Lecture Notes: Logic

Logical arguments are made up of two parts: premises and a conclusion. The conclusion is that which the argument seeks to prove. The premises are factual statements that support, and ultimately prove, the conclusion.

Let’s look at a syllogism as an example of a basic logical argument made up of two premises and a conclusion.

P1. The White House is in Washington, DC.

P2. The Oval Office is in the White House.

⸫ The Oval Office is in Washington, DC.

In this argument, the two premises prove the conclusion.

When examining a logical argument, we first look to whether it is valid, then determine if it is sound.

A valid argument is one in which the premises force the conclusion to be true; the logic flows. It is important to remember that neither the premises nor the conclusion need to actually be true in order for the argument to be valid. The if/and/then test is a simple way of determining validity without getting mixed up in truth (that comes later).

Let’s apply the if/and/then test to our original argument:

IF the White House is in Washington, DC

AND the Oval Office is in the White House,

THEN the Oval Office is in Washington, DC.

This logically flows. If the White House is in Washington, DC and the Oval Office is in the White House, then it stands to reason that the Oval Office is in Washington, DC. The argument is valid.

Now we test the argument’s soundness. A sound argument is a valid argument in which all the premises and the conclusion are actually true. Some quick research would show us that the premises are true; likewise for the conclusion. This argument is sound.

Let’s try another one.

P1. All philosophy teachers are millionaires.

P2. John is a philosophy teacher.

⸫ John is a millionaire.

First, we test validity—without worrying about whether the premises or conclusion are true. We only want to see if the logic flows, so we apply the if/and/then test:

IF all philosophy teachers are millionaires.

AND John is a philosophy teacher.

THEN John is a millionaire.

The logic of this argument flows; it passes the if/and/then test. If all philosophy teachers are millionaires and John is a philosophy teacher, then John would have to be a millionaire.

This valid argument, however is not sound. The first premise is not true. This is what we call proceeding from a false premise. If one premise is false, the argument is not sound. We can stop there, but for fun, let’s look at the conclusion. It is not true. I am not a millionaire.

So, we have examined two logical syllogisms, both of which are valid, one of which is sound. Let’s look at an invalid argument:

P1. All doctors have advanced college degrees.

P2. Allie has an advanced college degree.

⸫ Allie is a doctor.

When we apply the if/and/then test, we easily find this argument to be invalid:

IF all doctors have advanced college degrees

AND Allie has an advanced college degree,

THEN Allie is a doctor.

The logic of this argument does not flow. If all doctors have advanced college degrees and Allie has an advanced college degree, then it still does not necessarily mean Allie is a doctor. Allie might have a Master of Fine Arts degree. The premises do not force the conclusion to be true. This argument is invalid.

An invalid argument cannot be sound.

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