Categorical Logic

nLecture 10

nCategorical Logic

nCategorical statements

nWhy do we need Categorical Logic?

nPropositional logic does not cover all valid logical forms.

nIt is an important part of logic, but not the only part of logic.

nIs the following argument valid?

nAll human beings are mortal;

nSocrates is a human being;

nSo, Socrates is mortal

nThis is a valid argument. Can it be explained by propositional logic? Not quite.



1.All human beings are mortal;

2.Socrates is a human being;

3.So, Socrates is mortal

nAnalysis (Modus Ponens?):

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

b)Socrates is a human being;

c)So, Socrates is mortal

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



nAnother argument

nSome four-legged creatures are gnus.

nAll gnus are herbivores.

nTherefore, some four-legged creatures are herbivores.

nIs this argument valid?



nSome four-legged creatures are gnus.

nAll gnus are herbivores.

nTherefore, some four-legged creatures are herbivores.

nThis argument is valid, but it cannot be explained by propositional logic.

nWe need some kind of new logic.

nCategorical Logic

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

nThe arguments we have discussed are called syllogism.

nAristotle discovered all valid forms of syllogism.


nCategorical Logic

nCategorical logic studies inference between categorical statements.

nIt is the focus of today’s lecture to explain the structure of categorical statements.

nTypical inferences between categorical statements:


nCase: All As are Bs, All Bs are Cs, so all As are Cs.

nLatin Square:

nCase: All As are Bs; therefore it is not the case that some As are not Bs.

nAll As are Bs; so Some As are Bs.


nCase: All As are Bs; so All Bs are As.

nCategorical Statements

nStatements in categorical logic have a specific structure.

nAll human beings are mortal.

nSome four-legged creatures are gnus.

nThey have the following structure:

nquantifier + subject phrase + <link verb ‘be’> + predicate phrase

nAll categorical statements are structured like this.

nThe link verb ‘be’ is called ‘copula’.


nCategorical statements are different from statements in propositional logic: they have quantifiers to quantify a statement.


nAll human beings are mortal.

nSome people are rich.

nNo pigs are able to fly.

nHere, all, some, and no are quantifiers.

nSubject phrase and Predicate phrase

nSubject phrase and predicate phrase are understood as classes, i.e. a set of objects that fall in the phrase .

nHuman being: it is a class/set of all objects that are human beings.

nMortal: it is a class/set of all objects that are mortal.

nSocrates:  this is an one-man class, which contains only one person-Socrates.

nOrdinary language

nSometimes the predicate phrases are not explicitly referring to a class; but we can always transform them to its corresponding class.

nSometimes copula is not present either, and we can add one.


nI love apples. == I am the one who loves apples

nBees are angry == Bees are one of the things that are angry.

nFour Kinds of Categorical Statements

nAll S are P.

nIt asserts that every member of class S is a member of class P.

nSome S are P.

nAt least one member of S is also in the class P.

nNote the use of the word ‘Some’: it does not imply that there are more than one member of S is in P.

nNo S are P.

nNo member of S is in the class P.

nSome S are not P.

nAt least one member of S is not in the class P.

nTraditional Terminology

nIn traditional logic, each of four kinds of categorical statements is given a names.

nA: All S are P.  (Universal Affirmative)

nE: No S are P. (Universal Negative)

nI: Some S are P. (Particular Affirmative)

nO: Some S are not P. (Particular Negative)

nFurther explanations

nThere are two ways to characterize a categorical statement:

nQuality: whether the statement confirms (a confirmative) or negates (a negative);

nQuantity: whether it has a universal (all) or a particular quantifier (some).

nTypes of Categorical Statements

nA: all As are Bs: it is a Universal Confirmative.

nE: no As are Bs: it is the same as “All As are not Bs”; so it is a Universal Negative.

nI and O statements can be similarly understood.


nIn 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.

nThe standard form is also important for us to capture the argument relation between categorical statements.

nMissing Copula or Quantifier

nCopula or quantifiers are not present in these statements:

nDogs love meat.

nWorkers should get paid.

nSolution: add the missing parts (without changing the meanings of the statements)

nAll dogs are the animals that love meat.

nAll workers are the people who should get paid.

nTerms without Nouns

nSome statements may not have nouns as predicates:

nRoses are red;

nAll ducks swim.

