Logic Chp 4

Venn Diagram Rules for Non-Standard-Form Categorical Propositions

'X'=applies to particular propositions (i.e. "It is false the no A are B" and "it is false that all A are B") Shading=applies to universal propositions (i.e. "It is false that some A are not B" and "it is false that some A are B"); markings in venn diagrams are opposite to standard-form diagrams.

Parameter

A phrase that affects the structure, but not the meaning, of a statement. Parameters are used to translate singular propositions and include forms such as: "people identical to", "places identical to", "things identical to", "cases identical to", and "times identical to".

Categorical Proposition

A proposition (sentence that is either true or false) that consists of a quantifier, subject term, copula, and predicate term (all in the following order); asserts that either all part or part of the class denoted by the subject term is included or excluded from the class denoted by the predicate term; ex: "All or some or none of X are or are not Y."

Singular Propositions

A propositional statement that concludes something (or asserts) about a certain person, place, time, or thing; lacks a parameter, and typically has existential import.

Term Complement

A word or group of words that denote the class complement; a prefix such as "non-" is typically used when the class complement consists of a single word, and typically "things that are not" before class complements that consist of a group of words (ex: "People who drive cars" becomes "things that are not people who drive cars").

Conversion of Four Types of Categorical Propositions

A: "All S are P" becomes "All P are S" (OG: S circle is shaded; Con.: P circle is shaded) E: "No S are P" becomes "No P are S" (OG: middle circle is shaded; Con.: middle circle is shaded) I: "Some S are P" becomes "Some S are P" (OG: middle circle is marked with an 'X'; Con.: middle circle is marked with an 'X') O: "Some S are not P" becomes "Some P are not A" (OG: S circle is marked with an 'X'; Con.: P circle is marked with an 'X')

Contrapositive Categorical Proposition Forms

A: "All S are P" becomes "All non-P are non-A" [VALID/LOGICALLY EQUIVALENT] E: "No S are P" becomes "No non-P are non-S" [INVALID/LOGICALLY UNDETERMINED; everything outside of the venn diagram is shaded] I: "Some S are P" becomes "Some non-P are non-S" [INVALID/LOGICALLY UNDETERMINED; everything outside of the venn diagram is marked with an 'X'] O: "Some S are not P" becomes "Some non-P are not non-S" [VALID/LOGICALLY EQUIVALENT]

Obverse Statements

A: "All S are P" becomes "No S are non-P" E: "No S are P" becomes "All S are non-P" I: "Some S are P" becomes "Some S are not non-P" O: "Some S are not P" becomes "Some S are non-P" **Venn diagrams remain the same after obversion

Subcontrary Relation

According to the Aristotelian perspective of the traditional square of opposition; expresses partial opposition between determinate truth values between propositions I and O.

Unconditionally Valid

According to the modern/Boolean standpoint; arguments that are valid regardless if the terms denote existing things.

Immediate Inferences

Arguments that contain one premise the precedes the conclusion; validity for such arguments are tested through the MSO and assumes that the premise is true (ex: premise is an O proposition and the conclusion is an A proposition is claimed to be false; the contradictory relation of the two propositions renders the O proposition true, therefore the argument is valid).

Traditional Square of Opposition

Aristotelian perspective of the necessary relations between the four types of categorical propositions; due to Aristotle recognizing existential import in both universal (with the exception of universal propositions with things that do not actually exist) and particular propositions, there are more inferences in the square of opposition than the modern one possess.

Forms of Existential Fallacy

Begins with a universal proposition (premise), and ends with a particular proposition (conclusion). "All S are P. Therefore, some S are P." "It is false that some S are not P. Therefore, it is false that no S are P." "No S are P. Therefore, it is false that all S are P." "It is false that some S are P. Therefore, some S are not P."

Translating Ordinary Language Statements into Categorical Form: Terms without Nouns

Both the subject and predicate terms in a categorical proposition must include a plural noun or pronoun because it denotes the class; nouns and pronouns denote the class, while adjectives and participles connote class attributes. Ex: If a term in a categorical proposition only includes an adjective, then a plural noun or pronoun must be included to make the term denotative. So, "Some roses are red" becomes "Some roses are red FLOWERS (plural noun)."

Relations Between the Categorical Propositions as Interpreted by Aristotle

CONTRADICTORY: (A and O, and I and E props.) Opposition truth value. CONTRARY: (A and E props.) At least one, but not both, is false. SUBCONTRARY: (I and O props.) At least one, but not both, is false). SUBALTERNATION: (A and I, and E and O props.) Truth flows downward, falsity flows upward (so A and E props. are false).

