Shea Zellweger - Contribution

Contribution

Zellweger’s contribution to the field of logic is best demonstrated through his development of the X-stem Logic Alphabet (XLA). The XLA notation is a highly advanced extension of both Charles Sanders Peirce’s box-X notation (1902) and Warren Sturgis McCulloch’s dot-X notation (1942). It could be said that XLA (1961–62) is the evolutionary product of the comprehensive work of Peirce, McCulloch, and Zellweger, or PMZ as an acronym. The standard notation used today (dot Logical conjunction, vee Logical disjunction, horseshoe Material conditional representing and, or, if) is a lingering, overly abstract, unsystematically selected set of symbols that was primarily developed and used by Peano, Whitehead, and Russell, or by common acronym PWR. This already exposes the primary weakness. Dot, vee, horseshoe do not carry any information that identifies, specifies, and encodes the truth tables they represent, namely, TFFF, TTTF, and TFTT. In marked contrast, XLA is an intentionally engineered set of sixteen iconographic letter shape symbols specifically designed to improve the efficiency of learning and performing logical operations. Serving as a system of highly abbreviated mini truth tables, Zellweger’s claim is that XLA is not only much easier to learn. It is also much easier to use. In fact, when ten-base numerals are used without the abacus and when XLA is used without written laid out rows and columns of truth tables, ordinary operations in both notations are easier to perform during the act of computational writing.

It can be said that the current PWR symbols are to logic what Roman Numerals are to arithmetic. Roman Numerals (I, II, III) were cumbersome to use and only maintained a dominant role in arithmetic until 1202, when Leonardo Fibonacci in his work Liber Abaci, demonstrated that calculations with Hindu-Arabic numerals (1, 2, 3) were far more efficient. The lack of mental and written efficiency in the use of traditional PWR symbols may be because they are not icons. Therefore, these extremely abstract symbols cannot in writing visually depict the truth tables themselves, the simple geometric forms, the notational symmetry relations, and the isomorphic sets of interrelations inherent in logic. Conversely, the XLA symbols are iconographic and they possess a shape value. This enables complex logical operations to be performed through easy flips and rotations of the letter shape symbols themselves.

By design, the letter shape of each X-stem Logic Alphabet symbol visually embodies and displays its individual underlying logic truth table. In other words, after the simple and exact truth table code in the deep structure of XLA has been learned, operations performed on the letter shape symbols are equivalent to logical operations acting on highly abbreviated sets of mini truth tables. Consequently, those using XLA never have the need to interrupt their calculations to check rows and columns of laid out truth tables. This basic and central advantage of XLA over PWR is often not fully recognized, even by practiced logicians. Nonetheless, systems of notation evolve and improve over time (e.g. Roman Numerals to the Decimal System and Imperial Units to the Metric System).

In brief, XLA is described in two steps: (1) give the 16 binary connectives the right geometry, the right shape value anatomy; and (2) add the transformational physiology, namely, apply the algebra of simple symmetry groups to the 16 iconic letter shape symbols. Change comes with a whisper. This whisper presents a triple isomorphism. The mental operations are the same as the symmetry operations are the same as the logical operations. Said the other way around, the logical operations are the same as the symmetry operations are the same as the mental operations. Said again in a different order, the logical operations are the same as the mental operations are the same as the symmetry operations. Here we have a prime example of cognitive ergonomics at its best. The single act of performing any one automatically performs the other two.

Whether or not the (PMZ) (XLA) system, or something similar to it, replaces the traditional PWR symbols remains to be seen. Nonetheless, for researchers and semioticians, Zellweger's contributions to logic notation will most likely play a valuable role in future developments.