Categorical proposition
Encyclopedia
A categorical proposition
contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogism
s and both are discussed in Aristotle
's Prior Analytics
.
Categorical propositions are part of deductive reasoning
.
Examples:
The subject and predicate are called the terms of the proposition. The subject is what the proposition is about. The predicate is what the proposition affirms or denies about the subject. A categorical proposition thus claims something about things or ways of being: it affirms or denies something about something else.
Categorical propositions are distinguished from hypothetical propositions (if-then statements that connect propositions rather than terms) and disjunctive propositions (either-or statements, claiming exclusivity between propositions).
An important consideration is the definition of the word some. In logic, some refers to "one or more", which could mean "all". Therefore, the statement "Some S is P" does not guarantee that the statement "Some S is not P" is also true.
Distribution refers to whether all or some members of a class are affected by a proposition. Both subjects and predicates have distribution. If all members of a class are affected by a proposition, that class is distributed; otherwise it is undistributed.
An A proposition distributes the subject to the predicate, but not the reverse. Consider the following categorical proposition: "All dogs are mammals". All dogs are indeed mammals but it would be false to say all mammals are dogs. Since all dogs are included in the class of mammals, "dogs" is said to be distributed to "mammals". Since all mammals are not necessarily dogs, "mammals" is undistributed to "dogs".
An E proposition distributes bidirectionally between the subject and predicate. From the categorical proposition "No beetles are mammals", we can infer that no mammals are beetles. Since all beetles are defined not to be mammals, and all mammals are defined not to be beetles, both classes are distributed.
Both terms in an I proposition are undistributed. For example, "Some Americans are conservatives". Neither term can be entirely distributed to the other. From this proposition it is not possible to say that all Americans are conservatives or that all conservatives are Americans.
In an O proposition only the predicate is distributed. Consider the following: "Some politicians are not corrupt". Since not all politicians are defined by this rule, the subject is undistributed. The predicate, though, is distributed because all the members of "corrupt people" will not match the group of people defined as "some politicians". Since the rule applies to every member of the corrupt people group, namely, "all corrupt people are not some politicians", the predicate is distributed.
The distribution of the predicate in an O proposition is often confusing due to its ambiguity. When a statement like "Some politicians are not corrupt" is said to distribute the "corrupt people" group to "some politicians", the information seems of little value since the group "some politicians" is not defined. But if, as an example, this group of "some politicians" were defined to contain a single person, Albert, the relationship becomes more clear. The statement would then mean, of every entry listed in the corrupt people group, not one of them will be Albert: "all corrupt people are not Albert". This is a definition that applies to every member of the "corrupt people" group, and is therefore distributed.
In short, for the subject to be distributed, the statement must be universal (e.g., "all", "no"). For the predicate to be distributed, the statement must be negative (e.g., "no", "not").
Copi and Cohen state two rules about distribution of terms in valid syllogisms:
When these rules are not followed, a fallacy
or sophism
can ensue. Breaking the rules regarding distribution of the middle, major, and minor terms are respectively called the fallacy of the undistributed middle
, the illicit major
fallacy, and the illicit minor
fallacy.
Peter Geach
and others have criticized the use of distribution to determine the validity of an argument. It has been suggested that statements of the form "Some A are not B" would be less problematic if stated as "Not every A is B," which is perhaps a closer translation to Aristotle
's original form for this type of statement.
Quantifier (subject term) copula (predicate term)
are logical truths, but not all logical truths are tautologies.
Quantifiers have scope, namely, the first whole proposition, simple or compound, to their right. In this sense, they have the same scope as the negation sign. "Bx" is inside the scope of the quantifier in "(x)(Ax Bx)" but outside in "(x)Ax Bx".
Variables inside the scope of a quantifier are bound by that quantifier; otherwise they are free. More precisely, a variable is only bound by a quantifier on the same letter; hence "x" is bound in "(x)Mx" but not in "(y)Mx", even though it is inside the scope of the quantifier in both cases.
When a variable is within the scopes of two or more quantifiers, then it is bound by the most local (least global) quantifier on the same letter, if any. Hence, "x" is bound by "(x)" in "(y)[(Ay By) (x)Cx]" and "(x)(Ax·(x)Bx)".
