Introduction. Sentence logic or propositional logic is
concerned with simple propositions and the relation between them
and not with the structure of the sentences or propositions.
The validity of arguments and proofs is not just depended on the
relation between the propositions, but also on the internal structure
of the propositions. Class logic or predicate logic develops
this aspect of logic by analyzing the structure of propositions.
Tradional logic analyzed the structure of propositions into four parts:
subject part, predicate part, quantitative part, and qualitative part.
(a) The subject part is that part of a proposition about
which something is being asserted.
(b) The predicate part of a proposition is that part which
asserts something about the subject. Traditional logic interpreted
the predicate part of all propositions as asserting or predicating
some property of the subject. For example, the proposition "All
men are mortal" is interpreted as predicating the property
of mortality to all men. But all propositions are not of this
form. For example, the proposition "John loves Mary" asserts
a relation between John and Mary, not a property of loving to John.
(c) The quantitative part of a proposition refers to the
"grammatical number" of the subject. In English grammar,
the subject is either singular or plural. The word "John"
refers to a single entity, whereas "men" refers to a
class or set of entities. The words "all," "some,"
"one," "few," "many" indicate the
quantitative aspect of the proposition.
(d) The qualitative part of a proposition refers to that
aspect of a proposition which affirms or denies the subject-predicate
relation. For example, the proposition "John is a fool"
affirms the relation and "John is not a fool" denies
the relationship.
Propositional Functions. Propositions that refer to a specific individual or thing is called a singular proposition. For example, the propositions "John is a fool" and "Socrates is a man" refer to specific individuals, John and Socrates. A singular proposition has the general form "x is P," where x is the subject part and P is predicate part. This usually is written Px or P(x) and is read "x has the property P." When the subject x or the predicate P in Px is not specified, the sentence is called a propositional function, or a propositional form, or an open sentence. For example, the sentence "He is a fool" is a propositional function and is neither true or false until a specific individual is substituted for the pronoun "He." In general, Px or P(x) is a propositional function or form, which will not become a true or false proposition until the name of a specific individual or entity is substituted for the variables x or P. The variable symbol x is a "place holder" awaiting substitution of a proper name and is called an individual variable; and when the predicate P is unspecified, the symbol P is called a predicate variable. For example, the proposition "John is a fool" is symbolized Fj and "Socrates is a man" as Ms. Note that a small or lower case letter is used for specific subject and a large or capital letter is used for specific predicate. These specific, particular values are called constants, in contrast to a variable which can take any value or stand for any individual or property. Constants are substitution instances for variables, that is, a constant replaces a variable.
Quantification. If the subject of a statement refers to
a collection or class of objects and the statement discusses some
portion of the collection, all or part, the statement is said
to contain a quantifier. Consider the following statements.
All men are mortal.
Some women are intelligent.
If x and y are any numbers, then x + y = y + x is true.
The above statements all contain quantifiers. All is the
quantifier in the first sentence; some is the quantifier
in the second sentence; any is the quantifier in the last
sentence. The foregoing are examples of general propositions,
that is, they are propositions that assert something about some
(called existential propositions) or all individuals
of a certain kind (called universal propositions). Existential
propositions asserts that at least one individual of a certain
kind exist. But, unlike the singular proposition, they do not
single out a specific individual. Quantitiers require special
handling when forming the negation of these statements. For example,
the negation of the sentence "All men are gods" is
"It is false that all men are gods." But this means
"Some men are not god," and not "No men are gods,"
which means "All men are not gods." Notice that the
quantifier is changed from all to some when the
negation is formed. Sentences that refer to "all,"
"every," or "any" are said to be universally
quantified and the symbol (x) is called a universal quantifier,
where x refers to the subject part of the sentence that follows
the predicate symbol. The phrase "For all x," when it is
prefixed to a statement, indicates a universal quantifier. Some writers
use the symbol (∀x) for the universal quantifier.
Sentences that refer to "some," "at least one,"
"there is" are said to be existential quantified and the
symbol (∃x) is called an existential quantifier, where x refers
to the subject part of the sentence that follows the symbol. The phrase
"There is an x," when it is prefixed to a statement, indicates
an existential quantifier. The existential quantifier can be
interpreted to mean that there is nonempty subject set which
is a subset of a predicate set or which has the predicated property.
That is, a statement of the form "There is an x such that Px,"
asserts that an element of the subject set have the property P, and is
written (∃x)Px. Now the negation of a quantified sentence
can be written in symbols. The negation of an universal
sentence is ~[(x)Px] = (∃x)[~Px] and the negation of an existential
or particular sentence is ~[(∃x)Px] = (x)[~Px]. Remember that the
negation of a true proposition must be false. These pairs of
sentences are called contrdictories, of which one must
be true and the other false.
