty.

- Some immediate inferences are preserved: conversion for E and I propositions, contraposition for A and O propositions, and obversion for any proposition. But conversion by limitation and contraposition by limitation are no longer generally valid.
- Any argument that relies on the mistaken assumption of existence commits the existential fallacy.

- The result is to undo the relations along the sides of the traditional Square of Opposition but to leave the diagonal, contradictory relations in force.

- Diagrams and symbolizing techniques are useful in helping to visualize the relationships of categorical propositions.
- Venn diagrams are especially effective at exhibiting the relationships between classes by marking and shading overlapping circles.

The relationships among classes in the Boolean interpretation of categorical propositions can be represented in symbolic notation. We represent a class by a circle labeled with the term that designates the class.Thus the class S is diagrammed as shown below:

- To diagram the proposition that S has no members, or that there are no S’s, we shade all of the interior of the circle representing S, indicating in this way that it contains nothing and is empty. To diagram the proposition that there are S’s, which we interpret as saying that there is at least one member of S, we place an x anywhere in the interior of the circle representing S, indicating in this way that there is something inside it, that it is not empty.

- To diagram a standard-form categorical proposition, not one but two circles are required. The framework for diagramming any standard-form proposition whose subject and predicate terms are abbreviated by S and P is constructed by drawing two intersecting circles:

- If Tweety is a canary, then
- Tweety is a bird.
- Tweety is a canary.Valid/sound
- Tweety is a bird.
- If Tweety is a bird, then Valid/unsound
- Tweety is a canary.
- Tweety is a bird.
- Tweetyis a canary.
- If Tweety is a canary, then
- Tweety is a bird.Invalid
- Tweety is a bird.
- Tweety is a canary.
- Using if p then q
- Euler Diagram (circles)

- 1. What are the properties of A, E, I, and O propositions? Come up with examples of propositions for each of these types of standard form categorical propositions.
- 2. What do affirmative propositions have in common? What do particular propositions have in common? What about universal and negative propositions? How do the terms quality and quantity come into play in these considerations?
- 3. What is the difference between contraries and contradictories? Between contraries and subcontraries?
- 4. When does conversion result in valid inferences? Why does it work then, but not in other cases? Consider the same question with contraposition and obversion as well.
- 5. Why is existential import so problematic for Aristotelian logic? What changes does it require to the square of opposition?

- 1. What are the options for dealing with the question of existential import? Why should we adopt one option over the other?
- 2. What is the meaning of the traditional square of opposition? How does the position of each proposition exhibit the relationships between them? What inferences does it illustrate?
- 3. When using Venn diagrams, what do shading, overlapping, and “x” mean? How is each of the standard-form categorical propositions diagrammed using this method?
- 4. What does existential import entail? Which propositions have existential import? Why is this a problem for interpreting the traditional square of opposition?
- 5. What changes to the square of opposition result from the Boolean interpretation of existential import?