Laws of Logic and Reasoning

It is generally agreed that what we know as the Laws of Logic were first formulated by Plato and Aristotle during the 5th Century BC.  In his book Laws, Plato established a framework to promote a just and virtuous society.  This included a commitment to seeking objective truth and distinguishing between true and false beliefs.  It was Aristotle who wrote what we now recognize as the first Laws of Logic. Aristotle was concerned with logic used as the basis for law, philosophy, and science more so than fostering a perfect society.  These laws are still considered as the foundation for rational thought.

Plato’s Laws

Plato’s laws can be expressed as follows; 1. Nothing can become greater or less, either in number or magnitude, while remaining equal to itself.  2. It is impossible to have an increase or decrease of anything without addition or subtraction.  3. Nothing of a physical nature can exist without becoming and having become.  In other words, physical existence necessitates a caused beginning. This law is sometimes refered to The Law of sufficient Reason.

On the surface, these statements seems ridiculously simple.  However, the implications are far-reaching.  Taken as a whole, these laws express that physical existence cannot cause itself.  They also allude to any change in a physical object must be the result of forces outside of that object.

           Plato and Aristotle – a painting by Raphael Sanzio. Licensed under CC BY

Aristotle’s Laws

It was Aristotle who developed what we call today the 3 Laws of Thought.  These (along with a 4th law) make up the foundation of study in formal logic.  The laws are:
  1. The Law of Identity. This law states that everything is identical to itself.  Symbolically, if A represents a true statement then “A = A”. In other words, if statement A is a truth statement then it is always true when presented in its original context.  This also holds for a false statement.  If statement A is false then it is always false when presented in its original context. An example would be to say “George Washington was the first President of the United States under the Constitution” is a true statement. It is equal to “The first President of the United States under the Constitution was George Washington”.  The second statement is true because it is equal to the first which is true.  Caution must be given to make sure which “George Washington” is being referred to.  If the first statement is true, the second might be false if the person intended is a different “George Washington”.  In this case, even though the wording is the same, the statements are not equal because they do not pertain to the same context.

     

  2.  The Law of Non-Contridiction. This law asserts that a proposition (or statement) and its negation cannot both be true at the same time and in the same sense or location. For example, “I have only one cat and its name is Morris” and “I have only one cat and its name is not Morris” cannot both be true simultaneously in the same location. Symbolically, a negation is represented by ~.  So A and ~A cannot be both true at the same time or location.  Again, the importance of sense, time, and location cannot be over emphasized. “I have only one cat and its name is Morris” and “I have only one cat and its name is not Morris” can certainly both be true if the propositions are made at different times.

     

  3. The Law of the Excluded Middle.  Perhaps the most referenced of Aristotle’s laws, the Law of the Excluded Middle, states that for any proposition, either the proposition or its negation must be true; there is no third option.  The law gets its name from the fact there is no “middle ground” between true and false.  For example, “Today is Wednesday” or “Today is not Wednesday” must be true; there are no other possibilities.  Symbolically we could write A≠~A.  The Law of the Excluded Middle does not apply to A and ~B, A and ~C and so on.
 

In order to fully apply Aristotle’s Laws, there must be a way to compare one proposition, or statement, to another. It seems unlikely that Aristotle did not understand this so it may be that he simply took comparisions for granted. However, all studies of logic start off with the following:

         4. The Law of Quantitative Comparison.  This law states that given two entities A and B, both having a common       
          property P, then either A has more of property P than B,  or A has less of property P than B or A and B are indifferent to
          each other.  It is important to note here that comparisons are based on observations or assumed values.  This also
          introduces the very real possibility of bias when deciding on values and outcomes.  It also demands a consistent way
         of determing value.  Let’s say that A is a coffee cup in my house and B is an antique clock.  Both have value (which
          would be property P).   But how do you determine value?  Is it the resale amount in dollars, sentimental value, usage
          value? Either A or B could be of more value depending on the determining factor(s).

