**Deductive Logic**

Definitions

Geometry (and mathematics in general) is a very precise subject. Figures are classified according to certain properties, and definitions must be completely clear.

As in Plato’s dialog, *Meno*, we all have some notions of what certain figures are. For example, we know what a square is. However, some people might be misled by the way one is drawn. A baseball diamond is a square, but it is tilted at 45^{o} when viewed from home base.

But it is not a simple matter to define things. For example, we know that squares have 4 sides that are equal in length. But is that a good definition of a square? This figure has 4 sides that are the same length, but it isn't a square:

The statement, “a square is a 4-sided figure whose sides have the same length” happens to be a true statement, but it cannot serve as a definition of a square since it is not “reversible.” That is, if the sides of a 4-sided figure have the same length then that figure might not be a square.

A square also has to have its sides perpendicular. In other words, the definition of a square must include all important properties that distinguish a square from other 4-sided figures:

Definition: *A square is a 4-sided figure in which all sides are equal and pairs of sides are perpendicular.*

We can state 4 properties a good definition must possess:

1. It names the term being defined.

2. It identifies the category to which the term belongs.

3. It states the properties that distinguish the term from others in that category.

4. It is reversible.

In our definition of a square these properties are:

1. Term: **square**

2. Category: **4-sided figures**

3. Properties: **equal sides, perpendicular sides**

4. Reversible: No other 4-sided figures have those properties.

Conditional Statements

The statement, *all squares have 4 equal sides* happens to be true, though it cannot serve as the definition of a square. Another way to say this is, *if a figure is a square then it has 4 equal sides*. Such a statement (written in *if-then* form) is called a **conditional** statement. It consists of two parts: a statement between the words *if* and *then*, and a statement after the word *then*. The first part is called the **hypothesis** and the second the **conclusion**. Conditional statements always have the form:

*If* **hypothesis** *then* **conclusion**.

They are equivalent to “*all*” statements of this form:

*All* things that make the **hypothesis** true *are* things that make the **conclusion** true.

In this sense, the **hypothesis** and **conclusion** describe two categories or sets to which things belong, and both statements imply that the **hypothesis** category is ** contained in** the

For example:

*All* **A**’s *are* **B**’s or *If* it is an **A** *then* it is a **B**

both tell us that the **A**’s are a subset of the **B**’s:

The Converse and Biconditionals

The statements *All* **A**’s *are* **B**’s and *All* B’s *are* A’s say very different things, as can be seen from their Venn diagrams:

In the first case the **A**’s are contained in the **B**’s, while in the second case the **B**’s are contained in the **A**’s.

In *if-then* form, the statements

*If* it is an **A** *then* it is a **B** and *If* it is an **B** *then* it is an **A**

are just different statements. Each is called the **converse** of the other. The truth of one does not imply the truth of the other:

**True**: *If* a figure is a square *then* it has 4 equal sides.

**False**: *If* a figure has 4 equal sides *then* it is a square.

In some situations a statement and its converse are both true. In the case of “All **A**’s are **B**’s ** and** all

The shortened form of this statement is:

It is an **A** *if and only if* it is a **B**.

Such a statement is called a **biconditional** and is equivalent to:

*If* it is an **A** *then* it is a **B** *and**if* it is a **B** *then* it is an **A**.

Sometimes we state it this way:

*If* it is an **A** *then* it is a **B**, *and conversely*.

The "*and conversely"* part means the converse is also true.

Definitions are sometimes stated in *if-then* form, but are really *biconditionals* since they must be reversible.

For example, we might define ** perpendicular** this way:

*If* two lines are *perpendicular**then* they intersect at right angles*.*

What we really mean is, perpendicular lines are lines that intersect at right angles.

Or we could be precise and say:

Two lines are *perpendicular**if and only if* they intersect at right angles.

"*If and only if*" means both the statement and its converse are true.

Counterexamples and the Contrapositive

The statement, “all birds fly” happens to be false. This is because there are some birds, like the penguin, that don’t fly at all.

