Categorical Logic Categorical statements

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Categorical Logic Categorical statements
Lecture 10 Categorical Logic Categorical statements

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.

Analysis (Modus Ponens?):
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. But statement 1 and statement (a) are not quite the same thing.

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

We need some kind of new logic.
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.

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

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.

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’.

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.

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.

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.

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.

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)

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.

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.

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.

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.

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.

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’. All soldiers are warriors.

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’. No people are the people who love Ray.

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.

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

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.

‘Only’ & ‘only if’ ‘Only’? ‘only if’? Translation rule:
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.

‘The Only’ Case: Translation rule: Translation:
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.

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.

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.

#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. Is this an A-statement? Not really.

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.

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

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

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.