Editor | On 01, Oct 2007

By Abram Teplitskiy

Logic is a vital tool for inventors; it is usually defined as the science of correct thinking. Examine this ancient reasoning trick: “What you didn’t lose, you have. You didn’t lose a tail, therefore you have a tail.” Do you feel that you’ve been led into a trap?

Logical fallacies can ambush you in life. You need training to prevent mistakes. Take a second, and compare the vertical and horizontal segments enclosed between arrows shown in Figure 1. What segment is bigger? Check your guess with measurements. Prepare yourself – you will likely face more than one such illusion in your life.

Figure 1: Logic Test with Arrows

Now, try to perform some magic. Millions of steel balls are manufactured for ball bearings and each of these balls needs testing. What is a simple, inexpensive way to test millions of balls using nothing? It is not impossible. For rapid testing, balls for bearings are dropped onto a rigid plate and rebound depending on their quality – each ball “jumps” in a window, commensurate to its quality. It is an ideal solution – the balls test themselves.

Figure 2: Testing Ball Bearings Courtesy of Merle and Kelly Cunningham

There are many examples of “deformed” reflections in the world around us. Segments of equal length look as if they have different lengths; objects of equal weight are perceived as having different weights. With these deformed reflections, a person may even mistakenly find that she has two noses! Cross two fingers and move them along your nose – you will feel as if you have two noses.

Figure 3: “Deformed” Reflection Courtesy of Merle and Kelly Cunningham

Information contained in words can be affected by a distortion of our senses and, as a result, can have no “sense.” In the previous sentence, we already have used the word sense to represent two different meanings. Word structures can have ambiguity without directly using same words. Analyze a sentence, “This sentence is false.” Taking this statement as true, you have to consider it false.

Information contained in words can be affected by a distortion of our senses and, as a result, can have no “sense.” In the previous sentence, we already have used the word sense to represent two different meanings. Word structures can have ambiguity without directly using same words. Analyze a sentence, “This sentence is false.” Taking this statement as true, you have to consider it false. Or consider the statement, “All citizens of Crete are liars.” If we take this statement as true, the “one citizen” is also a liar; the above statement is a lie. Therefore, not all citizens of Crete are liars.

Such paradoxical situations regularly occur in real life. Let’s discuss Zeno’s paradox. If runners of a race begin to run towards the finish line, the law of the infinitesimal (this law states that any distance may be halved infinitely) would state that the runners would halve the distance to the finish line over and over. This would continue ad infinitum and the runners would never cross the finish line. However, we regularly watch runners cross the finish lines and this proves that actual infinities do not exist in the world in which we live.

Figure 4: Zeno’s Paradox Courtesy of Merle and Kelly Cunningham

You can easily solve the Achilles-tortoise paradox using simple mathematics.

V = the speed of Achilles
W = the speed of the tortoise
D = the starting distance between Achilles and the tortoise

The question is, in what time Achilles will run across distance D, having a speed advantage. The difference in speed is (V-W) and T = D / (V-W). Everyday experience shows that Achilles will easily outrun the tortoise. Isn’t experience the best judge of paradoxes? Broadly speaking, Zeno’s argument is false in that its conclusion overlooks the fact that an infinite number of parts can add up to a finite whole.

The Zeno paradox is similar to the color spectrum paradox. If you go from red to orange on a spectrum of colors, you can move in steps that are small enough so that you cannot detect any color difference between the first red and the second, between the second red and the third, and so on through a large number of “successfully” undetectable differences until you get to orange. One could conclude that there is, therefore, no real detectable color difference between red and orange. This obviously false conclusion takes you from a property of the component steps in a process (imperceptibility of change at each part of the process) to a presumed property of the whole product (imperceptibility of change for the entire process).

Problems can arise not only with the interpretations of observation, but with spoken and written language. Logic is a science of correct thinking. Two examples of ambiguity that challenge classic follow: 1) roadside sign: clean and decent dancing every night except Sunday and 2) dog advertisement: eats everything – very fond of children.

An example of classic logic is based on the “law of the excluded third,” which states that a person cannot be simultaneously a liar and not-liar. Another classic logic law, “law of identity,” requires univocality. These two examples represent the fact that classic logic is based on contradictory features, the “black or white” principle. This dualistic approach to reasoning is called dialectic logic.

A coin toss is the simplest example of a dualistic approach to reasoning. A coin has two sides – a head and a tail. You often see referees in sports flip coins while members of competing teams have to guess the outcome. In this example of dialectic logic, the system has two possible outcomes but in each case the participants can observe only one of them. Being inventive, we can observe both outcomes of this trial by adding a mirror to the toss.

Figure 5: Dialectic Logic– Coin Toss

There are many variations of logic that can prove useful to inventors. Explore them and share your results with other Student Corner readers!

Happy inventing!