Topic 11 Categorical Logic PDF

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Categorical Logic....

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Topic 11 (Deductive) Categorical Logic •

This is Shyam Ranganathan’s significantly revised version of revised slides by Jeff Vancha, University of Saskatchewan based on slides designed for Lewis Vaughn’s Power of Critical Thinking 4rth Ed. This mashup is protected by 29.2 of Canada’s Copyright Modernization Act of 2012.

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The purpose of these notes is to familiarize students with the topic explored in this course: critical thinking with a yogic foundation. On their own, they do not constitute a critical thinking textbook. Hence, they are not a substitute for the original text by Vaughn, nor are they a substitute for the slides that are published to be used in consort with this text. Unchanged text from previous versions concern matters of general knowledge in the field, for which no one has copyright.

Categorical and propositional logic 

Categorical logic (this topic) studies the relations between categories (kinds, well defined sets) of well-defined objects. 



Propositional logic (next topic) studies logical relations between propositions. 

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Tools are Venn diagrams and certain calculation rules.

Tools are truth values and truth-tables.

Deductive Reasoning: Categorical Logic, cont’d

Categorical logic 

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The basic unit of concern in categorical logic is the category, which may be either the subject or predicate of a thought. 

We study the relationship not between propositions but between the subject and predicate components of a proposition.



Examples: 

In the proposition “Paul is bald,” Paul is the subject, bald is the predicate.



In the proposition “No cats are reptiles,” cats is the subject, reptiles is the predicate.

Deductive Reasoning: Categorical Logic, cont’d

Individuals and Categories 

Subjects can be either individual entities (such as Paul) or categories of entities (such as the set of all bald persons).



When we say “Paul is bald,” categorical logic interprets this as, “Paul is a member of the category of bald entities.”

Dr. Shyam Ranganathan



When we say “Birds can fly,” this means “Birds belong to the category of entities that can fly.”



Categorical logic is thus all about category membership.

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Deductive Reasoning: Categorical Logic, cont’d

Categorical propositions 

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In categorical reasoning the propositions, or claims, of interest are categorical propositions. 

Categorical propositions make up simple assertions about the membership of categories (kinds, sets) of things.



For example: 

“All cows are herbivores.”



“No gardeners are plumbers.”



“Some business people are cheaters.”

Deductive Reasoning: Categorical Logic, cont’d

Why categorical logic? 

There are several reasons why categorical logic is studied. 

We often use this type of reasoning.



From the time of Aristotle to the nineteenth century, in the West, categorical logic was mistakenly believed to be all of logic; hence it is important for historical reasons, too.

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Problems with Categorical Logic



Categorical logic, while seemingly tracing the relationships between categories, really concerns the relationship between kinds: this is a specific variety of category. 

A kind is a category whose membership criterion is instantiated by its members.



For instance, red is a kind: all things that are in this category display redness.



Fruit Salad (a collection of different pieces of fruit) is not a kind: things that are part of fruit salad aren’t necessarily a collection of different pieces of fruit.

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Classes



Though not usually noted, a category like Fruit Salad, though not a kind, is a class.

Dr. Shyam Ranganathan



Classes are categories defined by membership criteria that do not have to instantiated by the members.

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Problems with Categorical Logic



The problem, given the distinction between classes and kinds, is that kinds might be defined in a manner that prohibit their members from being part of a class, and yet, a class definition can allow for it. 

Orange is a kind: pieces of orange share the essence of an orange, which is not a collection of different pieces of fruits. So pieces of orange can’t be fruit salad.



And yet, low and behold, given the class definition of fruit salad, pieces of orange can be part of a fruit salad.

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Problems with Categorical Logic



The problem, given the distinction between classes and kinds, is that kinds might be defined in a manner that prohibit their members from being part of a class, and yet, a class definition can allow for it. 

Orange is a kind: pieces of orange share the essence of an orange, which is not a collection of different pieces of fruits. So pieces of orange can’t be fruit salad.



And yet, lo and behold, given the class definition of fruit salad, pieces of orange can be part of a fruit salad.

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Historical Example

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In my own research, I discovered that the religion “Hinduism” is a class invented by the British (it’s membership criterion is: South Asian, no common founder). So one can be a Hindu and also a Christian, for instance.

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But when you look at “Christian” or most religions, they are kinds: and one can’t be a Christian and thereby a Hindu.

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This problem resulted in most all scholars trying to treat Hinduism as a kind, but then they were basically making stuff up about Hinduism.

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Why bother with Categorical Logic?



It has historical importance as noted.

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But also, understanding categorical logic is part of understanding its limitations.