A variable may occur more than once in an expression, free in some occurrences and bound in others, for example, "x" in "(x)Ax Bx". Hence it is imprecise to speak merely of free and bound variables. We must speak of free and bound occurrences of variables. In "(x)Ax Bx", the first occurrence of "x" is bound, because it is within the scope of the quantifier, but the second occurrence is free because it is outside that scope.
A variable may also occur freely with respect to one quantifier and bound with respect to another. For example, in "(x)Ax (x)Bx" the "x" in "Bx" is free with respect to the universal quantifier, bound with respect to the existential quantifier. So we must speak of free and bound occurrences of variables with respect to a given quantifier.
A quantifier that binds no variables is vacuous. For example, the universal quantifier is vacuous in "(x)Mz" and "(x)Ma" but not in "(x)Mx".
A general proposition is one with a quantifier; it can be existential or universal. A singular proposition lacks a quantifier and variables, and uses only constants, for example, "Ms". Singular and general propositions with no free variables are genuine propositions in the sense that they possess truth-values. By contrast, a propositional function has at least one free occurrence of a variable, for example "Hx". Therefore, propositional functions lack a truth-value; we can't tell whether the unfilled form " (blank) is human" is true or false until the blank (or free variable) is bound by a quantifier or replaced by a constant, that is, until the propositional function converted to a genuine proposition.
(Now that we know what a propositional function is, we can define quantifier scope more precisely: a quantifier's scope is the first whole proposition or propositional function to its right.)
One of the components of "(x)(Ax Bx)" is "Bx", which is a propositional function without truth-value. Hence we cannot determine the truth-value of the general proposition "(x)(Ax Bx)" using only the truth-values of the components. Hence, in predicate logic we give up truth-functionality. Hence, we give up methods for testing validity which depend on truth-functional propositions, such as truth tables.
There are two ways to convert a propositional function (like "Hx") into a proposition. First, the free variables may be bound by quantifiers; this is called generalization. Second, the free variables may be replaced by constants; this is called instantiation.
We will introduce four rules of inference for predicate logic. Universal generalization allows us to add the universal quantifier. Existential generalization allows us to add the existential quantifier. Universal instantiation allows us to remove the universal quantifier. Existential instantiation allows us to remove the existential quantifier. The two instantiation rules also allow us, after removing quantifiers, to replace form
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
s and both are discussed in Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's Prior Analytics
Prior Analytics
The Prior Analytics is Aristotle's work on deductive reasoning, specifically the syllogism. It is also part of his Organon, which is the instrument or manual of logical and scientific methods....
.
Categorical propositions are part of deductive reasoning
Deductive reasoning
Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...
.
Examples:
- Midshipman Davis serves on H.M.S. Invincible. (subject: Midshipman Davis; predicate: serves on H.M.S. Invincible)
- Some politicians are corrupt. (subject: politicians; predicate: corruptness)
- Nobody ever got fired for buying IBMIBMInternational Business Machines Corporation or IBM is an American multinational technology and consulting corporation headquartered in Armonk, New York, United States. IBM manufactures and sells computer hardware and software, and it offers infrastructure, hosting and consulting services in areas...
. (subject: people; predicate: getting fired for buying IBM)
The subject and predicate are called the terms of the proposition. The subject is what the proposition is about. The predicate is what the proposition affirms or denies about the subject. A categorical proposition thus claims something about things or ways of being: it affirms or denies something about something else.
Categorical propositions are distinguished from hypothetical propositions (if-then statements that connect propositions rather than terms) and disjunctive propositions (either-or statements, claiming exclusivity between propositions).
Quality, quantity and distribution
Categorical propositions can be categorized into four types on the basis of their quality, quantity, and distribution. These four types have long been named A, E, I and O. This is based on the Latin affirmo (I affirm), referring to the affirmative propositions A and I, and nego (I deny), referring to the negative propositions E and O.Quality
Quality refers to whether the proposition affirms or denies the inclusion of a subject within the class of the predicate. The two qualities are affirmative and negative. For instance, the A proposition ("All S is P") is affirmative since it states that the subject is contained within the predicate. On the other hand, the O proposition ("Some S is not P") is negative since it excludes the subject from the predicate.Quantity
Quantity refers to the amount of members of the subject class that are used in the proposition. If the proposition refers to all members of the subject class, it is universal. If the proposition does not employ all members of the subject class, it is particular. For instance, the I proposition ("Some S are P") is particular since it only refers to some of the members of the subject class.An important consideration is the definition of the word some. In logic, some refers to "one or more", which could mean "all". Therefore, the statement "Some S is P" does not guarantee that the statement "Some S is not P" is also true.