In traditional logic, the propositions (x)Px and (x)[~Px] are called contraries, since they might both be false but cannot both be true. The propositions (x)Px and (∃x)[~Px] are called subcontraries, which can both be true but cannot both be false. Finally, the truth of (∃x)Px is implied by the truth of (x)Px, and this pair is called subalternates. Similarly, for (x)[~Px] and (∃x)[~Px]. These relationships have been shown as a square array called the square of opposition.
| (x)Px | Contraries | (x)[~Px] |
| Subalternates | Contradictories | Subalternates |
| (∃x)Px | Subcontraries | (∃x)[~Px] |
Traditional logic emphasized four types of of subject-predicate propositions. They have been given letter names A, E, I, O, which is presumed to come from the Latin 'AffIrmo' and 'nEgO,' meaning "I affirm" and "I deny." These are called categorial propositions. A categorial proposition states a relationship between two classes or categories of things. 'S' repesents the subject class and 'P' represents the predicate class. The universal affirmative proposition (A) says that all of the S class is entirely included in the P class. The universal denial proposition (E) says that all of the S class is entirely excluded from the P class; that is, there is no overlap, or that the two classes are disjoint. The particular affirmative proposition (I) says that the that some of the S class is partially included in the P class, that there is some overlap between the classes. The particular denial proposition (O) says that some of class S is partially excluded from the P class. The O proposition is the negative of the A proposition and the E proposition is the negative of the I proposition ( and vice versa). That is, the A and O propositions are contradictories., and E and I propositions are also contradictories. But none of the other relationships discussed above connected with the square of opposition hold for the A, E, I, and O .propositions, even where it is assumed that there exists at least one individual in the universe. Modern logic rejects Aristotle's interpretation of the "All S is P" and "No S is P" as universal propositions. According to the traditional Aristotelian view, universals have existential import; the modern view rejects this view. It rejects the view that "All tigers are fierce" by itself implies that "Some tigers are fierce." This is deducible only from the conjunction "All tigers are fierce and there are tigers," not from "All tigers are fierce" alone. That is, universal propositions are construed as having no existential import. They consist exclusively in the denial of an existential proposition: A denies O and does not imply I, and E denies I and does not imply O. The truth of the universal A and E propositions does not imply the truth of the existential I and O propositions. As a consequent of this view, it is necessary to accept what appears at first sight to be a paradox, namely, that A and E are true together. There is one condition under which they are both true; the members of the subject class do not exist, (∃x)Sx is false. This is clear from the notation for A and E: ~(∃x)(Sx & ~Px) and ~(∃x)(Sx & Px). The universal propositions A and E do not assert existence. When it is said that "All S's are P's," this does not imply that there are any S's. This may seem odd in English. But very often when asserting an universal proposition it does not imply that anything in the subject class exists. If a sign says, "All trespassers will be shot," that is, (x)(Tx > Sx), this certainly does not mean that trespassers exist there. In fact, the sign was designed precisely to insure that there won't be any (one way or another). Thus it is not unusual to have a universal proposition in which no existence asserted, and, in logic, universal categorical propositions should never be thought of as asserting any kind of existence. Since the A and E propositions which are univeral quantification of complex propositional functions are true, the A and E propositions are not contraries. Since the I and O propositions which are existential quantification of complex propositional are false, the I and O propositions are not subcontraries. Thus, of the relationships on the classical square of opposition, only two are left; these are the contradictions between A and O, and E and I.
| Type | English Sentence | Formula | Negative Formula | English Sentence | Type |
|---|---|---|---|---|---|
| A | All S are P | (x)(Sx > Px) | ~(∃)(Sx & ~Px) | There are no S's which are not P's | ~O |
| E | No S is P | (x)(Sx > ~Px) | ~(∃x)(Sx & Px) | There is no S which is P | ~I |
| I | Some S is P | (∃x)(Sx & Px) | ~(x)(Sx > ~Px) | Not all S's are non-P's | ~E |
| O | Some S is not P | (∃x)(Sx & ~Px) | ~(x)(Sx > Px) | Not all S's are P's | ~A |
To understand exactly what a categorical proposition says, and
why it is symbolized the way it is, it is necessary to understand
what are classes or sets and how their members or elements are
related to them and properties. A property of an individual is
simply a characteristic, or attribute, of the individual. And
the individual is said to be a member of the set or class. A
set is a collection of objects or things and a class is a set
with a defining property which is a criterion that determines
what individuals are members of the class. Often the terms
"set" and "class" are used interchangeablly;
however, a set is just a collection of items that may or may not
have defining property, and a class is a specific kind of set that
always has a specific defining property. Membership in a class is
symbolized x ∈ P, where the symbol ∈ means
"is a member of," and the symbol x represents the individual
that is being denoted as belonging the class P. The symbol Sx is
equivalent to the symbols x ∈ S, and Px is equivalent to x ∈ P.
