PREDICATES AND QUANTIFIERS

In logic, predicates and quantifiers are seen as one of the essential parts and difficult for students to comprehend. Though it’s an advance branch in logic which needed be split for better aid in understanding the topic.

By the way of review, here is an example of each of the four main types of categorical statements or proposition.

NAME.

1. Universal Affirmative

2. Universal Negative

3. Particular Affirmative

4. Particular Negative

FORMS

1. All S are P

2. No S are P

3. Some S are P

4. Some S are not P

Example 

1. All trees are plants

2. No trees are plants

3. Some dogs are collie

4. Some dogs are not collies

Here are symbols interpretations

Operator 

1. (.)

2. (~)

3. (V)

Names 

1. Dot

2. Tilde

3. Vee

Translation 

1. And

2. Not

3. Or

Type of compound

1. Conjunction

2. Negation

3. Disjunction

Let’s consider these atomic sentence

Aristotle is a logician

From these, it can be say that the particular things or entity namely the property or attributes of being a logician.

Let the lower case letter “a” name the individual, Aristotle and the capital letter “L” stand for the predicate (is a logician).

From it can be symbolised as La. 

And if we used the lowercase letter “b” to the Roman philosopher Beothius, the statement “both Aristotle and Beothius ate logician will symbolised

La . Lb

Example: All men are Rational 

 (Mx —> Rx) 

(X) (Mx —>Rx)

a. Every human is rational

b. Each human is rational

c. Humans are rational

d. Any human is rational

e. If anything is human then it is rational

f. Anything that is a human is a rational

g. A thing is human only if it is rational

h. Only rationals are human

Another example.

No trees are animals 

(Tx: is tree; Ax: is animals).

It can be symbolize as

(X)( Tx—> ~Ax)

From the above symbol it means: for any x, if x is tree, then x is not animals.

Example

No professor are bus driver. 

(Px: is professor; Bx: is Bus driver )

Translation

(X)( Px—>~Bx)

It means for any x; if x is professor then it is not a bus driver.

From the example, No trees are animals can also be express in various ways of expressing universal negative English.

a. Nothing that is a tree is an animals

b. All trees are nonanimals

c. If anything is tree then it is not animal

d. Nothing is a tree unless it is not an animal

Also noted: universal affirmative can be expressed in a variety of ways in english. for example, each of the following is a stylistic variant of ” All humans are rational”.

The Four Quantifier Rules 

1. Universal Generalization

2. Universal Instantiation

3. Existential Generalization

4. Existential Instantiation

Universal Generalization

This is the first implicational rules.  (UI for Short). Following the argument illustrates the need for the rule example:

All  humans are mortals.

Socrates is human

Therefore, Socrates is mortal.

(Hx: x is human; Mx; x is mortal; S: Socrates )

1. (X) (Hx—> Mx)

2. HS

3. Ms

From these, it is not regards as conditioner, but a universally Quantifier statement. However the first premises do tell us that for every x, if x is human then x is mortal.  What goes for everything, goes for Socrate so from (1) we can infer

Hs—Ms      universal Instantiation (UI)

We call the rules of inference that permit to the moves Universal Instantiation

1. (x) (Hx—>Mx)

2. Hs                therefore Ms

3. Hs—>Ms         1. Is UI

4. Ms                     3, 2 is modus pollen

Exstential Generalization

The second rule for existential Generalization is (EG). The need for these rule is illustrated by the following arguments and proof.

Example

All humans are mortals. Socrate is human, therefore Some socrate is mortal.

(Hx: x is human; Mx: x is mortal; S: socrate

1.(X)(Hx—>Mx)

2. Hs                    therefore Œ(Mx)

3. Hs—>Mx

4. Œ (Mx).           1, UI.   3,2 MP. 4 EG

Existential Instantiation

Our third inference rule is existential Instantiation. The following arguments illustrated the need for these rule.

Example

All basketball players are atheletes. Some baseball players takes performance enhancing drugs. So, some atheletes takes performance enhancing drugs.

(Bx: is a baseball player; Ax: is atheletes; Dx: is performance enhancing drugs).

Solution

1. (X)(Bx—>Ax)

2. (ŒX)(Bx.Dx).      Therefore (ŒX)(Ax.Dx)

Explanation

The second premise tell us that at least one baseball player takes drugs, but those not tells us which one. However, if we know that someone satisfied this disruption, we can give that someone a name to go on talking about him.

Let’s call over our mystery person.”Balco Bob ” (b: Balco Bob) from 2 we can infer

3. Bb. Db

The remainder of the proof is now straight forward

4. Bb

5. Db

6. Bb—>Ab

7. Ab

8. Ab. Db

9. (ŒX)(Ax.Dx).

Note the number 9 is the conclusion.

Universal Generalization

Fourth inferential rule is universal Generalization (UG). Consider the following arguments and the accompanying proof, which illustrated a typical example of universal Generalization.

All trees are plants. All plants are living things so, all trees are living things.

(Jx: x is tree; kx: x is plant; Lx: x is living things).

1. (X) (Jx—>Kx)

2. (X) (Kx—>Lx)

3. Ja —>Ka                  1, UI

4. Ka—>La                 3,4  Hs

5. Ja —> La                5, UG

6. (X)(Jx—>Lx)