Distribution
| Statement | Quantity | Subject (P) | Quality | Predicate (S) |
|---|---|---|---|---|
| All P are S. | universal | affirmative | ||
| No P are S. | universal | negative | ||
| Some P are S. | particular | affirmative | ||
| Some P are not S. | particular | negative |
Distribution refers to whether all or some members of a class are affected by a proposition. Both subjects and predicates have distribution. If all members of a class are affected by a proposition, that class is distributed; otherwise it is undistributed.
An A proposition distributes the subject to the predicate, but not the reverse. Consider the following categorical proposition: "All dogs are mammals". All dogs are indeed mammals but it would be false to say all mammals are dogs. Since all dogs are included in the class of mammals, "dogs" is said to be distributed to "mammals". Since all mammals are not necessarily dogs, "mammals" is undistributed to "dogs".
An E proposition distributes bidirectionally between the subject and predicate. From the categorical proposition "No beetles are mammals", we can infer that no mammals are beetles. Since all beetles are defined not to be mammals, and all mammals are defined not to be beetles, both classes are distributed.
Both terms in an I proposition are undistributed. For example, "Some Americans are conservatives". Neither term can be entirely distributed to the other. From this proposition it is not possible to say that all Americans are conservatives or that all conservatives are Americans.
In an O proposition only the predicate is distributed. Consider the following: "Some politicians are not corrupt". Since not all politicians are defined by this rule, the subject is undistributed. The predicate, though, is distributed because all the members of "corrupt people" will not match the group of people defined as "some politicians". Since the rule applies to every member of the corrupt people group, namely, "all corrupt people are not some politicians", the predicate is distributed.
The distribution of the predicate in an O proposition is often confusing due to its ambiguity. When a statement like "Some politicians are not corrupt" is said to distribute the "corrupt people" group to "some politicians", the information seems of little value since the group "some politicians" is not defined. But if, as an example, this group of "some politicians" were defined to contain a single person, Albert, the relationship becomes more clear. The statement would then mean, of every entry listed in the corrupt people group, not one of them will be Albert: "all corrupt people are not Albert". This is a definition that applies to every member of the "corrupt people" group, and is therefore distributed.
In short, for the subject to be distributed, the statement must be universal (e.g., "all", "no"). For the predicate to be distributed, the statement must be negative (e.g., "no", "not").
Copi and Cohen state two rules about distribution of terms in valid syllogisms:
- The middle termMiddle termThe middle term must distributed in at least one premises but not in the conclusion of a categorical syllogism. The major term and the minor terms, also called the end terms, do appear in the conclusion.Example:...
must be distributed in at least one premisePremisePremise can refer to:* Premise, a claim that is a reason for, or an objection against, some other claim as part of an argument...
. - If the major termMajor termThe major term is the predicate term of the conclusion of a categorical syllogism. It appears in the major premise along with the middle term and not the minor term. It is an end term .Example:...
or the minor termMinor termThe minor term is the subject term of the conclusion of a categorical syllogism. It also appears in the minor premise together with the middle term. Along with the major term it is one of the two end terms.Example:...
is distributed in the conclusion, then it must be distributed in the premises.
When these rules are not followed, a fallacy
Fallacy
In logic and rhetoric, a fallacy is usually an incorrect argumentation in reasoning resulting in a misconception or presumption. By accident or design, fallacies may exploit emotional triggers in the listener or interlocutor , or take advantage of social relationships between people...
or sophism
Sophism
Sophism in the modern definition is a specious argument used for deceiving someone. In ancient Greece, sophists were a category of teachers who specialized in using the tools of philosophy and rhetoric for the purpose of teaching aretê — excellence, or virtue — predominantly to young statesmen and...
can ensue. Breaking the rules regarding distribution of the middle, major, and minor terms are respectively called the fallacy of the undistributed middle
Fallacy of the undistributed middle
The fallacy of the undistributed middle is a logical fallacy, and more specifically a formal fallacy, that is committed when the middle term in a categorical syllogism is not distributed in the major premise...