nSolution: replace them with a noun phrase or a noun clause

nAll roses are red things.

nAll ducks are animals that swim.

nSingular Statements

nWhat about statements with a singular subject term?

nG. W. Bush is a good president.

nGeorge loves Starbucks.

nRule: treat singular statement as A-statement.

nSingular term is an one-object class. Then it says about everything in the class.

nG. W. Bush is a good president == all people who are identical with G. W. Bush is a good president.

nGeorge loves Starbucks==all people who is identical with George love Starbucks.

nOther expressions for the Universal Quantifier


nEvery soldier is a warrior.

nEach one of you is responsible.

nWhoever is a doctor earns a lot of money.

nAny tiger can be dangerous.

nThese are all A-statements; these quantifiers are the same as ‘all’.

nEvery soldier is a warrior.

nAll soldiers are warriors.



nNobody loves Ray.

nNothing is better than pure love.

nNone of the animals are alive.

nRule: these are all E-statements. Treat these quantifiers as ‘no’.

nNobody loves Ray.

nNo people are the people who love Ray.



nMost students are honest.

nMany people are retiring late.

nA few dogs got killed.

nAt least one person is missing.

nAlmost all the cats have four legs.

nThere are government employees who are spies.

nRule: these are all I-statements. Treat all these quantifiers as ‘some’.

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



nNot all the rich people are smart.

nSome rich people are not smart.

nThere are government employees who are not qualified.

nSome government employees are not qualified.

nMost Democrats are not in favor of the war.

nSome Democrats are not  in favor of the war


nOther Structures

nNot all A are B  ≠  No A are B.

nNot all A are B:

nThis is an O-Statement: some As are not Bs.

nNo A are B

nThis is an E-Statements. No As are Bs.


nNot all Republicans support the war.

nNo Republicans support the war.

n‘Only’ & ‘only if’


nOnly rich people are invited.

n‘only if’?

nOnly if one studies logic one gets smart.

nTranslation rule:

nThese statements are A-statements. The term after ‘only’ or ‘only if’ is predicate term.

nOnly rich people are invited == all invited people are rich ones.

nOnly if one studies logic one gets smart. == all smart people are those who study logic.

n‘The Only’


nThe only guests invited are boys.

nCockroaches are the only survivors.

nTranslation rule:

nThese are A-statements; the term that occurs after ‘the only’ is the subject term.


nThe only guests invited are boys == all guests invited are boys.

nCockroaches are the only survivors == all survivors are cockroaches.

nTry these exercises in the book!

nPage 263, Ex. 7.2: Questions No. 4, 6, 8, 10, 14, 15.

nPage 263-4, Ex. 7.3: No. 10, 13, 14, 17.

nEx. 7.2

n#4 People who whisper lie.

nAll people who whisper are people who lie.


n#6 Only if something has a back beat is it a rock-and-roll song.

nAll rock-and-roll songs are things with a back beat.

nA-statement (note the predicate and the subject)

nOnly, only if: what follows are predicates of an A-statement

nThe only: what follows are subjects of an A-statement

n#8: Nothing that is a snake is a mammal.

nNo snakes are mammals. E-statement.


n#10: The only good human is a dead human.

nAll good humans are dead humans.


n#14: There is no excellence without difficulty.

nNo excellence is without difficulty.


nOr: All excellent things are things with difficulty.

n A-statement

n#15: Jonathan is not a very brave pilot.

nNo people who are identical to Jonathan are very brave pilots.


nIs this an A-statement? Not really.

nA statement vs. E-statement

nA statement: All S are P.

nE-statement: No S are P.

nWhat about?

nAll S are not P:   E-statement

nNo S are not P:   A-statement

nWhy? Venn diagrams show they are so.


nSome S are P.

nSome S are not P.

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

n10: People who love only once in their lives are shallow people. (Oscar Wilde)

nAll the people who love only once in their lives are shallow people.


n13: Many socialists are not communists.

nSome socialists are not communists.



n14: All prejudices may be traced to the intestines.

nAll prejudices are things that may be traced to the intestines.


n17: He that is born to be hanged shall never be drowned.

nAll people who are born to be hanged are people who shall never be drowned.

nA -statement

nNo people who are born to be hanged are the people who shall be drowned



nIn 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).

nIt is similar to the exercises we did in the previous slides.


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