# Posts Tagged ‘Proposition’

## 3 Value Logic

Posted by allzermalmer on May 10, 2013

I am going to use Polish Notation in expressing these truth tables of 3 value logic and 2 value logic. Lower case letters are variables: x, y, z, …
Capital Letters represent logical operators: N, A, K, E, C

Nx = ~x
Axy = x v y
Kxy = x & y
Cxy = x → y
Exy = x ↔ y

This notation is explicated in the text book Formal Logic by logican A.N. Prior in the late 1950’s to early 1960’s. The notationw as used by Polish logican Jan Lukasiewicz. Lukasiewicz was one of the first logicans to formally organize a three value logic. The logical matrix or logical matrices of both 2 value logic and 3 value logic are presented.

2 value logic uses 1 and 0.
3 value uses 1, 1/2, and 0.

1 stand for true.
1/2 stands for indeterminate.
0 stands for false.

I have put in bold those portions of truth tables in 3 value logic that do not have a similar truth table in 2 value.

Affirmation & Negation (x & Nx) : 2 value

• (1) If x = 1 then Nx = 0.
(2) If x = 0 then Nx = 1.

Affirmation & Negation (x & Nx) : 3 value

• (1) If x = 1 then Nx = 0.
• (2) If x = 1/2 then Nx = 1/2
• (3) If x = 0 then Nx = 1

Conditional (Cxy) : 2 value

• (1) If x = 1 and y = 1, then Cxy = 1
• (2) If x = 1 and y = 0, then Cxy = 0
• (3) If x = 0 and y = 1, then Cxy = 1
• (4) If x =0 and y = 0, then Cxy = 1

Conditional (Cxy) : 3  value

• (1) If x = 1 and y = 1, then Cxy = 1
• (2) If x = 1 and y = 1/2, then Cxy = 1/2
• (3) If x = 1 and y = 0, then Cxy = 0
• (4) If x = 1/2 and y = 1, then Cxy = 1
• (5) If x = 1/2 and y = 1/2, then Cxy = 1
• (6) If x = 1/2 and y = 0, then Cxy = 1/2
• (7) If x = 0 and y = 1, then Cxy = 1
• (8) If x = 0 and y = 1/2, then Cxy = 1
• (9) If x = 0 and y = 0, then Cxy = 1

Conjunction (Kxy) : 2 value

• (1) x = 1 and y = 1, then Kxy = 1
• (2) x = 1 and y = 0, then Kxy = 0
• (3) x = 0 and y = 1, then Kxy = 0
• (4) x = 0 and y = 0, then Kxy = 0

Conjunction (Kxy) : 3 value

• (1) If x = 1 and y = 1, then Kxy = 1
• (2) If x = 1 and y = 1/2, then Kxy = 1/2
• (3) If x = 1 and y = 0, then Kxy = 0
• (4) If x = 1/2 and y = 1, then Kxy = 1/2
• (5) If x = 1/2 and y = 1/2, then Kxy = 1/2
• (6) If x = 1/2 and y = 0, then Kxy = 1/2
• (7) If x = 0 and y = 1, then Kxy = 0
• (8) If x = 0 and y = 1/2, then Kxy = 0
• (9) If x = 0 and y = 0, then Kxy = 0

Disjunction (Axy) : 2 value

• (1) If x = 1 and y = 1, then Axy = 1
• (2) If x = 1 and y = 0, then Axy = 1
• (3) If x = 0 and y = 1, then Axy = 1
• (4) If x = 0 and y = 0, then Axy = 0

Disjunction (Axy) : 3 value

• (1) If x = 1 and y = 1, then Axy = 1
• (2) If x = 1 and y = 1/2, then Axy = 1
• (3) If x = 1 and y = 0, then Axy = 0
• (4) If x = 1/2 and y = 1, then Axy = 1
• (5) If x = 1/2 and y = 1/2, then Axy = 1/2
• (6) If x = 1/2 and y = 0, then Axy = 1/2
• (7) If x = 0 and y = 1, then Axy = 1
• (8) If x = 0 and y = 1/2, then Axy = 1/2
• (9) If x = 0 and y = 0, then Axy = 0

