Law of Syllogism and Law of Detachment

Many valid arguments fit the following pattern:

If a, then b.

If b, then c.

Therefore, if a then c.

The symbol →rightwards arrow means "implies." The three dots in the third line mean "therefore."

This pattern is called the Law of Syllogism . Although the name of the pattern may sound difficult, just remember that the Law of Syllogism is simply a pattern. Once you know the pattern, you can use it to determine whether an argument is valid.

There are other patterns besides the Law of Syllogism that guarantee an argument will be valid. Look at the pattern shown here.

This pattern is called the Law of Detachment . Like the Law of Syllogism, this law has a pattern that allows you to determine whether or not a given argument is valid.

The Law of Detachment

a→ba→b

aa

∴b

Remember the argument in the trapezoid illustration? That argument fits the Law of Detachment. If we use F for "four sides" and Q for "quadrilateral," we can write this argument as

If F, then Q.

F.

Therefore, Q

F → Q

If a plane figure has 4 sides, then it is a quadrilateral

A trapezoid is a plane figure with 4 sides

A trapezoid is a plane figure with 4 sides

Knowing the patterns that make an argument valid will help you avoid the mistake of confusing truth and validity. Remember, just because a conclusion is true doesn't mean the argument is valid.

Sometimes you can test an argument by substituting different words into the argument. Let's make a few changes in the spider argument. See what happens if we change "Arachnids" to "Dogs," "Spiders" to "Horses," and "8" to "4."

This argument is clearly invalid, but the pattern of the argument is the same as the spider argument. Both these arguments are invalid because the pattern is invalid.

An argument is a series of statements that is made up of two or more premises and a conclusion.

A valid argument is an argument in which a true conclusion is logically derived from true premises.

Valid arguments follow specific patterns such as the Law of Detachment and the Law of Syllogism.

Remember the argument in the trapezoid illustration? That argument fits the Law of Detachment. If we use F for "four sides" and Q for "quadrilateral," we can write this argument as

If F, then Q.

F.

Therefore, Q