We introduce the first logical implication with an example.

In Example 2.2.1 and Example 2.2.2, we saw that if we knew that both

p = 1,

and

p → q = 1,

then we knew that q = 1 as well. In other words, it seems as if we have that

[p ∧ (p → q)] ⟹ q.

We can verify this more formally using a truth table (note that we represent p as premise p₁ and we represent the proposition (p → q) as premise p₂, and we represent the conclusion q using the letter c.)

Indeed, since the column representing [p ∧ (p → q)] → q column consists entirely of 1s, we have that

[p ∧ (p → q)] ⟹ q.

In other words, whenever we know that p is true, and (p → q) is true, then q is true as well. As such, the argument

[p ∧ (p → q)] → q

is valid.

It is common to write this kind of argument out in a tabular form. We essentially just a list of all the premises p₁, p₂, p₃, …, p_n in a single column. We then add a horizontal line and then below that line, we write the conclusion c (in the example above, the conclusion c represents proposition q.) Right next to the conclusion, we add a ∴ symbol, which can be read as “therefore”, or “thus” on the left. This can be seen below, and in Figure 2.2.1.

Also note that because the conjunction operator ∧ is commutative, we could also present this argument as

[(p → q) ∧ p] → q,

and so we could also write this in tabular form as

This kind of argument is commonly referred to as “Modus Ponens”

There is another kind of valid argument that is directly related to the Modus Ponens argument. Again, we first demonstrate the argument with an example.

From Example 2.2.3, we see that if we have that

p → q = 1
¬q = 1 (meaning q = 0),

then we must have that ¬p = 1 (again, meaning that p = 0.) Using the language of arguments, if we have premises (p → q) and (¬q), then we have conclusion (¬p.)

Again, we examine a truth table to formally verify that this is a valid argument:

Again, we have another tautological true implication, meaning we have a logical implication:

[¬q ∧ (p → q)] ⟹ ¬p.

Thus, the argument

[¬q ∧ (p → q)] → ¬p

is a valid argument.

Just like with the previous argument, this argument can be written in a tabular form as presented in Figure 2.2.2 to the left. This argument is sometimes referred to as the “Modus Tollens” argument.

Both Modus Ponens and Modus Tollens have

p → q

as a premise. In some sense, we could think of the Modus Tollens argument as the “contrapositive” of the Modus Ponens argument.

From Example 2.1.2 in the previous section, we saw that an argument of the form

[(p → q) ∧ (q → r)] → (p → r)

is a valid argument. Such an argument is commonly referred to as the “Law of the Syllogism.” A tabular form of this argument is presented in Figure 2.2.3 at the left.

Here’s a truth table, showing that we do indeed get all 1s in the last column, thus demonstrating the validity of the Law of the Syllogism argument:

Again, we point out that because ∧ is commutative, the premises of the argument can be swapped, and you will still have a valid argument. This will be the last time this fact will be explicitly mentioned, though it will be implicitly used heavily throughout our work.

Naturally, we can keep chaining more and more propositions at the end of each of the premises’ implications. Here is an example involving 5 propositions labeled j, a, c, o, and b:

Not all of the arguments at our disposal are profound. Some may seem quite obvious. For instance, suppose we knew that proposition p is true, and that proposition q is true. Since they are both true, we also know that the conjunction (p ∧ q) is also true.

Therefore, we know that the argument

[(p) ∧ (q)] → (p ∧ q)

is a valid argument.

Why do we care about an argument as simple as this? Because as we develop more sophisticated arguments, propositions p and q may come up (either as premises of said argument, or as results derived from other premises.) When this happens, p and q can be combined in a conjunction, and that conjunction can be used to further develop the argument.

We will see how this simple argument, sometimes referred to as the “Rule of Conjunction” can be used strategically in the next section.

As Example 2.2.4 demonstrates, if we know that at least one of two propositions p and q is true, but that one of them is false, (say q), then the other proposition must be true (or else the disjunction would have been false.)

We can of course verify that this argument is valid by constructing an appropriate truth table:

So, this argument is valid. A tabular form of the argument is given in Figure 2.2.5.

Suppose we are given some proposition p. We would naturally want to know if p was a true proposition or a false proposition. How could we go about determining if p = 0 or if p = 1?

One thing we could try and do is to simply assume that p = 0 (meaning ¬p = 1.) If assuming p = 0 yields a contradiction F₀ then surely p ≠ 0 because true statements should never yield contradictions. Hence, we would have to have that p = 1.

We essentially get the argument

(¬p → F₀) → p

Notice that since there is only one premise

p₁: ¬p → 0,

(where we replace the general contradiction F₀ with it’s truth value of 0) we don’t have any conjunction operators present. We verify this argument is valid with the following truth table by noting that the last column consists entirely of 1s (same as always.)

The argument

[¬p → F₀] → p

is commonly known as the “Rule of Contradiction.”

We have presented five different kinds of arguments. These, arguments are commonly referred to as rules of inference because they allow you to infer, or deduce, a conclusion given a list of premises.

There are many more kinds of these rules of inference, and we present a sample of such rules of inference below.

Before we see how these rules of inference can be used, it’s important to take a step back to see what we’ve accomplished, but perhaps more importantly to see what we have not accomplished.

Notice that each of these rules of inference are logical implications. This is demonstrated by the fact that we used the single arrow ⟹ as opposed to the double arrow ⟺. Indeed, while some of the above arguments may contain logical equivalencies (such as the Law of Conjunction), most of these are not logical equivalencies.

For instance, examining the Law of the Syllogism, we have that

[(p → q) ∧ (q → r)] ⟹ (p → r).

However, note that when p = r = 0, and q = 1, we have that

(p → q) = 1
(q → r) = 0
(p → r) = 1

but

(p → q) ∧ (q → r) = 1 ∧ 0 = 0.

Hence, we have that

[(p → q) ∧ (q → r)] ⇐/⇒ (p → r).

As such, we have not shown that the expression [(p → q) ∧ (q → r)] can be replaced with the simpler expression (p → r). This is a subtle difference, but nonetheless an important difference.

We will see in the next section that both logical equivalencies and logical implications can be used in the development of arguments, but logical implications will not be helpful when trying to simplify complicated logical expressions.