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Propositions and Categories

Dr. Shyam Ranganathan

Terms 

The words in categorical claims that name categories (categories) of things are called terms. 

Each categorical proposition has both a subject term and a predicate term.



Example: “All cats are carnivores.” 

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The subject term here is cats, and the predicate term is carnivores.

Propositions and Categories

Introducing S and P 

We can express the form of such statements like this: “All S are P.” 

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By convention, S stands for the subject term in a categorical statement; P stands for the predicate term.

Propositions and Categories

A, E, I, and O 

The statement “All S are P” is one of four standard forms of categorical statements. 

All S are P. (All cats are carnivores.) 



No S are P. (No cats are carnivores.) 

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Particular (or Existential) Affirmative (I)

Some S are not P. (Some cats are not carnivores.) 

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Universal Negative (E)

Some S are P. (Some cats are carnivores.) 



Universal Affirmative (A)

Particular (or Existential) Negative (O)

Propositions and Categories

It’s all about category membership 

All categorical statements are about the membership of entities in a category.

Dr. Shyam Ranganathan



“All S are P” asserts that any member of the S category is included in the P category. 



“No S are P” asserts that no member of S is included in P. 

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e.g., all members of the category of cats are included in the category of carnivores.

e.g., no members of the category of cats are part of the category of herbivores.

Propositions and Categories

It’s all about category membership, cont’d 

“Some S are P” asserts that at least one member of the S category is also a member of the P category. 



e.g., some members of the category of plants are also members of the category of trees.

“Some S are not P” asserts that at least one member of the S category is not a members of the P category. 

e.g., some members of the category of ocean-dwellers are not members of the category of fish.

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Propositions and Categories

Noun phrases 

Examples so far have had individual words (nouns) naming a category. 





But subject and predicate terms can also be noun phrases. For example: 

Cats that live outdoors

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Animals with fur

So statements like this are also categorical: 

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Cats, herbivores, trees, etc.

“Some animals with fur shed in the house.”

Propositions and Categories

Who can be a term? 

The only things that can serve as subjects or predicates are: 

Nouns (mice, cars, rhymes, etc.)

Dr. Shyam Ranganathan

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

Pronouns (I, me, you, she, etc.)



Noun phrases (people with dandruff, old-fashioned computers)

Propositions and Categories

Adjectives 

Adjectives (red, carnivorous, pretty, etc.) have to be translated into a statement about a category: 

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“All cats are furry” should be read as, “All cats are furry things” or “Any cat belongs to the category of furry things.”

Propositions and Categories

The copula  The copula is a verb or verb phrase (present in some languages) that links the subject and the predicate. 

The copula is shown in bold in the following examples: 

“No cat is a bird.”



“All wombats come from Australia.”



“All whales are mammals.”



“Some fish are not sharks.”

Propositions and Categories Quantity 

Every categorical statement has a quantifier, a word that shows to how many members of a category the statement applies.

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There are two types of quantifiers: 

The universal quantifier: “all,” “none.”

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The particular quantifier: “some.”

Dr. Shyam Ranganathan

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Thus, the quantity of a categorical statement is whether it is universal or particular.

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Propositions and Categories

The meaning of “some” 

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In categorical (and predicate) logic, “some” means “at least one.” 

This could include “all”! (But not necessarily…)



In the case of continuous entities such as water, it means “a non-zero amount.”

The particular quantifier is also called the existential quantifier. 

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But that term is used mostly in predicate logic.

Propositions and Categories

To be or not to be 

A universal such as, “All mice love cheese,” does not assert the existence of cheese-loving mice. 

It just says, “If something is a mouse then it loves cheese.”



Why is this important? There can be universal statements that are true, but apply to nothing. (e.g. All Jedi Masters use the force.)

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A particular such as, “Some mice love Roquefort,” does assert the existence of at least one Roquefort-loving mouse.

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Propositions and Categories

Quality 

Categorical statements come in two qualities: affirmative and negative.

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An affirmative categorical statement affirms that one category is partially or totally included in another.





“All mice are rodents.”



“Some mice like cheese.”

A negative categorical statement denies that one category is partially or totally included in another. 

“No mouse can fly.”

Dr. Shyam Ranganathan



“Some mice do not get caught.”

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Translations and Standard Form

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Many categorical statements you’ll run into don’t seem to fit any of the four patterns. –

Part of the job of assessing categorical arguments is translating the categorical statements found “in the wild” into the tamer and clearer configurations of the standard forms.

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Translations and Standard Form, cont’d

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In order to check the validity of a categorical argument, we have to translate the claims into standard categorical form. –

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This will bring out the underlying structure of the statements.