Types of Reasoning

There are several types or styles of reasoning.  The two most prominent are deductive and inductive. Logic relies more on deductive than inductive.  Deductive reasoning is also called mathematical reasoning.  The process is to take a general truth and reason specific truths. Inductive reasoning is also called scientific reasoning and employs the reverse. Inductive reasoning uses what is known about specific instances in order to infer general truths. There is a major difference between deductive and inductive reasoning, specific truths deduced from general are always true.  However, general truths induced from specific have the possibility of being false.
 
Example: Science uses observation and experimentation to find specific truths.  A piece of metal is heated and it is observed to expand.  The same goes for many other substances.  So science concludes that “as matter is heated it expands and as it cools it contracts”.   This general rules is induced based on the behaviors of the materials observed.  The general rule works…usually.  But it is false for water.  As water cools and freezes, it expands rather than contracts.  This makes the general rule true most of the time but false part of the time.  This is not to diminish many of the accepted laws of science.  Newton’s Laws of Thermodynamics are assumed to be universally true because there has not been a observed contradiction.  However, the only way that science can absolutely “prove” something to be true is to examine every possibility.  This is usually impossible.  So science examines a limited number of instances and induces that the unobserved will abide by the same rules and patterns.  
 
Deductive reasoning is the same type that is often studied in geometry in high school.  For instance, known general absolute truths about right angles and triangles in a plane are used to “prove” the Pythagorean Theorem”.  Thus the Pythagorean Theorem is absolutely true regardless of the size or orientation.  Concluding specific from general leads to certainity, which is why any study of logic leans heavily on deductive reasoning. Deductive reasoning typically relies on certain rules of inference, called the Law of Detachment and the Law of Syllogism.
 
The Law of Detachment.  This law simply says that if a conditional statement (also known as an “if-then” statement) is true and the hypothesis (the “if” part of the statement) is true, then the conclusion (the “then” part of the statement) must be true. Symbolically, a conditional statement is represented by →.  “If A then B” would be written as A→B.  The Law of Detachment says that if A→B is true and A is true then B must be true. 
 
An example is the conditional statement “If a blue whale is the largest marine mammal in the world then it lives in the ocean”, which is true.  The hypothesis is “A blue whale is the largest marine mammal in the world”, which is also true. So, the conclusion “It lives in the ocean” must be true.
 
The Law of Syllogism
The law of syllogism, also known as reasoning by transitivity, states that if one conditional statement’s conclusion is the hypothesis of a second conditional statement, then a new conditional statement can be formed by combining the hypothesis of the first and the conclusion of the second. In simpler terms, if “if P then Q” and “if Q then R” are true, then “if P then R” is also true. Each of the given conditional statements is called a premise and the inferred statement is the conclusion. Symbolically a syllogism would take the form:
                                                                                                     A→B   (1st or Major Premise)
                                                                                                     C→A   (2nd or Minor Premise)
                                                                                                     C→B   (Conclusion)
The most famous example of a syllogism is:
                                                                                          All men are mortal.
                                                                                          Socrates is a man.
                                                                                          Socrates is mortal.

It is very important to distinguish between validity and truth.  A conclusion may be valid (meaning that the conclusion has been constructed correctly) but still be false if one or both premises are false. 
Example:
                                                                                        All men are fish.           (false Major Premise)
                                                                                        Socrates is a man.        (true Minor Premise)  
                                                                                        Socrates is a fish.         (valid but false Conclusion)
 
Similarily, it is also possible to have an invalid argument (even if both premises are true) and still have a true conclusion.
Example:
                                                                                     All men are mortal.                 (true Major Premise)
                                                                                     My dog Riley is not a man.    (true Minor Premise)
                                                                                     My dog Riley is mortal           (invalid but true Conclusion)
 
In this last example, we would say that the conclusion may be true but it is not logical, meaning that it does not follow.