, the illicit major
Illicit major
Illicit major is a logical fallacy committed in a categorical syllogism that is invalid because its major term is undistributed in the major premise but distributed in the conclusion.This fallacy has the following argument form:#All A are B...
fallacy, and the illicit minor
Illicit minor
Illicit minor is a logical fallacy committed in a categorical syllogism that is invalid because its minor term is undistributed in the minor premise but distributed in the conclusion....
fallacy.
Peter Geach
Peter Geach
Peter Thomas Geach is a British philosopher. His areas of interest are the history of philosophy, philosophical logic, and the theory of identity.He was educated at Balliol College, Oxford...
and others have criticized the use of distribution to determine the validity of an argument. It has been suggested that statements of the form "Some A are not B" would be less problematic if stated as "Not every A is B," which is perhaps a closer translation to Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's original form for this type of statement.
Schema
The general schema of categorical propositions is:Quantifier (subject term) copula (predicate term)
are logical truths, but not all logical truths are tautologies.
Quantifiers have scope, namely, the first whole proposition, simple or compound, to their right. In this sense, they have the same scope as the negation sign. "Bx" is inside the scope of the quantifier in "(x)(Ax Bx)" but outside in "(x)Ax Bx".
Variables inside the scope of a quantifier are bound by that quantifier; otherwise they are free. More precisely, a variable is only bound by a quantifier on the same letter; hence "x" is bound in "(x)Mx" but not in "(y)Mx", even though it is inside the scope of the quantifier in both cases.
When a variable is within the scopes of two or more quantifiers, then it is bound by the most local (least global) quantifier on the same letter, if any. Hence, "x" is bound by "(x)" in "(y)[(Ay By) (x)Cx]" and "(x)(Ax·(x)Bx)".
A variable may occur more than once in an expression, free in some occurrences and bound in others, for example, "x" in "(x)Ax Bx". Hence it is imprecise to speak merely of free and bound variables. We must speak of free and bound occurrences of variables. In "(x)Ax Bx", the first occurrence of "x" is bound, because it is within the scope of the quantifier, but the second occurrence is free because it is outside that scope.
A variable may also occur freely with respect to one quantifier and bound with respect to another. For example, in "(x)Ax (x)Bx" the "x" in "Bx" is free with respect to the universal quantifier, bound with respect to the existential quantifier. So we must speak of free and bound occurrences of variables with respect to a given quantifier.
A quantifier that binds no variables is vacuous. For example, the universal quantifier is vacuous in "(x)Mz" and "(x)Ma" but not in "(x)Mx".
A general proposition is one with a quantifier; it can be existential or universal. A singular proposition lacks a quantifier and variables, and uses only constants, for example, "Ms". Singular and general propositions with no free variables are genuine propositions in the sense that they possess truth-values. By contrast, a propositional function has at least one free occurrence of a variable, for example "Hx". Therefore, propositional functions lack a truth-value; we can't tell whether the unfilled form " (blank) is human" is true or false until the blank (or free variable) is bound by a quantifier or replaced by a constant, that is, until the propositional function converted to a genuine proposition.
(Now that we know what a propositional function is, we can define quantifier scope more precisely: a quantifier's scope is the first whole proposition or propositional function to its right.)
One of the components of "(x)(Ax Bx)" is "Bx", which is a propositional function without truth-value. Hence we cannot determine the truth-value of the general proposition "(x)(Ax Bx)" using only the truth-values of the components. Hence, in predicate logic we give up truth-functionality. Hence, we give up methods for testing validity which depend on truth-functional propositions, such as truth tables.
There are two ways to convert a propositional function (like "Hx") into a proposition. First, the free variables may be bound by quantifiers; this is called generalization. Second, the free variables may be replaced by constants; this is called instantiation.
We will introduce four rules of inference for predicate logic. Universal generalization allows us to add the universal quantifier. Existential generalization allows us to add the existential quantifier. Universal instantiation allows us to remove the universal quantifier. Existential instantiation allows us to remove the existential quantifier. The two instantiation rules also allow us, after removing quantifiers, to replace form