A categorical proposition is a part of deductive reasoning that contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former.[1] Categorical propositions occur in categorical syllogisms and both are discussed in Aristotle's Prior Analytics.
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).
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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 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 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.
| Name | Statement | Quantity | Quality | Distribution | |
|---|---|---|---|---|---|
| Subject | Predicate | ||||
| A | All S is P | universal | affirmative | distributed | undistributed |
| E | No S is P | universal | negative | distributed | distributed |
| I | Some S is P | particular | affirmative | undistributed | undistributed |
| O | Some S is not P | particular | negative | undistributed | distributed |
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").[2]
Copi and Cohen state two rules about distribution of terms in valid syllogisms:[3]
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.[4][5] 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,"[6] which is perhaps a closer translation to Aristotle's original form for this type of statement.[7]
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