Logical Truths and Inference

On the other hand, $Q$ is a true statement, regardless of what the world view we take. In fact, $Q$ remains valid irrespective of the truth-value of its component statements, namely, $a$ and $b$. Logicians call statements like $Q$ logical truths.

Inference

We use logical truths in logical inference. Logical inference is taking specific facts and concluding some other fact from these given facts. In logical inference, we do not verify the validity of the given facts. Our concentration is only focused on making sure that our conclusions are logically sound.

One of the best-known examples of logical inference is given by Aristotle, in which he establishes that Socrates is mortal. Aristotle considers the following two facts:

Firstly, we must note that the purpose of logical inference is not to verify the validity of the given facts. Therefore, Aristotle does not attempt to justify the validity of $F_1$ and $F_2$. The facts in inference are simply taken for granted by a logician.

The second important thing to notice is that Aristotle’s conclusion $C$ is a new proposition; that is,

$C \not \equiv F_1$, and $C \not \equiv F_2.$

The conclusion $C$ is a fact that was not known until Aristotle logically concluded it from $F_1$ and $F_2$.

To really understand what Aristotle was trying to explain, we have to imagine ourselves living in Greece in the times when he lived. The ancient Greeks had a concept of immortal gods in contrast to the mortality of men. Socrates was the most distinguished of Greek philosophers.

Aristotle is teaching us the inevitability of logic. Once the two facts $F_1$ and $F_2$ are accepted, logic asserts that we must also bear their logical conclusion, namely, $C$. Although we can wish or hope that Socrates was immortal. Yet, logicians will have to set aside their hopes and accept the mortality of Socrates.

We can ask what happens if some of the given “facts” are not true. Where will logical inference lead us? The answer to this question is captured in the common phrase:

“Garbage in, garbage out!”

In case the initial statements given as “facts” happen to be untrue, the result of applying logical inference is completely unpredictable. In fact, things can be far worse. If the facts are inconsistent, logic can lead us to conclude any statement!

Let’s consider Aristotle’s example closely. Aristotle concludes $C$, that is, “Socrates is mortal.” If we think about the two given facts, $F_1$ and $F_2$ together contain more information than $C$. $F_1$ is claiming the mortality of all men, not only Socrates. $F_2$ assigns the status of a man to Socrates. Therefore, Socrates will have all the attributes or properties common to all men (mortality being only one).

So by applying logic, Aristotle has actually reduced information. The conclusion $C$ alone, has strictly less information than $F_1$ and $F_2$. Initial instinct says that the reduction of information is not useful. With logic, we should be able to increase information. Both of these ideas are wrong!

A logical inference never increases the information given in the facts and, often, it reduces it. What is more spectacular is that the reduced information is often more helpful than the original facts. We illustrate this with two examples:

Consider a patient suspected of having an illness, and the doctor asks them to get several tests done. The doctor analyzes the tests and gives a verdict that the patient does not have any illnesses. The doctor uses the test results, medical knowledge, and logical reasoning to reach the verdict. The verdict has less information than the one contained in the tests. Any expert doctor can deduce the final verdict from the test. However, it is impossible to return to the test from the verdict. The reduced information is far more helpful to the patient and their loved ones.

For another example, consider the company records stored in a room. An expert can go through all these files and deduce that the company is in sound financial condition. This information is far more vital for the investors than the stacks of files.

Information reduction is important in decision-making. Logic is absolutely vital in reducing information and leading us to the right conclusions.