9

Algebra of logic

Prepared by L.D.Korovina

«When it have caught a fish, forget about the net. When it have caught a bird, forget about snare. When it have caught a beast, forget about a trap. When it have caught thought, forget about words. Would me find the person, who has forgotten words! I would talked to him.»

So is written in ancient China treatise «Chguan-Tszi».

Cited fragment was written hundreds years ago before appearance of modern logic as science. But it reflect, determinate extremely precisely essence of logic.

The science permitting to analyse reasoning, distracting from their content, paying attention only on the shape, selecting their structure, is termed as formal logic.

Some definitions:

logic is a formalization of reasoning;

logic is a formal language for deducing knowledge from a small number of explicitly, in detail stated premises (preconditions or hypotheses, axioms, facts).

Logic provides a formal framework, formal structure for knowledge representing. Logic differentiates between the structure and content of an argument.

There are some types of logic: propositional logic, predicate logic, probability logic.

In formal logic we test arguments that are universally true by their internal structure - arguments that are tautologies (propositional logic) or valid well-formed formulas (predicate logic).

Propositional logic – the formal system that uses propositional well-formed formulas; also known as statement logic, propositional calculus.

Predicate logic – extension of propositional logic.

Probability logic - the logic researching the expressions, which values are made in an interval between true and false. Probability is a degree of possibility of appearance of any specific event in a chain of various events in defined conditions, able to repeat many times.

We will study elements of propositional logic.

• Logic is a formalisation of reasoning.

• Logic is a formal language for deducing knowledge from a small number of explicitly, in detail stated premises (or hypotheses, axioms, facts).

• Logic provides a formal framework, formal structure for representing of knowledge.

• Logic differentiates between the structure and content of an argument.

A predicate is a sentence which contains a finite number of variables and becomes a statement if particular values are substituted for the variables. Objects in a domain have specific properties.

Before transition to consideration of objects and rules, it is necessary to define some ideas:

Premise [philosophical] – precondition, presupposition, prerequisite.

Deduction. If the conclusion is justified, based solely on the premises, the process of reasoning is called deduction. Example of deductive argument:

“Yalta is a port or a holiday resort. Yalta is not a port. Therefore, Yalta is a holiday resort.”

Inference. If the validity of the conclusion is based on generalisation from the premises, based on strong but inconclusive evidence, the process is called inference (sometimes called induction). Example of inductive argument:

“Most students who did not do the tutorial questions will fail the exam. Ali did not do the tutorial questions. Therefore Ali will fail the exam.”

Propositional logic.

Simple types of statements, called propositions, are treated as atomic building blocks for more complex statements

“Yalta is a port or a holiday resort. Yalta is not a port. Therefore, Yalta is a holiday resort.”

Basic propositions in the argument are:

p – Yalta is a port;

q – Yalta is a holiday resort.

In abstract form, the argument becomes:

p or q

Not p

Therefore q

Or, using symbolic designation:

p v q

¬p

\ q

Predicate logic.

A ‘predicate’ is just a property. Predicates define relationships between any number of entities [substances] using qualifiers:

" “for all”, “for every”;

$ “there exists”

Example:

Let P(x) be the property ‘if x is a triangle then the sum of its internal angles is 180°”.

In predicate logic:

" x P(x)

“For every x such that x is a triangle, the sum of the internal angles of x is 180°”.