Aristotelian Standpoint of Existential Import

Claims that ALL universal categorical propositions (A and E) about EXISTING things have existential import because they convey information about existing things denoted by the subject term; universal propositions relating to things that do not exist do not have existential import (ex: "Satyrs are interesting creatures" does not have existential import, whereas, "all pheasants are birds" does have existential import). Particular categorical propositions (I and O) are claimed to have existential import.

Boolean Standpoint of Existential Import

Claims that ALL universal categorical propositions (A and E) contain no existential import; claims that the propositions do not imply the existence of the things denoted by the subject term in all cases (ex: "no roses are daises" does not claim the existence of roses). Particular categorical propositions (I and O) are claimed to have existential import.

Copula

Connecting words such as "are", "are not", "have", "had", "is", "is not", etc.; in a categorical proposition, proceeds the subject term and precedes the predicate term; "copula"="couple".

Contrary Relation

Expresses partial opposition between determinate truth values between propositions A and E (as opposed to the modern square of opposition which concludes that A and E propositions have logically undetermined truth value); in other words, if an A proposition is false, then the adjacent E proposition must be true and vice-versa.

Existential Fallacy

Fallacy that occurs when a conclusion is attempted to be made with existential import (premises have existential import; actually exists) from premises that do not have existential import; arguments ALWAYS include a universal premise and a particular conclusion; begins with a universal proposition (premise), and ends with a particular proposition (conclusion).

"It is false that 'A'..." & "It is false that 'E'..."

False particular propositions

"It is false that 'O'..." & "It is false the 'I'..."

False universal propositions

Illicit Contraposition

Formal fallacy that occurs when a conclusion depends on the contraposition of an E or I proposition (logically undetermined). Ex: "Some S are P. Therefore, some non-P are non-S" and "No S are P. Therefore, no non-P are non-S."

Illicit Conversion

Formal fallacy that originates from arguments that depends on the conversion of logically undetermined (A and O) statements

Predicate Term

In a [standard form] categorical proposition proceeding from the copula; makes a claim about the subject(s). Ex: "Not all modern airplanes have jet engines", the predicate term is: "[things that have] jet engines".

Distribution

In a categorical proposition: "An attribute of the terms (subject and predicate) of propositions. A term is said to be distributed if the proposition makes an assertion about every member of the class denoted by the term; otherwise, it is undistributed." (taken from textbook pls I needed to plagiarize it <3).

Quantity

In a categorical proposition; asserts a statement that either some or all members belong to a class denoted by the subject term; is either universal or particular.

I Proposition

In a categorical proposition; particular+affirmative; "Some S are P", no distribution; the subject term S and the predicate term P are both undistributed.

O Proposition

In a categorical proposition; particular+negative; "Some S are not P", P is distributed; the subject term S is undistributed, and the predicate term P is distributed.

A Proposition

In a categorical proposition; universal+affirmative; "All S are P", 'S' is distributed; the subject term S is distributed, but the predicate term P is undistributed.

E Proposition

In a categorical proposition; universal+negative; 'No S are P", both are distributed; both the subject term S and the predicate term P are distributed.

Quality

In a categorical proposition; what it says about the class membership; is either negative or positive/affirmative.

Existential Import

In a categorical proposition; when it is denoted by the subject term that the subject class does in fact, exist (Ex: "All Tom Cruise's movies are hits", the subject term denotes that Tom Cruise has directed movies).

Obversion

In a standard-form categorical proposition; an operation that consists of (1) changing the quality (without changing the quantity; change the quantifier for A and E props.; change the copula for I and O props.), and (2) replacing the predicate term its term complement (words(s) that denote the class complement/everything outside of the denoted class); never contains a fallacy (i.e. "fallacy of obversion" does not exist).

Contraposition

In a standard-form categorical proposition; involves (1) switching the subject and predicate terms, and (2) replacing the subject and predicate terms with their term complement; only A and O contrapositive propositions are valid, while I and E statements are always invalid. Ex: "All whales are animals" becomes "All non-animals are non-whales."

