Categorical Logic

The Building Blocks: Categorical Statements

Categorical statements assert a relationship between two classes, groups, or categories. There are only four types of relation that can exist between two classes. The following claims illustrate these four relations:

Cats are mammals.

Cats are not reptiles.

Cats are long-haired animals.

Cats are not short-haired animals.

To be as clear as possible about the particular relation asserted by a given claim, it is often useful to "translate" it into a standard form. Standard form categorical renderings of the above sentences result in:

All cats are mammals.

No cats are reptiles.

Some cats are long-haired animals.

Some cats are not short-haired animals.

These "translations" may be further simplified by using letters in place of the two class terms:

All C are M.

No C are R.

Some C are L.

Some C are not S.

These simplified representations highlight what is meant by the form, as distinct from the content, of a categorical statement. Consider the pure forms:

All ____ are ____.

No ____ are ____.

Some ____ are ____.

Some ____ are not ____.

These standard form categorical statements have descriptive names; traditionally, they have also have been assigned one-letter names.

A: Universal Affirmative

E: Universal Negative

I: Particular Affirmative

O: Particular Negative

Practice:

Translate the following ordinary sentences (statements) into their proper standard form, as outlined above. The answers follow, using letters for the subject terms and predicate terms, so you may check your skills.

Note that asking and answering two questions will determine the Form of the statement, leaving you to "fill in the blanks" for the subject and predicate terms. Ask: (1) Is the claim universal or particular? And (2) Is it affirmative or negative?

Dolphins live in the Atlantic Ocean.

Designated hitters are baseball players.

Whales are not fish.

Not all flowers are sweet-smelling.

Answers:

I: Some D are A.

A: All D are B.

E: No W are F.

O: Some F are not S.

Tips for less obvious translations:

When you see… Use

whoever, wherever, always, when, all

anyone, never, etc.

a few some

if…then, provided…,…, etc. all or no

unless no (if not)

only, none but, none except all (& switch term order)

the only all

not every, not all some…are not

there is some

all but, all except, few two statements required

Practice: Rephrase the following statements in standard categorical form:

Whoever works the hardest, achieves the most.

A few flowers actually stink.

If it looks and quacks like a duck, it is a duck.

Only Price Club members get the two percent discount.

Vitamin B12 is the only vitamin that must be derived from an external source.

Not all "A" students thoroughly learned the material.

There is a candidate who would really accomplish something in office.

Answers, in simplified form:

All H are A.

Some F are S.

All Q are D.

All D are P.

All E are B.

Some A are not T.

Some C are A.

Representing Categorical Claims Visually: Venn Diagrams:

Overlap two circles.

Label one for the subject term and the other for the predicate term.

Use shading (for universal, A and E, claims) to eliminate content. Note that you will always shade a crescent for an A claim and a "football" for an E claim.

Place an "x" (for particular, I and O, claims) to specify content. For I claims, the "x" is placed in the "football," where the subject and predicate categories overlap. For O claims, the "x" is placed in the crescent of the subject term circle.

Practice: Your instructor will refer you to resources for illustration and practice.

IMMEDIATE INFERENCE:

Once you know whether a given categorical statement is true or false, you also know, without the addition of more information, the truth value of certain corresponding categorical claims. Consider:

If it is true that "All cats are mammals" then:

"No cats are mammals" is false,

"Some cats are mammals" is true, and

"Some cats are not mammals" is false.

Consider a second case in which the "A" claim is false:

Knowing that "All cats are great pets" is false, then:

"No cats are great pets" may be either true or false,

"Some cats are great pets" may be either true or false, and

"Some cats are not great pets" is true.

Practice: Try this yourself with the E, I, and O claims. Explain your answers. For example: If it is true that every cat is a mammal then it must be true that some cats are mammals.

Traditionally, these immediate inference relationships have been represented on a "Square of Opposition" and the various relationships have been named.

A-O and E-I represent contradiction.

If one is true, the other is false and vice versa.

A-I and E-O represent subalternation.

If A/E is true, I/O is true; if I/O is false, A/E is false.

A-E represent contrariety.

If one is true, the other is false.

I-O represent subcontrariety.

If one is false, the other is true.

