Categorical logic was discovered long long ago by Aristotle (384-322 BCE).

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Lecture 10

Categorical Logic

Categorical statements

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Why do we need Categorical Logic?

• Propositional logic does not cover all valid logical forms.
• It is an important part of logic, but not the only part of logic.
• Is the following argument valid?
• All human beings are mortal;
• Socrates is a human being;
• So, Socrates is mortal
• This is a valid argument. Can it be explained by propositional logic? Not quite.

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• Argument:

All human beings are mortal;

Socrates is a human being;

So, Socrates is mortal

• Analysis (Modus Ponens?):

If Socrates is a human being, then Socrates is mortal.

Socrates is a human being;

So, Socrates is mortal

• But statement 1 and statement (a) are not quite the same thing.

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• Another argument
• Some four-legged creatures are gnus.
• All gnus are herbivores.
• Therefore, some four-legged creatures are herbivores.
• Is this argument valid?

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• Argument:
• Some four-legged creatures are gnus.
• All gnus are herbivores.
• Therefore, some four-legged creatures are herbivores.
• This argument is valid, but it cannot be explained by propositional logic.
• We need some kind of new logic.

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Categorical Logic

• Categorical logic was discovered long long ago by Aristotle (384-322 BCE).
• The arguments we have discussed are called syllogism.
• Aristotle discovered all valid forms of syllogism.

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Categorical Logic

• Categorical logic studies inference between categorical statements.
• It is the focus of today’s lecture to explain the structure of categorical statements.
• Typical inferences between categorical statements:
• Syllogism:
• Case: All As are Bs, All Bs are Cs, so all As are Cs.
• Latin Square:
• Case: All As are Bs; therefore it is not the case that some As are not Bs.
• All As are Bs; so Some As are Bs.
• Conversion:
• Case: All As are Bs; so All Bs are As.

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Categorical Statements

• Statements in categorical logic have a specific structure.
• All human beings are mortal.
• Some four-legged creatures are gnus.
• They have the following structure:
• quantifier + subject phrase + <link verb ‘be’> + predicate phrase
• All categorical statements are structured like this.
• The link verb ‘be’ is called ‘copula’.

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Quantification

• Categorical statements are different from statements in propositional logic: they have quantifiers to quantify a statement.
• Consider:
• All human beings are mortal.
• Some people are rich.
• No pigs are able to fly.
• Here, all, some, and no are quantifiers.

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Subject phrase and Predicate phrase

• Subject phrase and predicate phrase are understood as classes, i.e. a set of objects that fall in the phrase .
• Human being: it is a class/set of all objects that are human beings.
• Mortal: it is a class/set of all objects that are mortal.
• Socrates: this is an one-man class, which contains only one person-Socrates.

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Ordinary language

• Sometimes the predicate phrases are not explicitly referring to a class; but we can always transform them to its corresponding class.
• Sometimes copula is not present either, and we can add one.
• Examples:
• I love apples. == I am the one who loves apples
• Bees are angry == Bees are one of the things that are angry.

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Four Kinds of Categorical Statements

• All S are P.
• It asserts that every member of class S is a member of class P.
• Some S are P.
• At least one member of S is also in the class P.
• Note the use of the word ‘Some’: it does not imply that there are more than one member of S is in P.
• No S are P.
• No member of S is in the class P.
• Some S are not P.
• At least one member of S is not in the class P.

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Traditional Terminology

• In traditional logic, each of four kinds of categorical statements is given a names.
• A: All S are P. (Universal Affirmative)
• E: No S are P. (Universal Negative)
• I: Some S are P. (Particular Affirmative)
• O: Some S are not P. (Particular Negative)

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Further explanations

• There are two ways to characterize a categorical statement:
• Quality: whether the statement confirms (a confirmative) or negates (a negative);
• Quantity: whether it has a universal (all) or a particular quantifier (some).
• Types of Categorical Statements
• A: all As are Bs: it is a Universal Confirmative.
• E: no As are Bs: it is the same as “All As are not Bs”; so it is a Universal Negative.
• I and O statements can be similarly understood.

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Translations

• In ordinary language, categorical statements are often not put in a standard form. So translations are often needed in order for us to have a precise and accurate understanding of these statements.
• The standard form is also important for us to capture the argument relation between categorical statements.

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Missing Copula or Quantifier

• Copula or quantifiers are not present in these statements:
• Dogs love meat.
• Workers should get paid.
• Solution: add the missing parts (without changing the meanings of the statements)
• All dogs are the animals that love meat.
• All workers are the people who should get paid.

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Terms without Nouns

• Some statements may not have nouns as predicates:
• Roses are red;
• All ducks swim.
• Solution: replace them with a noun phrase or a noun clause
• All roses are red things.
• All ducks are animals that swim.

