Boolean Chips Game of Logic

Armahedi Mahzar © 2015

Lately, I have found a new logical method which I called as arithmologic. Phenomenologically speaking, the arithmologic represents any logical expression as arithmetical expression, which is a string of signed alphamerics, and represents the logical reasoning with the process of discarding pairs of differently signed similar alphamerics.

The arithmologic method is so easy, that it can be taught to any primary school students. However, the method can be made easier by replacing signed alphamerics string with collection of different things, such as colored chips, and do the reasoning by playing the game which is nothing but the reduction of the collection by removing paired chips, so it can be taught to a kindergarten kid.

So let us start by forming chips collection to represent Aristotelian fundamental propositions.

Representing Logical Expression with Chips

Aristotle using verbal string of words to represent logical expression such as IF a THEN b. George Boole in 19^th century used string of algebraic symbols to represent the mentioned logical expression as 1-a+b. In the 20^th  century, George Spencer-Brown used a containment of forms    to express the same logical expression as  . Later, Louis Kauffman in his Box Algebra represent the same logical expression as. Finally, I replaced Kauffman letters with colored chips, to get Object Logic algebra the representation of the logical expression IF a THEN b is 

.

Now, we can construct an arithmologic game that simulates Boolean arithmetic by representing TRUE or 1 by black chip
 and variables by colored chips. Other representations for logical expressions is shown in the following table

  

The Categorical Proposition of Aristoteles

If a is represented by RED chip, b by a GREEN chip and 1 by a BLACK chip, the the four fundamental categoric proposition of Aristotle can be represented as it is shown in the following table.

Simulating the Syllogism Validity Proof

Reasoning by syllogism now can be simulated by three steps algorithm

    1. Combined all the representation of premises with AND or  as the combinator.

2. Dispose all chip pairs   where the red chip is the symbol of any colored chip.

   3. Read the rest as conclusion. If the rest is containing two color chips then the syllogism is valid. Otherwise it is invalid.

Proving Valid Syllogisms

To prove the Barbara syllogism, IF all m is p AND all s is m THEN all s is p, we represent s, m and p with red, green and blue chips and represent the conjunction of premises as the chips configuration above the horizontal line in the picture below. 

By discarding opposite pairs of chips, we will get the chips configuration below the line which can be read as the conclusion of the syllogism.

The proof the validity of all 15 syllogisms can be derived with the help of the following table.

Beside the 15 valid syllogisms without any assumption of the existence of a certain term, there are 9 valid moods of syllogism containing existential assumption.

For example, the validity of Barbari syllogism which is containing one assumption of the existence of the subject term can be proven like this, where the the third proposition is Iss is represented by the following picture. 

In the proof, we just eliminate the oppositional pair off chips above the horizontal line to get the chip configuration below the line.

The proof the validity of all 9 existential syllogisms can be shown in the following table.

 

Afternotes

The chips game can be used to prove hypothetical and disjunctive syllogisms of the Stoic logician. In fact it can be used to prove any Boolean tautology. So, it is shown that a game of concrete object can simulate any logical proof in abstract algebraic symbols.

The objects chosen in this blog are colored chips. However the colored chips can be replaced with any objects and the black chips can be replaced with any sheets of paper. For example, the colored chips are replaced with colored marbles and the black chips are replaced with closed cards.

In this blog, logic is formulated with Boolean algebraic symbols. However logic can also be represented by Peircean existential graphs or Sommersian arithmetical symbols. Both Peircean pictorial and Sommersian literal formulations can also be simulated with similar game of concrete things. The new games is simpler than the Boolean game described here.

All the logic games of concrete things are so easy to play that it can be taught to any kindergarten kid. Surely, we just teach them the rules of formation and transformation of the things arrangement without the logical interpretation.

Once they are skilled in the logic game playing, the algorithm will be deeply entrenched in their subconscious so it will facilitate their logical skill in later ages. Hopefully, the games can also enhance their IQ like the WFF’N PROOF game created by professor Layman E. Allen in the Yale University.