[see Square of Opposition]

Immediate Inference and Venn Diagrams: Although Venn diagrams are primarily used to evaluate cases of mediate inference, i.e., categorical arguments having two premises and a conclusion, they may also be used to evaluate immediate inference, where the conclusion is drawn from only one premise. For example, consider the following argument:

No alligators are crocodiles. (true)

Therefore, some alligators are crocodiles. (false)

[see diagram]

CATEGORICAL SYLLOGISMS:

Once you have identified an argument as a (deductive) categorical one, you will want to evaluate it for validity and for the truthfulness of its premises. Our focus here, however, will be solely on the validity requirement. Proceed as follows:

Put each premise and the conclusion in standard categorical (A, E, I, O) form. (This radical clarification of the premise claims will also make it easier to judge them true or false.)

Make sure there are three and only three distinct terms.

(One of these terms will occur in each premise, and is referred to as the Middle Term. The other two terms will each occur once in the conclusion and once in a premise.)

Using S as a placeholder for the subject term of the conclusion, P as a placeholder for the predicate term of the conclusion, and M as the argument’s middle term, we have the following bare bones syllogism:

All M are P Major Premise

All S are M Minor Premise

All S are P Conclusion

Note that categorical arguments may be precisely identified by specifying two characteristics, referred to as Mood and Figure.

The mood of the above argument is AAA, as every claim in it is an A claim, a Universal Affirmative statement. Simply identify premises and conclusion as A, E, I, or O statements (in top down order) and you have the mood of the argument. Identify the location of the Middle Term and you have the Figure of the argument. The above argument is in the first figure, notated thus: AAA-1

There are only four locations the middle term can occupy, namely:

First Figure ___ M ___ P

___ S ___ M

___ S ___ P

Second Figure M

Third Figure M

Fourth Figure M

PRACTICE: Complete the above. Construct several arguments and identify them by mood and figure.

Using Venn Diagrams to Evaluate Categorical Arguments for Validity:

Overlap three circles, one for each term in the argument.

Mark the diagram so that it represents the claims made by each premise. For a universal claim, shade the appropriate region – either a football or a crescent. For a particular claim, place an x in the appropriate region, i.e., in either a football or a crescent. Note: If the football or crescent region is entirely unshaded, the x will be placed on the line bisecting the football or crescent. If the region is partially shaded, the x will be placed in the unshaded sub-region of the football or crescent.

Having completed the argument (premises) diagram, separately create a two-circled Venn diagram for the conclusion.

Finally, answer the following question: Is the conclusion mark, which I see in my conclusion diagram, already present in the three-circled premises diagram? If yes, the argument is valid; if no, the argument is invalid.

Note on how this works: A conclusion can only NECESSARILY (which is what validity requires) follow from premises if it is really ALREADY IMPLICITLY CONTAINED in the premises. So, if we can picture the information claims of the premises, we can simply look and see whether that picture also pictures the conclusion claim.

Practice: Your instructor will direct you to resources for illustration and practice.

PROPOSITIONAL LOGIC

The Building Blocks: Simple Statements and Four Connectives:

Simple: Atlanta is an international city.

The door is ajar.

Negation: Atlanta is not an international city.

The door is not ajar.

Conjunction: Atlanta is an international city and the city of trees.

The door is ajar and no one is at home.

Disjunction: Atlanta is an international city or the city of trees.

The door is ajar or the light is playing tricks on my eyes.

Implication: If Atlanta is an international city then it is the city of trees.

If the door is ajar then someone must be home.

While sentences, of course, may be combinations of the above, any combination will itself always be either a negation, a conjunction, a disjunction, or an implication. For example:

If Atlanta is an international city and the city of trees, then people either consider Atlanta a highly desirable place to live or they are provincial in their outlook.

This sentence is fundamentally an implication, an "if…then" statement. Its antecedent is a conjunction and its consequent is a disjunction. This structure is clearer if we simplify it, using convenient letters to stand for the simple statements contained in the sentence. Thus: If I and T, then D or P.

Take a moment and make your own simplifications of the above statements.

Simplifying statements in this way makes their meaning as clear as possible and makes it easier to determine whether they are true or false. It also makes it easier to determine whether, in argument, one claim (the conclusion) really follows from others (the premises).

Question: When is a negation true?

Answer: When its simple statement is false.