Translations and Standard Form, cont’d

Pattern of categorical statements 

All standard-form categorical statements (in English) have the following pattern:

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Quantifier ― Subject Term ― Copula ― Predicate Term

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Terms

Identify the terms 

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Ensure that they designate categories by turning adjectives into category descriptions: 

[Original]: “All dogs are loyal.”



[Translation]: “All dogs are loyal individuals.”

Terms, cont’d

Order the terms 

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Put subject and predicate in the standard order: 

[Original]: “Beyond the mountains stood the redwood trees.”



[Translation]: “The redwood trees stood beyond the mountains.”

Terms, cont’d

Dr. Shyam Ranganathan

“Only” 

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The words “only” and “only if” precede the predicate term in an A-statement: 

[Original]: “Only palm readers are wise advisors.”

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[Translation]: “All wise advisors are palm readers.”



[Original]: “Only if something is a music file is it an MP3.”

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[Translation]: “All MP3s are music files.”

Terms, cont’d

“The only” 

The words “the only” precede the subject term in an A-statement: 

[Original]: “Hamburgers are the only real junk food.”



[Translation]: “All real junk foods are hamburgers.”

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[Original]: “The only crimes prosecuted are murders.”

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[Translation]: “All prosecuted crimes are murders.”

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“The only” = “all”!

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Terms, cont’d

Singular Statements 

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Treat singular terms (such as names) as if they were categories: 

[Original]: “Jamie Foxx is an actor.”

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[Translation]: “Anyone identical to Jamie Foxx is an actor.”

This is awkward but it’s the best we can do with categorical logic. 

(Singular statements are handled in a more natural way in predicate logic.)

Quantifiers •

Universal quantifiers may be non-standard: –

[Original]: “Every hockey players is an athlete.”

–

[Translation]: “All hockey players are athletes.”

Dr. Shyam Ranganathan

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–

[Original]: “Whoever is an artist is a genius.”

–

[Translation]: “All artists are geniuses.”

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[Original]: “The lion is a carnivore.”

–

(This is idiomatic in English, to use “the” as a preposition for a category.)

–

[Translation]: “All lions are carnivores.”

Quantifiers, cont’d

Negative universals 

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These are usually pretty obvious: 

[Original]: “Nothing for sale is truly valuable.”



[Translation]: “No items for sale are truly valuable.”

Quantifiers, cont’d

Implicit (unexpressed) quantifiers 

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Usually an unexpressed quantifier is a universal: 

[Original]: “Sharks are good swimmers.”

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[Translation]: “All sharks are good swimmers.”

But sometimes an unexpressed quantifier is probably meant to be a particular: 

[Original]: “Trent students are radicals.”

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[Translation]: “Some Trent students are radicals.”

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You have to be sensitive to context.

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Quantifiers, cont’d

Some nonstandard particular quantifiers • 

“There are” and “there exist”: 

[Original]: “There are (there exist) government workers who are spies.”



[Translation]: “Some government workers are spies.”

Dr. Shyam Ranganathan

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

Translate “most” as “some”:



[Original]: “Most movie stars are snobs.”



[Translation]: “Some movie stars are snobs.”

Quantifiers, cont’d

Some nonstandard particular quantifiers, cont’d 

Translate “several” as “some”: 

[Original]: “Several politicians are space aliens.”



[Translation]: “Some politicians are space aliens.”

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Diagramming Categorical Statements

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You can graphically represent the relationship between subject and predicate terms with the use of Venn diagrams. –

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The diagrams consist of overlapping circles, each one representing a category specified by a term in a categorical statement.

Diagramming Categorical Statements, cont’d

A, E, I, and O 

There are four basic kinds of categorical statements: A, E, I, and O. 

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So, there are exactly four basic kinds of Venn diagrams.

Diagramming Categorical Statements, cont’d

Venn diagrams for A-statements 

(Below, the shaded areas are empty)



All S are P

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Diagramming Categorical Statements, cont’d

A-statements explained 

This diagram means that all members of the S category are also members of the P category.

Dr. Shyam Ranganathan



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Example: “All wombats are marsupials.” 

The shaded area is empty.



That is, there do not exist any wombats that are not marsupials.

Diagramming Categorical Statements, cont’d

Venn diagrams for E-statements 

No S are P

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Diagramming Categorical Statements, cont’d

E-statements explained 

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Here the overlap is shaded (empty). 

Example: “No mice are reptiles.”



This is means that no members of S are also members of P.

Diagramming Categorical Statements, cont’d

Venn diagrams for I-statements 

(Below, the ‘x’...