Categorical Logic
Categorical statements

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Propositional logic does not cover all valid logical forms.
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It is an important part of logic, but not the only part of logic.
Is the following argument valid?
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All human beings are mortal;
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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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1.
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Argument:
All human beings are mortal;
Socrates is a human being;
So, Socrates is mortal a) b) c)
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
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Some four-legged creatures are gnus.
All gnus are herbivores.
Therefore, some fourlegged creatures are herbivores.
Is this argument valid?

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

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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 studies inference between categorical statements .
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It is the focus of today’s lecture to explain the structure of categorical statements.
Typical inferences between categorical statements:
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Syllogism:
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Case : All As are Bs, All Bs are Cs, so all As are Cs.
Latin Square:
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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:
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Case: All As are Bs; so All Bs are As .

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

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

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

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

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

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There are two ways to characterize a categorical statement:
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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
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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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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.
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The standard form is also important for us to capture the argument relation between categorical statements.

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

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

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What about statements with a singular subject term?
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G. W. Bush is a good president.
George loves Starbucks.
Rule: treat singular statement as A-statement .
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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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Cases
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Every soldier is a warrior.
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Each one of you is responsible.
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Whoever is a doctor earns a lot of money.
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Any tiger can be dangerous.
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These are all A-statements; these quantifiers are the same as ‘ all ’.
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Every soldier is a warrior.
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All soldiers are warriors.

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

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Cases
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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’.
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There is a significant difference between ‘most’ ‘many’ and
‘some’, but it is beyond our concern here.

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

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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’?
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Only rich people are invited.
‘only if’?
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Only if one studies logic one gets smart .
Translation rule:
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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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Case:
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The only guests invited are boys.
Cockroaches are the only survivors .
Translation rule:
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These are A-statements ; the term that occurs after ‘ the only ’ is the subject term.
Translation:
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The only guests invited are boys == all guests invited are boys.
Cockroaches are the only survivors == all survivors are cockroaches.

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

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#4 People who whisper lie.
All people who whisper are people who lie.
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A-statement
#6 Only if something has a back beat is it a rock-androll song.
All rock-and-roll songs are things with a back beat.
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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.
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A-statement.
#14: There is no excellence without difficulty.
No excellence is without difficulty.
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E-statement.
Or: All excellent things are things with difficulty.
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A-statement
#15: Jonathan is not a very brave pilot.
No people who are identical to Jonathan are very brave pilots.
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E-statement.
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Is this an A-statement? Not really.

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

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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.
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A-statement
13: Many socialists are not communists.
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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.
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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.
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A -statement
No people who are born to be hanged are the people who shall be drowned
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E-statement

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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).
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It is similar to the exercises we did in the previous slides.