Topic: Valid arguments

(Логически обоснованные доводы. Use of reason to decide something or persuade someone)

Definition: An argument is a finite collection A₁, A₂, …, A_n of statements (simple or compound), followed by a statement A. We denote an argument by the notation:

The horizontal bar stands for the word “therefore”. Each of the statements A₁, A₂,.., A_n is called a premise of the argument. The statement A is called the conclusion of arguments.

Note: In practice we are interested only in arguments for which the premises are true. Because arguments like:

A₁ = “My father was Alexander the Great.”

A₂ = “My mother was the English Queen Victoria.”

A = “I am your present king.”

Only make us feel pity for the speaker.

Definition: An argument A₁, A₂, …, A_n ∴ A is valid if whenever on assignment of truth values to the statement variables makes the premises A₁, A₂, …, A_n true, then it also makes the conclusion A true.

Another name and definition of a valid argument

Definition: Given a propositional formula G. Let a_1, a_{2, …,} a_n, be propositional variables of G. Then the interpretation of G is an assignment of truth values to a_1, a_{2, …,} a_n in which every a_i is assigned T or F but not both.

Definition: Given formulas F_1,F₂,…,F_n and a formula G. G is said to be a logical consequence of F_1,F₂,…,F_n (or logically follows form F_1,F₂,…,F_n ) if and only if for any interpretation I in which F₁ ˄ F₂ ˄… ˄ F_n is true,G is also true.(Another way to say the same: if F₁ ˄ F₂ ˄… ˄ F_n → G is a tautology).

F_1,F₂,…,F_n are called axioms(or postulates, premises) of G.

If G is a logical consequence of F₁, F₂,…,F_n , then the formula ((F_{1 ˄} F_{2 ˄}… _˄ F_n) → G) is called a theorem. G is also called the conclusion of the theorem.

In mathematics and in other fields, many problems are started as problems of proving theorems.

Ex: Consider the formulas:

F₁ = (p → q), F₂ = ~q, G = ~p

Show that G is a logical consequence of F₁ and F₂ . (= Show that argument

p → q, ~q ∴ ~p is valid.)

Method 1: By a truth table show that G is true for every interpretation where F_{1 ˄} F₂ is true (same: F_{1 ˄} F₂ → G is a tautology)

p	q	p → q	~q	(p → q) ˄~q	~p	((p → q) ˄~q) →~p
F	F	T	T	T	T	T
F	T	T	F	F	T	T
T	F	F	T	F	F	T
T	T	T	F	F	F	T

if premises are true, then conclusion is also true Is a tautology

Method 2: Transform analytically [((p → q) ˄ (~q)) →~p] to a tautology: ((p → q) ˄~q) → (~p) = ~((p → q) ˄(~q)) ˅(~p) = ~((~p ˅ q) ˄(~q)) ˅(~p) =

~(~p ˄ ~q) ˅ (q ˄ ~q)) ˅(~p) = ~(~p ˄ ~q ˅ с) ˅ (~p) = p ˅ q ˅ (~p) =

(p ˅ ~p) ˅q = t ˅ q = t (tautology).

An argument that is not valid is said to be invalid.

Ex 1: consider the argument

p → q

p

q

An example of such an argument is:

p → q = “If today is Saturday then I do not have to go to the school.”

p = “Today is Saturday”.

q = “Therefore, I do not have to go to school”.

In order to decide whether this argument is valid or not, we consider the following truth table.

It is in the last line of the table, that p → q and p are both true. But in this case, the conclusion q is also true. Hence, according to the definition, the argument is valid.

This simple, but important argument is called modus ponens.

Example: Arguments that looks like modus ponens but which is not.

p ® q = “If today is Saturday then I do not have to go to school”.

q = “I do not have to go to school.”

p = “Therefore, today is Saturday.”

Is the argument valid or not?

p	q	p ®q
F	F	F
F	T	T
T	F	F
T	T	T

In these two rows both premises are true. But the conclusion p takes two values: true and false in these rows. Hence, the argument is invalid.