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Singular Statements

• What about statements with a singular subject term?
• G. W. Bush is a good president.
• George loves Starbucks.
• Rule: treat singular statement as A-statement.
• Singular term is an one-object class. Then it says about everything in the class.
• G. W. Bush is a good president == all people who are identical with G. W. Bush is a good president.
• George loves Starbucks==all people who is identical with George love Starbucks.

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Other expressions for the Universal Quantifier

• Cases
• Every soldier is a warrior.
• Each one of you is responsible.
• Whoever is a doctor earns a lot of money.
• Any tiger can be dangerous.
• These are all A-statements; these quantifiers are the same as ‘all’.
• Every soldier is a warrior.
• All soldiers are warriors.

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E-Statement

• Cases:
• Nobody loves Ray.
• Nothing is better than pure love.
• None of the animals are alive.
• Rule: these are all E-statements. Treat these quantifiers as ‘no’.
• Nobody loves Ray.
• No people are the people who love Ray.

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I-Statement

• Cases
• Most students are honest.
• Many people are retiring late.
• A few dogs got killed.
• At least one person is missing.
• Almost all the cats have four legs.
• There are government employees who are spies.
• Rule: these are all I-statements. Treat all these quantifiers as ‘some’.
• There is a significant difference between ‘most’ ‘many’ and ‘some’, but it is beyond our concern here.

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O-Statements

• Cases:
• Not all the rich people are smart.
• Some rich people are not smart.
• There are government employees who are not qualified.
• Some government employees are not qualified.
• Most Democrats are not in favor of the war.
• Some Democrats are not in favor of the war

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Other Structures

• Not all A are B ≠ No A are B.
• Not all A are B:
• This is an O-Statement: some As are not Bs.
• No A are B
• This is an E-Statements. No As are Bs.
• Case:
• Not all Republicans support the war.
• No Republicans support the war.

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‘Only’ & ‘only if’

• ‘Only’?
• Only rich people are invited.
• ‘only if’?
• Only if one studies logic one gets smart.
• Translation rule:
• These statements are A-statements. The term after ‘only’ or ‘only if’ is predicate term.
• Only rich people are invited == all invited people are rich ones.
• Only if one studies logic one gets smart. == all smart people are those who study logic.

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‘The Only’

• Case:
• The only guests invited are boys.
• Cockroaches are the only survivors.
• Translation rule:
• These are A-statements; the term that occurs after ‘the only’ is the subject term.
• Translation:
• The only guests invited are boys == all guests invited are boys.
• Cockroaches are the only survivors == all survivors are cockroaches.

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Try these exercises in the book!

• Page 263, Ex. 7.2: Questions No. 4, 6, 8, 10, 14, 15.
• Page 263-4, Ex. 7.3: No. 10, 13, 14, 17.

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Ex. 7.2

• #4 People who whisper lie.
• All people who whisper are people who lie.
• A-statement
• #6 Only if something has a back beat is it a rock-and-roll song.
• All rock-and-roll songs are things with a back beat.
• A-statement (note the predicate and the subject)
• Only, only if: what follows are predicates of an A-statement
• The only: what follows are subjects of an A-statement
• #8: Nothing that is a snake is a mammal.
• No snakes are mammals. E-statement.

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• #10: The only good human is a dead human.
• All good humans are dead humans.
• A-statement.
• #14: There is no excellence without difficulty.
• No excellence is without difficulty.
• E-statement.
• Or: All excellent things are things with difficulty.
• A-statement
• #15: Jonathan is not a very brave pilot.
• No people who are identical to Jonathan are very brave pilots.
• E-statement.
• Is this an A-statement? Not really.

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A statement vs. E-statement

• A statement: All S are P.
• E-statement: No S are P.
• What about?
• All S are not P: E-statement
• No S are not P: A-statement
• Why? Venn diagrams show they are so.
• Compare:
• Some S are P.
• Some S are not P.

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Ex. 7.3: No. 10, 13, 14, 17

• 10: People who love only once in their lives are shallow people. (Oscar Wilde)
• All the people who love only once in their lives are shallow people.
• A-statement
• 13: Many socialists are not communists.
• Some socialists are not communists.
• O-statement

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• 14: All prejudices may be traced to the intestines.
• All prejudices are things that may be traced to the intestines.
• A-statement.
• 17: He that is born to be hanged shall never be drowned.
• All people who are born to be hanged are people who shall never be drowned.
• A -statement
• No people who are born to be hanged are the people who shall be drowned
• E-statement

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Quiz

• In this quiz you will be give a set of categorical statements in English, and you are asked to translated them into standard form of a categorical statement, and indicate its type (A, E, I, O).
• It is similar to the exercises we did in the previous slides.