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Modern Logic

What is this Tutorial About?

This tutorial expands on the material covered in Compound Indication by looking at it from the perspective of Modern Logic. Modern logic offers a more systematic and comprehensive analysis of clauses (or propositions) and how atomic propositions can combine to form compound ones. This, however, is a very brief overview and presents only the major discussions.

 

Propositional Logic

Propositions & Their Negations

A proposition is a complete thought which is either true or false. It cannot be both true and false, nor can it be neither. In Arabic, a proposition is termed قضية حملية. Propositions are represented with capital letters, as in the following example.

 

P

The proposition P: it is raining today

 

A proposition can be negated using the symbol of negation, as in the following example.

 

P

it is not raining today

 

Combining Propositions Using Logical Connectives
To Form Compound Propositions

Two or more propositions can be combined to form another (more compound) proposition. For example, it is raining today and Im getting wet can be combined to form a new statement. We can say, for instance, it is raining today and Im getting wet, or Im getting wet because its raining today, and so forth. A compound proposition, in Arabic, is termed قضية شرطية.

 

There are two main ways in which we can combine propositions (and their negations); conjunction and disjunction. These are summarized in the table below.

 

 

Name

Symbol

Explanation

Conjunction

AND

P Q

The compound proposition is true only when both P and Q are true

Disjunction

OR

P Q

The compound proposition is true when both P and Q are true, or when either one is true

 

Consider the two propositions it is raining today and Im getting wet. The compound proposition P Q, it is raining today and Im getting wet is true only when it is true that it is raining today and it is true that Im getting wet. Similarly, the compound proposition P Q, it is raining today or Im getting wet is true when it is both raining today and Im getting wet, it is raining today but Im not getting wet, or it is not raining today but Im still getting wet.

 

Some further examples appear in the table below. Let P represent the statement I studied for the test, and let Q represent the statement I failed the test.

 

Formula

Explanation

P Q

I didnt study for the test, and I failed the test

Q P

Either I passed the test, or I didnt study for it

(P Q) (P Q)

I studied for the test and I passed it, or I didnt study for it and I failed it

 

Notice that Q is internally negated; failing a test is the opposite or negation of passing it. So when we negate Q, I didnt fail the test, we end up with an affirmative proposition, I passed the test. When a statement is internally negated like this, it is termed معدولة in Arabic.

 

More Advanced Connectives

The combination of negation and the logical connectives of conjunction and disjunction form a set of three manipulations. These three manipulations are enough to represent any and all propositions that can ever be conceived. However, it is sometimes useful and natural to have more complex logical connectives. Below is a table that extends the basic connectives to introduce new ones.

 

Name

Symbol

Equivalent To

Explanation

 

 

Basic Set

 

 

 

NOT

P

 

negation of P

AND

P Q

 

true only when both P and Q are true

OR

P Q

 

true when one or both of P and Q is true

 

 

Other Useful Connectives

NAND

P Q

(P Q)

it is not the case that P and Q are both true

NOR

P Q

(P Q)

neither is P true, nor is Q true

XOR

P Q

(P Q) (P Q)

true when only one of P or Q is true

IMPL

P Q

P Q

if P then Q

XNOR

P Q

(P Q) (Q P)

if P, and only if P, then Q

 

Some examples of the complex connectives are given below. Let P represent the statement I studied for the test, and let Q represent the statement I failed the test.

 

Formula

Explanation

P Q

it is not the case that I studied for the test and failed it

P Q

I didnt study for the test but I passed it

P Q

I passed the test only if I studied for it

 

Logical Equivalence

It is important to be able to determine when two logical propositions are equivalent to one another. For example, how can we be sure that P Q is the same as P Q? To give a concrete example, how can we be sure that the following two propositions are equivalent: if God exists then he is good and either God doesnt exist, or he is good.

 

There are two ways to compare propositions. The first is by using their truth tables. A truth table is a chart in which we have all possible combinations of truth and falsehood for our propositions (4 in the case of 2 propositions), and we calculate the resulting truth or falsehood of the compound proposition. For example, in order to write a truth table for P Q, we would start as follows.

 

P

Q

P Q

true

true

 

true

false

 

false

true

 

false

false

 

 

Then we calculate the truth value of the compound proposition, as follows.

 

P

Q

P Q

true

true

true

true

false

false

false

true

true

false

false

true

 

What we are saying here is that, for example:

         if it is indeed raining today, and I am indeed getting wet, then if it is raining today then Im getting wet is a true statement

         if it is indeed raining today, but I am not getting wet, then if it is raining today then Im getting wet is a false statement

         if it is not raining today, but I am indeed getting wet, then if it is raining today then Im getting wet is still true. This is because its not even raining today so whos to say that the statement is false?

         if it is not raining today, and I am not getting wet, then if it is raining today then Im getting wet is still true. This is because its not even raining today so whos to say that the statement is false?

 

Now the truth table for P Q is as follows.

 

P

Q

P

P Q

true

true

false

true

true

false

false

false

false

true

true

true

false

false

true

true

 

Notice that the truth column for P Q is exactly the same as that of P Q. This proves that the two logical propositions are equivalent. Whenever one is true, the other is also true, and whenever one is false, the other is also false.

