Mathematical logic ( besides known as symbolic logic ) is a subfield of mathematics with close connexions to computing machine scientific discipline and philosophical logic. The field includes both the mathematical survey of logic and the applications of formal logic to other countries of mathematics. The consolidative subjects in mathematical logic include the survey of the expressive power of formal systems and the deductive power of formal cogent evidence systems.
Mathematical logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional survey of logic. Before this outgrowth, logic was studied with rhetoric, through the syllogism, and with doctrine. The first half of the twentieth century saw an detonation of cardinal consequences, accompanied by vigorous argument over the foundations of mathematics.
The history of logic is the survey of the development of the scientific discipline of valid illation ( logic ) . While many civilizations have employed intricate systems of logical thinking, and logical methods are apparent in all human idea, an expressed analysis of the rules of logical thinking was developed merely in three traditions those of China, India, and Greece. Of these, merely the intervention of logic falling from the Grecian tradition, peculiarly Aristotelean logic, found broad application and credence in scientific discipline and mathematics. Logic was known as dialectic or analytic in Ancient Greece.
Aristotle ‘s logic was further developed by Islamic and so mediaeval European logisticians, making a high point in the mid-fourteenth century. The period between the 14th century and the beginning of the 19th century was mostly one of diminution and disregard, and is by and large regarded as waste by historiographers of logic. Logic was revived in the mid-nineteenth century, at the beginning of a radical period when the topic developed into a strict and formalized subject whose example was the exact method of cogent evidence used in mathematics.
CONTRAPOSITIVE AND CONTRADICTION
Contraposition is a logical relationship between two propositions, or statements. For illustration, take the undermentioned ( true ) proposition: “ All birds are animate beings ” . We can repeat that as: “ If something is a bird, it is an animate being ” . The contrapositive is: “ If something is non an animate being, so it is non a bird ” . In mathematics and logic, the contrapositive is ever guaranteed to be true, every bit long as the original proposition was true. If the original proposition is false, the contrapositive will ever be false as good.
The contradiction is “ There exists a bird that is non an animate being ” . If the contradiction is true, the original proposition ( and by extension the contrapositive ) is untrue. Here, of class, the contradiction is untrue.
It appears clear that if something is in A, it must be in B, every bit good. We can paraphrase that as
It is besides clear that anything that is non within B can non be within A, either. This statement, is the contrapositive.
Therefore we can state that,
To reason, for ANY statement where A implies B, so not-B ALWAYS implies not-A. Proving or confuting either one of these statements automatically proves or disproves the other. They are absolutely logically tantamount.
A proposition Q is implicated by a proposition P when the following relationship holds
In common footings, this states “ If P so Q ” , or, “ If Socrates is a adult male so Socrates is human. ” In a conditional such as this, P is called the ancestor and Q the consequent. One statement is the contrapositive of the other merely when its ancestor is the negated consequent of the other, and vice-versa. The contrapositive of the given illustration statement would be
That is, “ If not-Q so not-P ” , or more clearly, “ If Q is non the instance, so P is non the instance. ” Using our illustration, this is rendered “ If Socrates is non human, so Socrates is non a adult male. ” This statement is said to be contraposed to the original, and is logically tantamount to it. Due to their logical equality, saying one is efficaciously the same as saying the other, and where one is true, the other is besides true ( likewise with falseness ) .
Strictly, a contraposition can merely be in the above signifier of two simple conditionals. However, it is common to name two more complex statements contraposed if they are the same apart from incorporating a contraposition. Therefore, or “ All P ‘s are Q ‘s ” is contraposed to or “ All non-Q ‘s are non-P ‘s ” .
SIMPLE PROOF BY CONTRADICTION
Suppose that it is given that
This means that it is given that if A is true, so B is true, and it is besides given that B is non true. We can so demo that A must non be true by contradiction. For if A were true, so B would hold to besides be true ( given ) , but it has been given that B is non true, so we have a contradiction. Therefore A is non true ( presuming that we are covering with concrete statements that are either true or non true ) , and we have that
We can use the same procedure the other manner unit of ammunition. Let it be given that
We besides know that B is either true or non true. If B is non true, so A is besides non true. However, it has merely been given that A is true, so presuming that B is non true leads to contradiction and must be false. Therefore, B must be true, and we have proved that:
Uniting the two proved statements gives:
Which makes them logically tantamount.