Q: When is a conjunction true?

A: When both its simple statements (conjuncts) are true.

Q: When is a disjunction true?

A: When at least one of its simple statements (disjuncts) is true.

Q: When is an implication, a conditional statement, true?

A: This one is a little odd. It is certainly true when both its simple statements, the antecedent and consequent, are true. It is certainly false when its antecedent is true and its consequent is false. It is also treated as true whenever its antecedent is false.

Symbols are sometimes used to further simplify the expression of statements. Typically:

negation = ~

conjunction = &

disjunction = v

implication = >

Truth Tables are sometimes used to illustrate the above. Your instructor will refer you to resources on this.

Note that words other than "not" sometimes express negation, words other than "and" often express conjunction, words other than "or" may express disjunction, and words other than "if…then" often express implication. For example:

None of the lottery tickets were winners. ~W

Studying is hard but also satisfying. H & S

Your desert choices are pie and cake. P v C

You will do well provided you study. S > W

(Note that here the antecedent and consequent are in reverse order in the "untranslated" sentence.)

The word "unless" may be rendered as "or" or as "if not…then."

Consider this sentence: Unless the Braves improve their bullpen pitching, they will lose the World Series.

This sentence may be rephrased as "Either the Braves improve their bullpen pitching or they will lose the World Series."

Put simply: I v L (improve or lose)

The same sentence may be written as "If the Braves do not improve their bullpen pitching then they will lose the World Series."

Put simply: ~ I > L

SIX VALID ARGUMENT FORMS IN PROPOSITIONAL LOGIC:

Only a little reflection should reveal to you that we daily use (or confuse and misuse) certain recurring patterns of reasoning. Knowing these patterns, really knowing them, will directly improve your reasoning and protect you from the mis-reasoning of others.

Modus Ponens: Affirming the Antecedent:

If A then B.

Yes, A.

So, B.

If Jake goes to the concert then he’ll miss our study session.

Jake is going to the concert.

So, Jake will miss our study session.

G > M

Modus Tollens: Denying the Consequent:

If A then B.

Not B

So, not A

If the car runs then there is gas in the tank.

There is no gas in the tank.

So, the car does not run.

R > G

Pure Hypothetical Syllogism:

If A then B.

If B then C.

Therefore, If A then C.

If Jake goes to the concert then he’ll miss our study session.

If he misses our study session, he’ll likely fail the exam.

So, If Jake goes to the concert, he’ll likely fail the exam.

G > M

M > F

G > F

Disjunctive Syllogism: Denying a Disjunct:

Either A or B.

Not A

Therefore, B.

Either Michael Jordan retires now or his reputation will suffer.

He is not going to retire.

So, his reputation will suffer.

R v S

Two Very Common Invalid Argument Form: Watch out for these: And recall again what it means for an argument to be valid: It means that it is impossible for the conclusion to be false when the premises are true.

Affirming the consequent:

If the car runs, there is gas in the tank.

There is gas in the tank.

So, the car runs.

R > G

Denying the antecedent:

If the car runs, there must be gas in the tank.

The car is does not run.

Therefore, the gas tank must be empty.

R > G

PRACTICE:

Construct arguments of your own for each of the above forms, four valid and two invalid.
Find an example or two of each of the valid forms. You will, undoubtedly, need to rephrase the arguments you find to make their structure (and meaning) clear.

Two Common but more Complex Argument Forms:

Constructive Dilemma:

A v B

A > X

B > Y

X v Y

Destructive Dilemma:

~X v ~Y

A > X

B > Y

~A v ~B

PRACTICE:

Try your hand at making up arguments that follow the form of the dilemmas. Start by simply replacing the letters with real claims. Then, through a little trial and error modification of the claims, the argument will begin to make sense. Soon, you will have constructed two very nice examples of these forms of reasoning.

DISTINGUISHING VALID FROM INVALID FORMS BY VISUAL INSPECTION ALONE: Remember that an argument is valid if and only if it exhibits one of the six valid forms identified above. Your instructor will refer you to resources for this practice.

TRUTH TABLES MAY BE USED TO DETERMINE THE VALIDITY OF PROPOSITIONAL ARGUMENTS OF ANY LENGTH OR OMPLEXITY.

Your instructor will refer you to resources on this.