Principles of Logic 3: Conditional Reasoning

Overview

Conditional reasoning involves arguments with if-then statements. There are two valid rules of conditional reasoning: modus ponens and modus tollens. And there are two invalid forms of conditional reasoning: Affirming the consequent and denying the antecedent. Learning to distinguish the valid forms from the invalid forms is essential to constructing and recognizing good arguments.

Conditional Reasoning

Conditional reasoning is any form of reasoning that involves conditionals. A conditional is an if-then statement. Let's look at an example:

If Omar misses work then he will lose his job.

The clause that follows the 'if' is called the antecedent. In the above conditional [Omar misses work] is the antecedent. The clause that follows the 'then' is called the consequent. In the above sentence, [he will lose his job] is the consequent.

It's important to notice that the antecedents are determined by whether they follow the 'if' NOT by the order of the conditional. Let's look at an example:

Omar will lose his job if he misses work.

In the above sentence [he misses work] is still the antecedent because it follows the 'if'. The fact that the sentence order is rearranged has not bearing on which clause is the antecedent. The antecedent always follows the 'if'.

Symbolization
Before we learn the rules, we're going to learn how to symbolize sentences. This will allow us to identify patterns more easily as well as reduce the amount of writing we have to do!

To symbolize if-then statements take the first letter or letters of a key term in the antecedent, draw an arrow, then write the first letter or letters of the key terms in the consequent. Any time you see a negation in a clause, indicate it with a ~.

Negations are: not, can't, won't, couldn't, wouldn't, isn't, aren't, didn't, doesn't, etc..

Example:

If I'm LATE for work then I won't GET a raise. = L —> ~G.

The 'L' is for 'if I'm late for work'; the 'G' is for 'then I get a raise' the ~ negates G making it 'then I won't get a raise'; the —> indicates it's an if-then statement.

Let's try one more:

If you don't KNOW how to symbolize sentences then you should PAY attention = ~K—>P

Modus Ponens vs Affirming the Consequent

Suppose your friend says to you: If I don't text you by 5pm then I'll be at the gym. Later that day you look at the clock. It's 5:30 and still no text from your friend. Assuming what your friend said earlier is true, where is your friend? At the gym.

Let's formalize and symbolize the argument:
(P1). If I don't TEXT you by 5pm then I'll be at the GYM.
(P2). (The friend) didn't TEXT you by 5pm.
(C). Therefore, the friend is at the GYM.

Symbolized:
(P1). ~T—>G
(P2). ~T
(C). G

The above inference pattern is called modus ponens. Any argument that has this form is valid. The basic form of modus ponens looks like this:

(P1). If [antecedent] then [consequent].
(P2). [antecedent].
(C). Therefore, [consequent].

In textbooks, it's typically symbolized like this:

(P1). P->Q
(P2). P
(C). Q

Let's look at a similar form of argument called affirming the consequent:

(P1). If I don't TEXT you by 5pm then I'll be at the GYM.
(P2). I'm at the GYM.
(C). Therefore, I didn't TEXT you by 5pm.

Let's symbolize it:

(P1). ~T—>G
(P2). G
(C). ~T

Recall the test for validity: If I assume the premises to be true, does the conclusion necessarily follow? Is the above argument valid? The correct answer is "no" because it's possible for all the premises to be true and the conclusion false. That is, the premises, even when assumed to be true don't guarantee the truth of the conclusion.

If you're not sure why, let's work through it. All you know from the premises is that if I don't text you then I'll be at the gym. But the above premises don't preclude me from going to the gym before 5pm. They also don't preclude me texting you from the gym either. In short, (P1) only tells you what to believe if I don't text, it doesn't tell you what to believe if I'm at the gym.

The above form of reasoning is called affirming the consequent (AC). I affirm the consequent when I have a conditional and as my second premise I affirm the consequent (e.g., "I'm at the gym"). Compare this with the valid form, modus ponens, where my second premise is the antecedent NOT the consequent.

