## Conditional Propositions

**Conditional Propositions**

Conditional propositions are compound propositions connected by the words “**If…then**” or just “**then**.” As we learned in the previous discussion titled “Propositions and Symbols Used in Symbolic Logic (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), the symbol for “**if…then**” is a horseshoe. Consider the example below:

**If** it rains today, **then** the road is wet. (p, q)

If we let ** p** stand for “It rains today” and

**for “The road is wet,” then the example above is symbolized as follows:**

*q**p ***⊃*** q*

Please note that the proposition that precedes the connective horseshoe (**⊃**) is called the “antecedent” and the proposition that comes after it is called “consequent.”

Please note as well that there are cases wherein the words “**if…then**” is not mentioned in the proposition, yet the proposition remains a conditional one. Consider the example:

**Passage of the law **means** morality is corrupted.** (p, q)

If we analyze the proposition, it is very clear that it is a conditional proposition because it suggests a “cause and effect” relation. Thus, the proposition can be stated as follows:

**If** the law is passed, **then** morality will be corrupted.

If we let **p** stand for “The law is passed” and **q** for “Morality will be corrupted,” then the proposition is symbolized as follows:

*p ***⊃*** q*

It is also important to note that sometimes the antecedent is stated after the consequent. If this happens, then we have to symbolize the proposition accordingly. Let’s take the example below.

**Morality would be corrupted should the abortion law is passed**. (p, q)

If we analyze the proposition, it is clear that the antecedent is “**Abortion law is passed**” and the consequent is “**Morality would be corrupted**.” Hence, the proposition “Morality would be corrupted should the abortion law is passed” is symbolized as follows:

*q ***⊃*** p*

Again, as I already pointed out in my previous discussion, the variables provided after the proposition represent the propositions in the entire proposition respectively. Thus, in the statement

**Morality would be corrupted should the abortion law is passed. **(p, q)

the variable ** p** stands for “Morality would be corrupted” and

**stands for “The abortion law is passed.” Again, since**

*q***q**is our antecedent and

**p**is our consequent, and since in symbolizing conditional propositions we need to write the antecedent first and then the consequent, so the proposition “Morality would be corrupted should the abortion law is passed” is symbolized as follows:

*q* **⊃*** p*

**Rules in Conditional Propositions**

- A conditional proposition is
**false**if the antecedent is**true**and the consequent**false**. - Thus, other than this form, the conditional proposition is true.

The truth table below illustrates this point.

The truth table above says:

- If
is*p***true**andis*q***true**, then*p***⊃**is*q***true**. - If
is*p***true**andis*q***false**, then*p***⊃**is*q***false**. - If
is*p***false**andis*q***true**, then*p***⊃**is*q***true**. - If
is*p***false**andis*q***false**, then*p***⊃**is*q***true**.

As we can see, the rules in conditional propositions say that the only instance wherein the conditional proposition becomes **false** is when the antecedent is **true** and the consequent **false**. Let us consider the example below.

**If **it rains today, **then** the road is wet.

Now, the first row in the truth table above says that ** p** is

**true**and

**is**

*q***true**. So, obviously,

*p***⊃**

**is**

*q***true**. This is because, if it is

**true**that “it rains today,” then it must also be

**true**that “the road is wet.”

The second row says that ** p** is

**true**and

**is**

*q***false**. So,

*p***⊃**

**must be**

*q***false**. This is because if it is true that “it rains today” then it must necessarily follow that “the road is wet.” However, it is said that

**is**

*q***false**, that is, the road is not wet; hence, the conditional proposition is false. Again, it is impossible for the road not to get wet if it rains.

The third row says ** p** is

**false**and

**is**

*q***true**. If this is the case, then

*p***⊃**

**is**

*q***true**. This is because if it is false that it rains today (in other words, it does not rain today), it does not necessarily follow that the road is dry. Even if it does not rain, the road may still be wet because, for example, a fire truck passes by and spills water on the road.

Lastly, the fourth row in the truth table above says ** p** is

**false**and

**is**

*q***false**. If this is the case, then

*p***⊃**

**is**

*q***true**. This is because, based on the example above, it says “it does not rain today” and the “road is not wet.” So, obviously, the conditional proposition is true.

We have provided a video for all our posts in Symbolic Logic. If you are interested, please visit the following:

1. Propositions and Symbols Used in Symbolic Logic (see https://www.youtube.com/watch?v=OdUbiNZVG1s)

2. What is Philosophy (see https://www.youtube.com/watch?v=nRG-rV8hhpU)