To demonstrate (or “prove”) that a statement like this is false, all we need to do is supply an example that shows it is false. Such an example is called a **counterexample**. *Penguin* is a counterexample to “all birds fly.”

In general, a counterexample is any example that shows some statement is false.

As discussed above, *if-then* statements are equivalent to *all* statements, and both can be represented by Venn diagrams. The statement, "*all* **A**’s *are* **B**’s" is diagrammed as:

This shows that the **A**’s are a subset of the **B**’s. As a consequence, there are no **A**’s that are **non-B**’s (no part of the **A**-set is outside of the **B**-set). Another way to diagram this is to make a 2 by 2 table with the **A**’s and **non-A**’s as rows, and the **B**’s and **non-B**’s as columns. We darken the cell corresponding to **A**’s *and* **non-B**’s to indicate this cell is empty (there is no overlap):

If we look at the **A** row, the only cell containing elements (not darkened) is in the **B** column. This tells us that all **A**’s are **B**’s.

Likewise, if we look at the **non-B** column, the only cell containing elements is in the **non-A** row. From this we see that the following statement is also true:

*All* **non-B**’s *are* **non-A**’s.

In other words, the two statements "*All* **A**’s *are* **B**’s" and "*All* **non-B**’s *are* **non-A**’s" are equivalent.

In *if-then* form, these statements are equivalent:

*If* something is an **A** *then* it is a **B** and *If* something is ** not** a

Equivalent statements like this are called ** contrapositives**. To form the contrapositive of a conditional statement, you negate both the hypothesis and conclusion and switch them:

**Statement:** *If* **hypothesis** *then* **conclusion**.

**Contrapositive:** *If* *not* conclusion*then* ** not hypothesis**.

Here’s a common-sense example:

**Statement:** If it is raining then the streets are wet.

**Contrapositive:** If the streets are **not** wet then it is **not** raining.

Think about it. Both statements say the same thing, but in different ways.

Syllogisms

A syllogism is a logical argument in which a conclusion is reached based on some known facts which are usually stated in *if-then* or *all* form. This kind of logic played a central role in ancient Greek mathematics and is the essence of proof in mathematics today. Aristotle (around 350 BC) wrote extensively about logic and syllogisms, and this forms the basis of western culture.

Here is an example of a syllogism. If we know the following facts:

1. All teachers are in the library.

2. Everyone in the library is watching Dr. T’s lecture.

3. Ms. Jones is a teacher.

Then we can reach any of the following three conclusions:

(a) Ms. Jones is in the library.

(b) Ms. Jones is watching Dr. T’s lecture.

(c) All teachers are watching Dr. T’s lecture.

In other words, we just follow the string of statements in a common-sense way. However, we must be cautious in some cases. Many people confuse a statement with its converse. For example, if we also know this statement is true:

4. Mr. Smith is in the library.

then we can **not** conclude that Mr. Smith is a teacher. The first statement says all teachers are in the library, but it does not say that everyone in the library is a teacher.

Sometimes you have to use the contrapositive to reach a conclusion. The contrapositive is important to understand, because it is the basis of the logic used in certain proofs. Such proofs are called “indirect” since they rely not on the given form of a statement, but on its contrapositive.

The logic of an indirect proof is essentially this: If you want to prove **B** follows from **A**, try assuming **B** is **not true** and show that this leads to the conclusion that **A** is **not true**. Thus to prove “If **A** then **B**” you instead prove “If **not B** then **not A**,” which is equivalent.

Here is an example of the use of the contrapositive (or an indirect proof) to arrive at a conclusion. Suppose the following two facts are true:

1. Everyone in the library is watching Dr. T’s lecture.

2. Murphy is not watching Dr. T’s lecture.

We can conclude that Murphy is not in the library, because the first statement is equivalent to, “If someone is in the library then that person is watching Dr. T’s lecture,” and this in turn is equivalent to its contrapositive, “If someone is not watching Dr. T’s lecture then that person is not in the library.”