Conversion

In a standard-form categorical proposition; involves switching the subject term and predicate term. Ex: The converse of "No dogs are reptilians" is, "No reptilians are dogs". [Quantifier] P [copula] S. (Only S and P are switched while the quantifier and copula remain the same)

Translating Ordinary Language Statements into Categorical Form: Singular Propositions

In order to translate a singular categorical proposition, a parameter must be introduced into the the statement to make it a universal, standard-form categorical proposition. Ex: "Socrates is mortal" becomes "All people identical to Socrates are people who are mortal", this however, makes causes the new form to lack existential import (according to Boole because it is universal) unlike the original.

Contradictory Relation

Not to be confused with contrary relation; same rules apply to the traditional square of opposition as the modern one (i.e. If the corresponding A or O, or E and I statement is false, then the corresponding statement is true and vice-versa).

Subject Term

Noun or noun phrase in a [standard form] categorical proposition; denotes the subject or subjects that are ABOUT to be claimed. Ex: "All cacti are succulents", "cacti" is the subject term".

Quantifier

Precedes the subject term in a categorical proposition, and expresses how much the subject term is included or excluded from the predicate class; ex: "all", "no", "none", "some", etc.

Class Complement

Refers to everything outside of the denoted class; ex: the class complement of "water" is everything outside of the class of "water" (i.e. stones, boats, etc.).

Affirmative Proposition

Relating to quality of a categorical proposition; asserts class membership (i.e. "All S are P", and "Some S are P").

Negative Proposition

Relating to quality of a categorical proposition; denies class membership (i.e. "No S are P", and "Some S are not P").

Particular Proposition

Relating to quantity in a categorical proposition; asserts something about one some ("some" means at least one in logic) members of the subject class (i.e. "Some S are P", and "Some S are not P").

Universal Proposition

Relating to quantity in a categorical proposition; asserts that either all members belong to a certain class (i.e. "All S are P", and "No S are P").

Modern Square of Opposition

Represents the relations between the four standard-form categorical propositions from the Boolean standpoint; A and O, and I and E propositions always have the opposite truth value (i.e. "contradictory relation"); the truth value of adjacent propositions can not be inferred and are said to have "logically undetermined truth value" (i.e. nothing can be inferred about the truth value of an 'E' proposition from an 'A' proposition).

Boolean Venn Diagrams

Shading=empty; 'X'=one particular thing exists in that area. A:"All S are P"=only 'S' is shaded. E:"No S are P"=only overlapping circle is shaded. I:"Some S are P"=only overlapping circle is marked with 'X'. O:"Some S are not P"=only 'S' is shaded.

Translating Ordinary Language Statements into Categorical Form: Nonstandard Verbs without a Copula

Some non-standard categorical propositions do not include a copula at all, and therefore, does not include the verb "to be". To translate these propositions into standard form, "are" or "are not" is included between the subject and predicate classes. Ex: "Some birds fly south during the winter" becomes "Some birds are animals that fly south during the winter."

Translating Ordinary Language Statements into Categorical Form: Nonstandard Verbs

Standard-form categorical propositions must include a copula that consists of the verb "are" and "are not", however, ordinary language typically incorporates other forms of the verb "to be" as their copula such as "will", "would", and "don't". Ex: "Some college students will become educated" becomes "Some college students are people who will become educated."

Logically Undetermined

Statements that do not have the same truth value; in conversion of a categorical proposition, A and O statements are logically unrelated to their converse statement (does not necessitate that their converse statements lack truth value, rather, that logic alone cannot determine what it is).

Logically Equivalent

Statements that have the same truth value; in conversion of a categorical proposition, E and I statements are logically equivalent to their converse statement.

Vacuously True

Truth that results from a categorical proposition that lacks members in the subject class; does not result from the relationship between two logically undetermined propositions (A and E), but only because of the vacant subject class; only refers to A and E propositions.

Vacuously False

Truth that results from a categorical proposition that lacks members in the subject class; occurs only in I and O propositions.

Rules of the Boolean Venn Diagram

Universal propositions are not marked with an 'X' because the diagram does not state anything about its existence. " The A proposition asserts that no members of S are outside P. This is represented by shading the part of the S circle that lies outside the P circle. The E proposition asserts that no members of S are inside P. This is represented by shading the part of the S circle that lies inside the P circle. The I proposition asserts that at least one S exists and that S is also a P. This is represented by placing an X in the area where the S and P circles overlap. This X represents an existing thing that is both an S and a P. Finally, the O proposition asserts that at least one S exists, and that S is not a P. This is represented by placing an X in the part of the S circle that lies outside the P circle. This X represents an existing thing that is an S but not a P."