Biconditional (Exy) : 2 value

• (1) If x = 1 and y = 1, then Exy = 1
• (2) If x = 1 and y = 0, then Exy = 0
• (3) If x = 0 and y = 1, then Exy = 0
• (4) If x = 0 and y = 0, then Exy = 1

Biconditional (Exy) : 3 value

• (1) If x = 1 and y = 1, then Exy = 1
(2) If x = 1 and y = 1/2, then Exy = 1/2
(3) If x = 1 and y = 0, then Exy = 0
(4) If x = 1/2 and y = 1, then Exy = 1/2
(5) If x = 1/2 and y = 1/2, then Exy = 1
(6) If x = 1/2 and y = 0, then Exy = 1/2
(7) If x = 0 and y = 1, then Exy = 0
(8) If x = 0 and y = 1/2, then Exy = 1/2
(9) If x = 0 and y = 0, then Exy = 1

## Truth of Reasoning and Truth of Fact

Posted by allzermalmer on October 26, 2012

“All that which implies contradiction is impossible, and all that which implies no contradiction is possible.” G.W. Leibniz

“I assume that every judgement (i.e. affirmation or negation) is either true or false and that if the affirmation is true the negation is false, and if the negation is true the affirmation is false; that what is denied to be true-truly, of course- is false, and what is denied to be false is true; that what is denied to be affirmed, or affirmed to be denied, is to be denied; and what is affirmed to be affirmed and denied to be denied is to be affirmed. Similarly, that it is false that what is false should be true or that what is true should be false; that it is true that what is true is true, and what is false, false. All these are usually included in one designation, the principle of contradiction.” G.W. Leibniz

“There are . . . two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible; truths of fact are contingent and their opposite is possible. When a truth is necessary, its truth can be found by analysis, resolving it into more simple ideas and truths, until we come to those which are primary. It is thus, that in Mathematics speculative Theorems and practical Canons are reduced by analysis to Definitions, Axioms, and Postulates. In short, there are simple ideas, of which no definition can be given; there are also axioms and postulates, in a word primary principles, which cannot be proved, and indeed have no need of proof, and these are identical propositions, whose opposite involves an express contradiction.” G.W. Leibniz

So Leibniz obtains all knowable propositions or statements to be divided based on the principle of contradiction. The truth of statements is divided into two realms. This also deals with what people can know, or knowability. It basically says that
“For each statement, if statement is knowable, then statement is either truth of reasoning or truth of fact. For each statement, if statement is truth of reasoning, then statements affirmation is logically possible and statements negation is logically impossible. For each statement, if statement is truth of fact, then statements affirmation is logically possible and statements negation is logically possible.”

A truth of reasoning is always true and not possible it is false. It is logically impossible that it is false. The negation of a truth of reasoning is an impossible statement or impossible proposition. It is self-contradictory. A truth of fact is not always true and possible it is false. It is logically possible that it is true or logically possible it is false. Truth of Reasoning is Logically Necessary and Truth of Fact is Logically Contingent.

“For each statement, if statement is Truth of Fact, then statement is an empirical claim. For each statement, if statement is Truth of Reasoning, then statement is not an empirical claim. For each statement, if statement is Truth of Reasoning, then statement is non-empirical claim. For each statement, if statement is Truth of Fact, then statement is not non-empirical claim.”

What also happens to come from this is that Truth of Facts do not entail or lead to Truth of Reasoning, and Truth of Reasoning do not entail or lead to Truth of Fact. This means that Truth of Facts do not imply or entail non-empirical claims and Truth of Reasoning do not imply or entail empirical claims. This means that statements of experience are not non-empirical claims and means statements of experience are empirical claims.