Symbolic Logic

In any discipline one seeks to establish facts and to draw conclusions based on observations and theories. One can do so deductively or inductively. In inductive reasoning, one starts with many observations and formulates an explanation that seems to fit. In deductive reasoning, one starts with premises and, using the rules of logical inference, draws conclusions from them. In disciplines such as mathematics, deductive reasoning is the predominant means of drawing conclusions. In fields such as psychology, inductive reasoning predominates, but once a theory has been formulated, it is both tested and applied through the processes of deductive thinking. It is in this that logic plays a role.

Basic to deductive thinking is the word "implies," symbolized by "=>." A statement p=> q means that whenever p is true, q is true also. For example, if p is the statement, "x is in Illinois," and q is the statement "x is in the United States," then p=> q is the statement, "If x is in Illinois, then x is in the United States."

In logic as well as in ordinary English, there are many ways of translating p=> q into words: "If p is true, then q is true" "q is implied by p;" "p is true only if q is true;" "q is a necessary condition for p;" "p is a sufficient condition for q."

The implication p => q has a truth table some find a little perplexing:

The perplexing part occurs in the next to last line where a false value of p seems to imply a true value of q. The fact that p is false does not imply anything at all. The imolication says only that q is true whenever p is. It doesn't say what happens when p is false. In the example given earlier, replacing x with Montreal makes both p and q false, but the implication itself is still true.

Implication has two properties which resemble the reflexive and transitive properties of equality. One, p=> p, is called a "tautology." Tautologies, although widely used, do not add much to understanding. "Why is the water salty?" asks the little boy.

"Because ocean water is salty," says his father.

The other property, "If p=> q and q=> r, then p=>r," is also widely used. In connecting two implications to form a third, it characterizes a lot of reasoning, formal and informal. "If we take our vacation in January, there will be snow. If there is snow, we can go skiing. Let's take it in January." This property is called a "syllogism."

A third property of equality 1 "If a = b, then b = a," called symmetry may or may not be shared by the implication p=>q. When it is, it is symbolized by the twoheaded arrow used earlier, "p q." p q means (p=> q) Λ (q=> p). It can be read "p and q are equivalent;" p is true if and only if q is true;" "p implies and is implied by q;" "p is a necessary and sufficient condition for q;" and "p implies q, and conversely."

In p=> q, p is called the "antecedent" and q the "consequent." If the antecedent and consequent are interchanged, the resulting implication, q=> p, is called the "converse." If one is talking about triangles, for example, there is a theorem, "If two sides are equal, then the angles opposite the sides are equal." The converse is, "If two angles are equal, then the sides opposite the angles are equal."

If an implication is true, it is never safe to assume that the converse is true as well. For example, "If x lives in Illinois, then x lives in the United States," is a true implication, but its converse is obviously false. In fact, assuming that the converse of an implication is true is a significant source of fallacious reasoning. "If the battery is shot, then the car won't start." True enough, but it is a good idea to check the battery itself instead of assuming the coverse and buying a new one.

Implications are involved in three powerful lines of reasoning. One, known as the Rule of Detachment or by the Latin modus ponendo ponens, states simply "If p=> q and p are both true, then q is true." This rule shows up in all sorts of areas. "If x dies, then y is to receive $100,000." When x dies and proof is submitted to the insurance company, y gets a check for the money. The statements p=> q and p are called the "premises" and q the "conclusion."

A second rule, known as modus tollendo tollens, says if p=>q is true and q is false, then p is false. "If x ate the cake, then x was home." If x was not at home, then someone else ate the cake.

A third rule, modus tollerdo ponens, says that if p V q and p are true, then q is true. Mary or Ann broke the pitcher.

Ann did not; so Mary did. Of course, the validity of the argument depends upon establishing that both premises are true.

It may have been the cat.

Another type of argument is known as reductio ad absurdum, again from the Latin. Here, if one can show that ~p=> (q Λ ~q), then p must be true. That is, if assuming the negation of p leads to the absurdity of a statement which is both true and false at the same time, then p itself must be true.