 

The other way to show that two propositions are equivalent is through the equivalence rules. This is by far the more powerful technique in showing equivalence between two or more propositions. The popular rules are summarized in the table below.

 

is equivalent to

Example

P

P

it is not the case that it is not raining

it is raining

P Q

Q P

it is raining and I am wet

I am wet and it is raining

P Q

Q P

I am on land or I am drowning

I am drowning or I am on land

(P Q) R

P (Q R)

I studied and wrote the test, and I passed

I studied, and I wrote the test and passed

(P Q) R

P (Q R)

either the sun or the moon is out, or its pitch black

the sun is out, or either the moon is out or its pitch black

P (Q R)

(P Q) (P R)

I wrote the test, and I used either a pen or a pencil

I wrote the test and used a pen, or I wrote the test and used a pencil

P (Q R)

(P Q) (P R)

either Im working, or Im sick and tired

Im working or Im sick, and Im working or Im tired

(P Q)

P Q

it is not the case that either I passed the test or I failed it

I didnt pass the test nor did I fail it (i.e. I didnt write it)

(P Q)

P Q

it is not the case that I studied for the test and passed it

either I didnt study for it or I didnt pass it (or both)

P P

always true

either its raining or its not

note: propositions that are always true are called tautologies

P P

always false

its raining and its not raining

note: propositions that are always false are called contradictions

P P

P

it is cloudy or its cloudy

it is cloudy

P P

P

I am tired and I am tired

I am tired

P Q

P Q

if the sun is out then the day is present

either the sun is not out or (if it is) the day is present

 

As an example, lets show that the following two propositions are equivalent: if God exists and he has no beginning, then he has no end, and if God exists and he has an end, then he has a beginning. We are going to show that if you believe one of these, then you believe both of them because they are logically equivalent.

 

In order to do this, lets start with the following propositions:

 

P: God exists

Q: God has a beginning

R: God has an end

 

Then we construct the first proposition as follows:

 

(P Q) R

if God exists and he has no beginning, then he has no end

 

And the second as follows:

 

(P R) Q

if God exists and he has an end, then he has a beginning

 

Now we use the equivalence laws to show that the two propositions are equivalent:

 

 

(P Q) R

(P Q) R

((P Q) R)

(P Q R)

(P R Q)

((P R) Q)

(P R) Q

(P R) Q

 

Predicate Logic

A predicate is a generalized proposition that has the capacity to take one or more parameters. It uses these parameters and substitutes them for components of the sentence. For instance, P Zaid hit Amr is a proposition, but P(x) x hit Amr or P(x, y) x hit y or P(x, y, t) x hit y at time t are all predicates. We can say that propositions, then, are special types of predicates that take no parameters. Parameters are usually represented with lower case letters.

 

Because predicates take parameters, we can add quantification. Predicates can be quantified in two ways; universally or existentially. A universal quantification asserts that the predicate is true for all possible values of a parameter. And an existential quantification asserts that the predicate is true for at least one value of a parameter. For example, let P(x) represent the statement x hit Amr. We can then say: for all x, P(x). This means that for all people x, x hit Amr. Or more simply, everyone hit Amr. As a further example, we can say: there exists an x such that P(x). This means that there is someone who did not hit Amr.

 

A universally quantified statement is known, in Arabic, as a موجبة كلية or a سالبة كلية, depending on the affirmation. And an existentially quantified statement is called a موجبة جزئية or a سالبة جزئية. Below is a chart that summarizes these concepts. Let P(x) represent x hit Amr, let Q(x) represent x is upset, and let R(x, y) represent x likes y.

 

 

Symbol

Example

Universal (all)

x (P(x) Q(x))

everyone who didnt hit Amr is upset

Existential (exists)

xy R(x, y)

it is not the case that everyone has someone who likes them

 

Notice that the logical relationships of AND, OR, IMPL, etc can be applied to quantified predicates just as they can apply to non-quantified predicates as well as propositions. Notice also that negation can be applied to a quantifier.

 

When negation is applied to a quantifier, we can simplify the expression by pushing the negation into the scope of the quantifier. When we do this, we must flip the quantifier; universal becomes existential and existential becomes universal. This is summarized in the table below.

 

Original

Negation Pushed In

x P(x)

x P(x)

x P(x)

x P(x)

 

An example can illustrate this process more clearly.

 

Original

Negation Pushed In

it is not the case that every human is good

there are some humans that are not good

it is not the case that some vegetables are good

all vegetables are not good

 

 

This process is reversible as well. If the entire predicate in the scope of a quantifier is negated, we can pull the negation in front of the quantifier provided we flip the quantifier. The table below demonstrates this.

 

Original

Negation Pushed In

x P(x)

x P(x)

x P(x)

x P(x)

 

As an exercise, lets formalize the theory of cause and effect using predicate logic. Let O(x) represent the statement event x has occurred, and let C(x, y) represent the statement event x was caused by event y. Then:

 

fg (O(f) (C(f, g) O(g))

 



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