Conclusion Mathematical Logic

To prove the new argument or proof, the truth must be shown as a result of another group of statements, each of which may be accepted as true or previously verified. The revelations received by the truth without needing proof are called axioms. For example, "Two different lines can not intersect at more than one point".

In proving a proposition or deriving an outcome of the known truths the argumentation pattern is used, namely by drawing conclusions from known statements called premises based on logical principles, namely ponen mode, tollens mode and syllogism.

Conclusions are said to be legitimate, if the conjunctions of the premises have concluded implications. Conversely, if the conjunctions of the premises have no implication then the argument is said to be false or illegitimate. Thus, a conclusion is said to be valid if the premises are true then the conclusions are also true.

1. Ponen mode

The ponen mode is an argument in the form of the following:

"If p → q is true and p is true then q is true"

In the form of diagrams can be presented as follows:

Example of Ponen Mode

Premise 1: If a child diligently learns, then he passed the test
Premise 2: Ahmad is a diligent child
Conclusion: ∴Ahmad passed the exam

To test valid or not parse-drawing inference can be used truth table. Ponen mode argument "If p → q is true and q is true then q is true" can be written in the form of implication, that is:

[(p → q) ∧ q] → q

This conclusion is said to be valid if it is a tautology. The truth table of the form is as follows:

Information :
T : True
F : False

From the table above it appears that [(p → q) ∧ q] → q is a tautology. So the argument or conclusion of the ponen mode form is valid.

2. Tollens mode

The tollens mode is an argument in the form of the following:

"Jika p → q benar dan ~q benar maka ~p benar"

In the form of diagrams can be presented as follows:

Example of Tollens Mode

Premise 1: If it is Sunday, then Budi is on an excursion
Premise 2: Budi is not on an excursion
Conclusion: ∴ it is not Sunday

To test valid or not conclusion by tollens mode can be used truth table. Ponen mode argument "If p → q is true and ~ q true then ~ p is true" can be written in the form of implication, that is:

[(p → q) ∨ ~q] → q

This conclusion is said to be valid if it is a tautology. The truth table of the form is as follows:

Information :
T : True
F : False

From the table above it appears that [(p → q) ∨ ~ q] → q is a tautology. So the argument or conclusion of form tollens mode is valid.

3. Silogism

Silogism is an argument shaped as follows:

"If p → q is true and q → r is true then p → r is true"

In the form of diagrams can be presented as follows:

Example of Silogism

Premise 1: If you study hard, then you go to class
Premise 2: If he goes to class, he will buy a bicycle
Conclusion: ∴If you study hard, you will buy a bicycle

To test valid or not silogism conclusion can be used truth table. The silogism argument "If p → q is true and q → r is true then p → r true" can be written in the form of implication, that is:

[(p → q) ∧ (q → r)] → (p → r)

This conclusion is said to be valid if it is a tautology. The truth table of the form is as follows:

Information :
T : True
F : False

From the table above it appears that [(p → q) ∧ (q → r)] → (p → r) is a tautology. So the argument or conclusion of the silogism form is valid.

The conclusion does not depend on the fairness or not the meaning of the conclusion as a statement but on the truth value of the conclusion.

Arguments whose conclusions are meaningful but are not obtained by using logical principles, then the conclusions are invalid.
Some of the arguments to which the conclusions are unusual but are obtained by using the principles of logic hence the conclusions are valid.

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