QUANTIFIERS AND LOGICAL INFERENCE

Adapted from Patrick J. Hurley, A Concise Introduction to Logic

(Belmont: Thomson Wadsworth, 2008).

Predicate Logic

Before I go on to explain quantifiers, first let me address different ways of symbolizing statements. Previously, we used one letter to symbolize one statement. But there is another way to symbolize certain kinds of statements that are relevant to quantifiers. We can also symbolize statements by symbolizing the predicate and subject separately.

E.g. Luna is a cat. = Cl (C= is a cat; l=Luna)

Predicate Logic

The relationship has a parallel in functional or quantificational statements. Luna is only an instance of a cat. There can be many cats. ‘Cats’ is the universal category governing whether a constant or variable counts as one of the things the term refers to, and Luna is the particular instantiation or constant.

Quantifiers

Instead of ‘Luna is a cat’, what if I wanted to say something about all cats or some cats as a category? This is when we use quantification. Let’s say I want to say ‘All cats are animals’. The symbol Ac does not cut it because the lower case ‘c’ represents only an individual. When I want to say something about all cats (or no cats), I need to use the symbol (x).

Universal Statements

x symbolizes the quantifier in universal statements:

All cats are animals. = (x)(Cx Ax)(Translation: For every x, if x is a cat, then x is an animal.)

No cats are dogs. = (x)(Cx Dx)(Translation: For every x, if x is a cat, then x is not a dog.)

Universal Statements

The previous slide contains statements about x such that for EVERY x, if x is a cat, then it is an animal. It is not just about one individual, but about the whole category of cats, dogs and animals.

See the next slide for Existential Statements.

Existential Statements

x symbolizes the quantifier in existential statements.

Some apples are red. = (x)(Ax Rx)(Translation: There exists an x such that x is an apple and x is red.)

Some apples are not red. = (x)(Ax Rx)(Translation: There exists an x such that x is an apple and x is not red.)

Examples of universal and existential statements.

Statement Symbolic translation

There are happy marriages. (x) (Mx • Hx) Every pediatrician loses sleep. (x) (Px ⊃ Lx) Animals exist. (x)Ax Unicorns do not exist. ~(x)Ux Anything is conceivable (x)Cx Sea lions are mammals. (x) (Sx ⊃ Mx) Egomaniacs are not pleasant (x) (Ex ⊃ ~Px)

companions. A few egomaniacs did not arrive (x) (Ex • ~Ax)

on time. Only close friends were invited (x) (Ix ⊃ Cx)

to the wedding.

Exercise

Translate the following statements into symbolic form. Avoid negation signs preceding quantifiers. The predicate letters are given in parentheses.

1. Elaine is a chemist. (C) 2. Nancy is not a sales clerk. (S) 3. Intel designs a faster chip only if Micron does.

(D) 4. Some grapes are sour. (G, S) 5. Every penguin loves ice. (P, L) 6. There is trouble in River City. (T, R) 7. Tigers exist. (T)

Logical Inference

There are a number of logical inferences that we use in every day language, logic, and math. Logic has names for them. Once you identify them, it makes those inferences explicit. Your textbook mentions six such inferences, and I will give them labels: modus ponens, modus tollens, disjunctive syllogism, conjunction, simplification, and addition.

Modus Ponens

We have run across this inference before. The rule of modus ponens is that if you have a conditional and assert the antecedent, you may infer the consequent.

E.g. If it rains, then the ground gets wet.It is raining.Therefore, the ground is wet.

Modus Ponens

This inference can be symbolized this way:

R WR W

Modus Tollens

This is an inference that also works off a conditional, but it asserts something about the consequent. If you have a conditional, and assert the negation of the consequent, you can infer the negation of the antecedent.

E.g.If it rains, then the ground gets wet.The ground did not get wet.Therefore, it did not rain.

Modus Tollens

This inference can be symbolized this way:

R W~ W~ R

Disjunctive Syllogism

In the disjunctive syllogism, if you have a disjunction and negate one side of the disjunctive, you may infer the non-negated side of the disjunctive.

E.g.Either the chalk is black or the chalk is white.The chalk is not black.Therefore, the chalk is white.

Disjunctive Syllogism

This inference can be symbolized this way:

B V W~BW

Conjunction

In this type of inference, if you have two established statements, then you may also conjoin them to make a single statement.

E.g.Chalk is white. Snow is white. Chalk and snow are white.

Conjunction

This inference may be symbolized this way:

CSC S

Simplification

In this inference, if you have two statements that are conjoined into a complex statement, you may infer one part of the complex statement.

E.g.Chalk and snow are white.Chalk is white.

Simplification

This inference may be symbolized this way:

C SC

Addition

In this inference, if you have a statement, you may properly add a disjunction without falsifying the original statement.

E.g.Chalk is white.Chalk is white or Rudolf ate my homework.

Addition

The previous inference can be symbolized this way:

CC V R

Exercise

Fill in the missing premise and give the inference that justifies the conclusion. (1) 1. B v K

2. ______3. K ____

(2) 1. N S2. ______3. S ____

(3) 1. K T2. ______3. ~K ____

Exercise – Fill in the blanks.

(1) 1. ~A2. A v E3. ______ ____4. ~A E ____

(2) 1. T2. T G3. (T v U) H4. ___________ ____5. H ____

(3) 1. M2. (M E) D3. E4. __________ ____5. D ____