Editor | On 07, Jan 2008

By Abram Teplitskiy

Logic allows for different viewpoints and, thus, can help lead to more solutions for inventive problems.

Deductive Reasoning

People who are completely sure they solved the secret behind perpetual motion submit claims for “eternal engines.” Their conclusion is based on the following type of reasoning: “Everyday I see sunrise and sunset. My parents said that they constantly witnessed these events, my grandparents said the same, so we can conclude that our planet’s rotation around the Sun is eternal, and it is in perpetual motion.” Because the human life is, unfortunately, rather short compared to astronomic time, basing a conclusion on such a small fragment of planetary time is unwise. In logic, it means that the induction is incomplete, because we are not able to check all possible options of future time. To satisfy perpetual motion believers, a natural “perpetual-like” generator – the sunflower; it follows the rotation of the Sun.

Deductive conclusions are true if their premises are true. The deductive relationship between premises and conclusions is called a syllogism. A classical example of valid syllogism is: Any man is a living organism. Socrates is a man. Therefore, Socrates is a living organism.

Logic and Contradictions

Logic successfully guided Galileo in his reasoning about the speed of falling bodies. At that time, Aristotle’s hypothesis that heavy bodies fall faster than lighter (e.g., a cannonball should fall faster than a small bullet) was believed. Galileo conducted a logical experiment. Supposing that Aristotle was right, Galileo developed two premises about the speed of these two objects falling together. Recognizing that the weight of these bodies tied together is greater than their separate weight, Galileo supposed:

1. The heavier body (the cannonball) will speed up the fall of the bullet.
2. The lighter body (the bullet), falling slower, will slow down the fall of the cannonball.

These two premises are contradictory. Galileo solved this contradiction with a paradoxical conclusion: all bodies, light or heavy, must fall with equal speed when they are unaffected by air resistance. Later, Galileo’s student Evangelist Torricelli, pumped out air from a standpipe, and let objects of different weight (such as a feather and a bullet) fall in vacuum. The experiment proved Galileo’s logical conclusions.

Now consider the so-called “barber paradox.” In a village, authorities established a rule: “A local barber can shave those, and only those, who do not shave themselves.” One day the barber said: “I am in a muddle and I don’t know how to get out of it. The council says I have to shave only those who do not shave themselves. What about me? If I shave myself, I belong to the self-shavers and therefore I am not allowed to shave myself. If I do not shave myself, I am one of the others, and therefore I must shave myself. What do I do?” There is a simple solution: the barber must go outside his village, where the village’s rules are not in force, to shave. This solution is simple and sounds like a job, but could work.

More examples of reasoning are found in Sir Arthur Conan Doyle’s stories about Sherlock Holmes. One of his stories, “Sign of Four,” started the “Holmes Deductive Method.” Holmes concluded that Dr. Watson visited the post office to send a telegram. His conclusion was based on observations: Watson’s shoes were covered with reddish clay, and this type of clay exists only around the post office; therefore, Watson visited the post office.

Logical thinking can be further broken down into:

1. Analysis: In the inventive field, analysis can be defined as the highlighting of all elements, conditions, component ties, etc. in a problem. In analysis, other similar problems’ elements are also examined and evaluated. Analysis helps to clarify the problem.
2. Synthesis: Synthesis is a logical operation that combines different elements (highlighted during analysis) in a new arrangement. Additional knowledge, such as solutions to similar problems, personal experience, etc. is the basis of transforming the elements of the problem into a solution.

The first step of both analysis and synthesis is based on finding differences and similarities of objects. By looking at something’s significant features, objects can be divided into different groups – classified. For example, all numbers can be divided in two groups as even or odd numbers. Alternatively, numbers can be classified as prime or composite. Compare the circles shown in Figure 1 – how can they be classified?

Figure 1: Classification Test Courtesy of Misha Teplitskiy

Look at the triangle in the left of Figure 2. Now, look at the shifted orientation of the triangle on the right. If a person has the opportunity to compare the initial and final placings of circles the solutions should be easy. In this case, look for a solution backwards – this is a powerful approach for inventive problem solving!

Figure 2: Changing Coins Courtesy of Misha Teplitskiy

The simplest way to rearrange the coins is shown in Figure 3.

Figure 3: Coin Rearrangement Courtesy of Misha Teplitskiy

Changing Objects

The first law of dialectic logic is that quantity becomes quality. In other words, by taking one grain as a first class of objects, by adding grain after grain that one grain becomes another class of objects – a pile. This is an example of one system (single grain) transferring to another system (pile of grains) without having definite boundaries.

In dialectic logic, objects or processes can exist in two contradictory states, exhibiting different properties in different times or spaces. Sir Isaac Newton concluded from his investigations of light and reflections of light streams that light is a flow of particles. Other scientists considered light to be a wave due to the effects of light diffraction and interference. Who was right? There was a longstanding discussion and the solution was unexpected – light has both particle and wave properties! It was the first time that such dualism had shown its reliability in physics. The contradictory properties coexisted in light’s nature, but are presented separately in time. Sometimes contradictory features form a friendly union in nature; the chameleon, for example, can change its color depending on how dangerous a situation is.

Exercises in Dialectic Logic

1. Outcrop opposites in objects. Select 20 objects of choice and outcrop what is “good” and “bad.”
2. A schoolbag needs to be enough big to carry all necessary books and notebooks, but also small enough to not overload students. Propose effective solutions for the schoolbag problem.
3. Create a chain of positive and negative features of some objects. In Aesop’s fable “The Ant and The Grasshopper,” the grasshopper was flying and singing during summer while the ant was working hard and storing food for winter. Some will think that it is good that the ant works hard, and it is bad that the grasshopper did not store any food for winter. But some people select a different good – the grasshopper’s summer singing brought a lot of pleasure to the country’s occupants. Present a different chain and think about its positive and negative features.
4. Prehistoric people needed to carry bludgeons that needed to be heavy to kill an animal, but also light to be easily carried. Catapults must be heavy to sling big stones, but light to facilitate operation. Create different examples of similar contradictions.

Happy inventing!