Here's one more example contrasting modus ponens and affirming the consequent:

Modus Ponens

(P1) If it's raining then there are clouds.
(P2) It's raining.
(C) There are clouds.

Symbolization:
(P1) R—>C
(P2) R
(C) C

Affirming the Consequent

(P1) If it's raining then there are clouds.
(P2) There are clouds.
(C) Therefore, it's raining.

Symbolization
(P1) R>C
(P2) C
(C) —R

In the modus ponens version, the conclusion must be true and so the argument is valid. In the second version, the conclusion isn't necessarily true. If there are clouds it doesn't follow necessarily that it's raining. It's possible for there to be clouds without rain. So, when an argument has the modus ponens form it will always be valid regardless of the truth of the premises(because validity and premise acceptability aren't related). An argument that follows the form of affirming the consequent will never be valid.

Modus Tollens vs Denying the Antecedent

Modus tollens (MT) and denying the antecedent (DA) are two other forms of conditional reasoning are often confused. Let's look at them both and see which is valid.

Let's start with an intuitive example:

(P1). If it's RAINING then there are CLOUDS.
(P2). There are no CLOUDS.
(C). Therefore, it isn't RAINING.

Symbolization (careful of negatives!):
(P1). R—>C
(P2). ~C
(C). ~R

The above argument follows the form of modus tollens: If I negate the consequent I can conclude the negation of the antecedent.

Let's now look at Denying the Antecedent:

(P1). If it's RAINING then there are CLOUDS.
(P2). It isn't RAINING.
(C). Therefore, there are no CLOUDS.

Symbolization (careful with negatives!):
(P1). R—>C
(P2). ~R
(C). ~C

Does the conclusion follow necessarily from the premises? Nope. The important thing here is to understand why not. Most of us just know from experience that there are cloudy days without rain. But what makes the argument invalid has to do with logic, not our knowledge of the world. The first premise only tells us about what follows from the fact that it's raining. It doesn't tell us what follows from 'it isn't raining'. Using only the premises we've been provided, we can't draw any logical inferences from 'it isn't raining'.

Let's turn to a less intuitive argument to further examine the logical relations in modus tollens and denying the antecedent.

(P1). If you don't exercise you'll feel weak.
(P2). I don't feel weak.
(C). Therefore I exercised. (Technically, "I didn't not exercise" but we eliminate the double negation).

(P1). ~E—>W
(P2). ~W
(C). E/~~E (either is fine).

The above form follows modus tollens and is valid. We negated the consequent which allows us to infer the negation of the antecedent. Notice that our original antecedent was already negated (If you don't exercise). This means we have to add one more negation to follow the modus tollens rule. However, when we have two negations side by side, in natural language they cancel each other out…To make things simpler, that's what we do.

Let's look at denying the antecedent:

(P1). If you don't exercise you'll feel weak.
(P2). I exercised (or I didn't not exercise).
(C). Therefore, I don't feel weak.

Symbolization:
(P1). ~E—>W
(P2). E/~~E
(C). ~W

The above argument is invalid. The premises tell us what will happen if I don't exercise but they don't tell us what will happen if I do. Is it logically possible to exercise and to feel weak? Yup. Maybe you exercised but you also ate some poisonous mushrooms. You'll feel weak even though you exercised. In fact, exercising after eating poisonous mushrooms might make you feel weaker! (Pro tip: Don't eat poisonous mushrooms then go to the gym).

The point is that denying the antecedent isn't a valid argument form. Even if we assume the premises to be true it's still logically possible for the conclusion to be false.

Summary of Argument Forms

We can summarize the relationship between the four argument forms and validity in the chart below:

Argument Form Valid vs Invalid
Modus Ponens (MP) Valid
Modus Tollens (MT) Valid
Affirming the Consequent (AC) Invalid
Denying the Antecedent (DA) Invalid
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License