Inference theory in discrete mathematics

The interference theory can be described as the analysis of validity of the formula from the given set of premises.

Structure of an argument

An argument can be defined as a sequence of statements. The argument is a collection of premises and a conclusion. The conclusion is used to indicate the last statement, and premises are used to indicate all the remaining statements. Before the conclusion, the symbol ∴ will be placed. The following syntax is used to show the premises and conclusion:

Premises: p1, p2, p3, p4, ….., pn
Conclusion: q

If (p₁∧ p₂ ∧ p₃ ∧ p₄ ∧ …… ∧ p_n) → q indicates a topology, in this case, the argument will be termed as valid otherwise, it will be termed invalid. The following expression is used to show the argument:

Valid Argument:

A valid argument can be described as an argument where if all their premises are true, then their conclusions will also be true.

For example:

This argument belongs to a form that is described as follows:

The above form is valid. It does not matter what propositions are substituted for the variables. This type of form is known as the valid argument form. From the above definition, if a valid argument form consists

premises: p1, p2, p3, p4, ….., pn
conclusion: q

then (p₁ ∧ p₂ ∧ p₃ ∧ …. ∧ p_k) → q is a tautology. That means ((p → q) ∧ p) → q is a tautology.

To understand this concept, we have to learn about quantifiers, which are described as follows:

Quantifiers:

Quantifiers can be described as a collection of statements used to determine the truth of elements of a given predicate. It also contains the predicate, which can be described as a statement used to have a specific number of variables (terms). The quantifiers are basically of two types, which are described as follows:

• Existential quantifiers are indicated by the symbol ∃. In general form, it is represented by words some, there exits, or at least one.
• Universal quantifiers are indicated by the symbol ∀. It is represented by words every, any, all or each.

Rules of Inference

We can construct a more complicated valid argument with the help of using simple arguments, which work as the building blocks. If we are talking about the usage of arguments, then there are some simple arguments that have been established as valid and very important. These types of arguments are known as the Rules of inference. There are various types of Rules of inference, which are described as follows:

1. Modus Ponens

Suppose there are two premises, P and P → Q. Now, we will derive Q with the help of Modules Ponens like this:

Example:

Suppose P → Q = "If we have a bank account, then we can take advantage of this new policy."

P = "We have a bank account."

Therefore, Q = "We can take advantage of this new policy."

2. Modus Tollens

Suppose there are two premises, P → Q and ¬Q. Now, we will derive ¬P with the help of Modules Tollens like this:

Example:

Suppose P → Q = "If we have a bank account, then we can take advantage of this new policy."

¬Q = "We cannot take advantage of this new policy."

Therefore, ¬P = "We don't have a bank account."

3. Hypothetical Syllogism

Suppose there are two premises, P → Q and Q → R. Now, we will derive P → R with the help of Hypothetical Syllogism like this:

Example:

Suppose P → Q = "If my fiancé comes to meet me, I will not go to office."

Q → R = "If I will not go to office, I won't require to do office work."

Therefore, P → R = "If my fiancé come to meet me, I won't require to do office work."

4. Disjunction Syllogism

Suppose there are two premises ¬P and P ∨ Q. Now, we will derive Q with the help of Disjunction Syllogism like this:

Example:

Suppose ¬P = "Harry birthday cake is not strawberry flavored."

P ∨ Q = "The birthday cake is either red velvet flavored or mixed fruit flavored."

Therefore, Q = "The birthday cake is mixed fruit flavored."

5. Addition

Suppose there is a premise P. Now, we will derive P ∨ Q with the help of Addition like this:

Example:

Suppose P be the proposition, "Harry is a hard working employee" is true

Here Q has the proposition, "Harry is a bad employee".

Therefore, "Either Harry is a hard working employee Or Harry is a bad employee".

6. Simplification:

Suppose there is a premise P ∧ Q. Now, we will derive P with the help of Simplification like this:

Example:

Suppose P ∧ Q = "Harry is a hard working employee, and he is the best employee in the office".

Therefore, "Harry is a hard working employee".

7. Conjunction

Suppose there are two premises P and Q. Now, we will derive P ∧ Q with the help of conjunction like this:

Example:

Suppose P = "Harry is a hard working employee".

Suppose Q = "Harry is the best employee in the office".

Therefore, "Harry is a hard working employee and Harry is the best employee in the office".

8. Resolution

Suppose there are two premises P ∨ Q and ¬P ∨ R. Now, we will derive Q ∨ R with the help of a resolution like this:

Example:

Suppose P → Q = "If my fiancé comes to meet me, I will not go to office".

¬ P ∨ R = "If my fiancé did not come to met me, I won't require to do office work".

Therefore, Q ∨ R = "Either I will not go to office or I won't require to do office work".

9. Constructive Dilemma

Suppose there are two premises (P → Q) ∧ (R → S) and P ∨ R. Now, we will derive Q ∨ S with the help of a constructive dilemma like this:

Example:

Suppose P → Q = "If my fiancé will come to meet me, I will not go to office".

R → S = "If my relatives will come, I will tell my employees that I will come".

P ∨ R = "Either my fiancé will comes to meet me or my relatives will come".

Therefore, Q ∨ S = "Either I will not go to office or I will tell my employees that I will come".

10. Destructive Dilemma

Suppose there are two premises (P → Q) ∧ (R → S) and ¬Q ∨ ¬S. Now, we will derive ¬P ∨ ¬R with the help of a Destructive dilemma like this:

Example:

Suppose P → Q = "If my fiancé comes to meet me, I will not go to office".

R → S = "If my relatives come, I will tell my employees that I will come".

¬Q ∨ ¬S = "Either I will go to office or I will tell my employees that I will not come".

Therefore, ¬P ∨ ¬R = "Either my fiancé will not come to meet me or my relatives will not come".

Rules of Inference with Quantifiers

There are some other rules of inference with quantifier statements, which are described as follows:

1. Universal Instantiation

Suppose there is a premise ∀x P(x). Now, we will derive P(c) with the help of Universal Instantiation like this:

2. Universal Generalization

Suppose there is a premise P(c) for any arbitrary c. Now, we will derive ∀x P(x) with the help of a Universal generalization like this:

3. Existential Instantiation

Suppose there is a premise ∃x P(x). Now, we will derive P(c) for some element c with the help of Existential Instantiation like this:

4. Existential Generalization

Suppose there is a premise P(c) for some element c. Now, we will derive ∃x P(x) with the help of Existential generalization like this:

Example of rules of inferences

Example 1:

If my fiancé comes to meet me, then I will be happy.

If my fiancé does not come to meet me, then I will go to office.

If I go to office, then I will complete my work.

Can we conclude, "If I am not happy, then I will complete my work"?

Solution: We will simplify this discussion by identifying propositions and using the variables of propositional to represent them like this:

P := My fiancé come to meet me

Q := I will be happy

R := I will go to office

S := I will complete my work

After putting the above propositional variables, we will get the following premises and conclusion like this:

Now we will use the rules of inference so that we can deduce the conclusion with the help of a given hypothesis like this:

Example 2:

An employee in my office has not completed his daily work

Everyone in my office completed his monthly files.

Can we conclude, "Someone who completed his monthly files has not completed his daily work"?

Solution: We will simplify this discussion by identifying propositions and using the variables of propositional to represent them like this:

C(x) := x is a employee in my office

B(x) := x has completed his daily work

P(x) := x completed his monthly files

After putting the above propositional variables, we will get the following premises and conclusion like this:

Now we will use the rules of inference for quantifiers so that we can deduce the conclusion with the help of a given hypothesis like this: