The disjunction (logical addition, logical “or”)

"It is not clear, though will kill,

Whether it is snow

or the falcon

Runs of the white pigeons

By asters..."

It is cite from the poem by Eugeni Leonidze «A Song of the first snow». Poetic image is not scientific prediction.

In expressly strict shape: «It is snowing or the falcon runs the white pigeons».

Before us two partial propositions are, coherent union «or». Certainly, if is true both first (it is snowing) and second (the falcon runs the white pigeons), all folded of them the proposition as a whole will be true also.

But let is true even first of them. Even if second is false, it is necessary to recognize all composite proposition as a whole as true.

Same there will be it and then, when the second proposition is true, and first is false.

And only in that case, when both are false, the proposition derived from them by means of union "OR" will be false also.

A (snow)	B ( falcon runs pigeons)	A or B
T	T	T
T	F	T
F	T	T
F	F	F

Logic operation «or», with which one we just were acquainted, is termed as a disjunction.

With help of disjunction third proposition will be derivation from two propositions: it is true, when one of two initial propositions is true even, and it is false, when both are false.

Disjunction is designated as A+B or A V B.

It is read as “A or B”.

The implication (logical consequence)

One of the chapters «A Universal history" treated by «Satiricon», is dedicated to ancient country Sparta (or Laconica) and to the orders, customs it inhabitants, and, certainly, their peculiar style to express: «They from childhood were inured to speak laconic, that is shortly and hardly... So, one spartanic women, returning the cock to cook-girl for freeing, said laconic: «will be overroasted — will be punished».

It can not be translated into English more precisely (not laconic):

«If it will be overroasted — then I will punish you».

The composite proposition of such structure names as an implication of those two partial propositions, from which one it is derived. From them first, the supplied word «if», is named as premise, or foundation, second, begin with a word that, — corollary, or conclusion.

The composite proposition of such structure names as an implication of those two partial propositions, from which one it is derivative.

Implication is designated as A→B or A B or A B.

It is read as “if A then B”, “if A else B”;”A implicated B”; “A inclinate B”.

From them first, the supplied word «if», is named as premise [foundation, reason, antecedent] second, begin with a word «then», — corollary [conclusion, consequence].

A truth table for strict definition of the relevant logic operation also called as an implication.

A	B	If A then B
T	T	T
T	F	F
F	T	T
F	F	T

The composite proposition of such structure names as an implication of those two partial propositions, from which one it is derivation.

To construct it, it is necessary to analyze all possible outcomes of a history with the cock and cook.

The cock is overroasted – cook is punished. That is both premise and corollary are true. Then according to promising of mistress (and severe laconic women knew how to hold a word!) all composite proposition is true.

The cock is overroasted, but cook has escaped punishment. That is the premise is true, and corollary is not. At those orders, which one present on laconic kitchens, such is absolutely eliminated, it is manifestative lie.

The cock is not overroasted, cook is not punished. True.

The cock is not overroasted, but cook is punished. Too true: that does not happen under a hot arm in the house, where severe laconic woman leads!

It may be such in mathematical logic too.

It is example: “If every of two values divides on three without residual, then sum of these values divides on three without residual too”. Let’s show to the variants.

It is possible that both basis and consequence are true. 6 and 9 divide on 3, and sum 6+9=15 too. It is possible that both basis and consequence are false. 2 and 5 don’t divide on 3, and it’s sum 2+5=7 don’t divide on 3. It is possible that basis is false, but consequence is true. 4 and 8 don’t divide on3, but it’s sum 12 divides. But it is impossible us to find example that two values divides on three, but it’s sum don’t divides.

Result: implication of two propositions is false when and only when, when basis is true and consequence is false.

So, what is happened? To negation an implication of two propositions A and B (it is wrong, that from A that follows B) is what to state: A and not B.

This construction from two propositions is constructed with the help of previous logic operations conjunction and negation, and consequently it is uneasy to establish, what truth values accepts this composite proposition depending on that, the trues or are false its partial components. Let's construct relevant implications.

As before: the implication of two propositions is false in that and only in the event that the premise is true, and the conclusion is false.

Verity of the implication of two propositions in the case of false premise means that consequence of false is all anything.

All conclusions to output one statement from other and components an essence of any logic reasoning are based on properties of implication.