| 1. | The Universal Affirmative, A: | (∀x)[(x ∈ S) > (x ∈ P)] | (∀x)[Sx > Px] |
| 2. | The Universal Denial, E: | (∀x)[(x ∈ S) > ~(x ∈ P)] | (∀x)[Sx > ~Px] |
| 3. | The Particular Affirmative, I: | (∃x)[(x ∈ S) & (x ∈ P)] | (∃x)[Sx & Px] |
| 4. | The Particular Denial, O: | (∃x)[(x ∈ S) & ~(x ∈ P)] | (∃x)[Sx & ~Px] |
Thus the A proposition "All men are mortal" can be put in the form "For all x, if x is man, then x is a mortal." And the E proposition "All men are not fools" in the form "For all x, if x is man, then x is not a fool." And the I proposition "Some men are fools" in the form "For some x, x is a man and x is a fool." And the O proposition "Some men are not fools" in the form "For some x, x is a man and x is not a fool."
Quantification was above applied to propositions with one individual
variable; this quantification is called monadic. But quantification
can applied to propositions with more than one variable and is
called polyadic. Specifically the quantification applied
to propositions with two variables is dyadic. For example,
x loves y is a two variable propositional function or form and
can be written in symbols Lxy, where L is the dyadic relation
"loves." The propositional function can be univerally
quantified as follows: "For all x and for all y, x loves y,"
and in symbols: "(∀x)(∀y)Lxy."
This propositional function can also be existentially quantified:
"For some x and for some y, x loves y,"
and in symbols: "(∃x)(∃x)Lxy."
When an one variable propositional function Px is quantified,
only two distinct propositions are possible: "(∀x)Px"
and "(∃x)Px." The number of distinct possible propositions
increases to four for a two variable propositional function:
| (a) Everyone loves everyone. | (∀x)(∀y)Lxy. |
| (b) Everyone loves someone. | (∀x)(∃y)Lxy. |
| (c) Someone loves everyone. | (∃x)(∀y)Lxy. |
| (d) Someone loves someone. | (∃x)(∃y)Lxy. |
In order to handle quantitified statements in a proof, four fundamental quantified rules must be added to the rules of inteference to provide additional methods for proving quantified statements.
(1) Universal Specification. Consider the universal quantified proposition "For all x, x is P." If the statement is true, all elements x must be contained in P, that is, have the property P. It seems reasonable to say that if every element in P, then a particular element, say a, must also be in P. For example the classic Socrates Proof: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. This can be written symbolically, (x)[Hx > Mx] & Hs. Therefore, Ms. Logically the problem is making the transition from quantified to the specific singular proposition. This problem can be solved by introducing the following argument which will be assumed valid. This rule of inference or argument is called Universal Specification, which in symbols is (x)Px. Therefore, Pa. Using this rule the Socrates Proof can be shown to be valid.
| Proof: | 1. | (x)[Hx > Mx] | First premise. |
|---|---|---|---|
| 2. | Hs | Second premise. | |
| 3. | Hs > Ms | 1, Universal Specification. | |
| 4. | Therefore, Ms | 2, 3, modus ponens. |
(2) Universal Generalization. Consider the following argument:
No human is infallible. All teachers are human.
Therefore, no teacher is infallible.
This argument can be written symbolically,
(x)[Hx > ~Ix] & (x)[Tx > Hx], Therefore, (x)[Tx > ~Ix].
Using Universal Specification, let us prove its validity.
| Proof: | 1. | (x)[Hx > ~Ix] | First premise. |
|---|---|---|---|
| 2. | (x)[Tx > Hx] | Second premise. | |
| 3. | Ha > ~Ia | 1, Universal Specification. | |
| 4. | Ta > Ha | 2. Universal Specification. | |
| 5. | Ta > ~Ia | 3, 4, hypothetical syllogism. | |
| 6. | therefore, (x)[Tx > ~Ix] | 5, ? |
(3) Existential Specification. Similar considerations
for Universal Specification leads to this rule of inference.
An existential quantified statement (∃x)Px will be true
if, and only if, there is in fact some element, say a, in the set X
such that Pa is true. Since in the proof the premise is assumed to
be true, it is reasonable to assume that there must be a specific,
but possibly unknown, element in the set X, say a, such that Pa
is true. Thus the following rule of inference, called Existential
Specification, is assumed to be valid. In symbols,
(∃x)Px. therefore Pa for some a ∈ X.
(4) Existential Generalization. Similar considerations
for Universal Generalization leads to this rule of inference.
If a ∈ X is an element such that Pa is true, then it can be
concluded that the statement (∃x)Px must be true.
Note that the element a is restricted to a specific, possibly unknown,
element in the set X. This rule of inference, called Existential
Generalization, is assumed to be valid. In symbols,
Pa for some a ∈ X. therefore (∃x)Px.