Inference: Modus Ponens and Modus Tollens

Inference

Let’s call a proposition with known truth value to be a fact. Establishing the truth value of some proposition from the given facts is called inference.

Examples

Let’s look at some examples to improve our understanding of inference.

As $F_2$ and $F_4$, both are facts, hence $C_1$ is true. Siam is a pigeon, and a pigeon is a bird; therefore, we can conclude that Siam is a bird.

Similarly, from $F_3$ and $F_4$, we can infer $C_2$. Siam is a pigeon, and pigeons are white or gray; therefore, Siam is white or gray.

Here,

$$H = \{F_1, F_2, F_3, F_4\}.$$

The conclusions $C_1$ and $C_2$ are inferred from $H$. We can call $H$ the set of hypotheses. Further, note that;

$C_1 \notin H$, and $C_2 \notin H$.

Let’s look at another example:

---

Let’s assume the following facts:

• $F_1$: Ruby and Julia are going together.
• $F_2$: Ruby is going to campus.

---

From the facts stated above, we can infer the following conclusion:

• $C_1$: Julia is going to campus.

As $F_1$ and $F_2$ are valid, $C_1$ should also be valid. Ruby and Julia are going together, and Ruby is going to campus. It means Julia is also going with Ruby to the campus.

Here,

$$H = \{F_1, F_2\}.$$

The conclusion $C_1$ is inferred from $H$. Further, $C_1 \notin H$.

Inferences can be valid or invalid. Let’s look at an example of invalid inference.

Given that $F_1$ and $F_2$ are valid, is $C_1$ also valid? From $F_1$, we can not infer that anything with four legs is a lion. Hence, we can not conclude $C_1$ from the given facts.

Only those inferences are valid in logic that are based on some established rules. We call templates of such established rules in logic as rules of inference. These rules of inference are building blocks for making sound arguments in mathematics.

Next, let’s look at the rules of inference and how each one is logically valid.

Modus ponens

Let’s assume that for arbitrary propositions, $q_1$ and $q_2$, we know that $q_1 \Rightarrow q_2$ is true. Further, we know that $q_1$ is true. Then, we can conclude that $q_2$ is true.

To understand it further, look at the following truth table:

Row Number	$q_1$	$q_2$	$q_1 \Rightarrow q_2$
1	T	T	T
2	T	F	F
3	F	T	T
4	F	F	T

Given that $q_1 \Rightarrow q_2$ is not false means we are not talking about row number two of the truth table. Further, when we assume that $q_1$ is not false, this means we are not talking about row numbers three and four. The only possibility left is row number one, where $q_2$ is true.

Therefore, whenever we have the facts,

$q_1 \Rightarrow q_2$ is true, and $q_1$ is true, we can conclude that, $q_2$ is true.

This inference rule is called modus ponens, which is Latin for “method for affirming.” We also refer to it as the rule of detachment or the law of detachment.

Another way to look at the validity of modus ponens is by noticing that the following compound proposition is a tautology.

$$\left(q_1 \land \left(q_1 \Rightarrow q_2\right)\right)\Rightarrow q_2.$$

The above conditional proposition is true, and its hypothesis is true. Hence, its conclusion has to be true.

We can verify this by the following truth table:

$q_1$	$q_2$	$q_1 \Rightarrow q_2$	$q_1 \land \left(q_1 \Rightarrow q_2\right)$	$\left(q_1 \land \left(q_1 \Rightarrow q_2\right) \right)\Rightarrow q_2$
T	T	T	T	T
T	F	F	F	T
F	T	T	F	T
F	F	T	F	T

Examples

Let’s look at a few examples to understand modus ponens further:

Then, by modus ponens, we can conclude that the following proposition is true.

• $G$: The ground is wet.

For our next example, we take the following propositions.

Then, by using modus ponens, we can conclude the following fact:

• $A_J$: Alina is feeling joyful.

It is important to note that we can not conclude that $A_E$ is true if we know that $A_E\Rightarrow A_J$ and $A_J$ are true.

Modus tollens

Let’s assume that for arbitrary propositions, $q_1$ and $q_2$, it is given that $q_1 \Rightarrow q_2$ is true. Further, we know that $\neg q_2$ is true. Then, we can conclude that $\neg q_1$ is true.

To understand it further, look at the following truth table.

Row Number	$q_1$	$q_2$	$\neg q_1$	$\neg q_2$	$q_1 \Rightarrow q_2$
1	T	T	F	F	T
2	T	F	F	T	F
3	F	T	T	F	T
4	F	F	T	T	T

Given that $q_1 \Rightarrow q_2$ is not false means we are not talking about row number two of the truth table. Further, assuming $\neg q_2$ true means we are talking about row number four. In row number four, $\neg q_1$ is also true.

Therefore, whenever we have the facts;

$q_1 \Rightarrow q_2$ is true, and $\neg q_2$ is true, we can conclude that, $\neg q_1$ is also true.

This inference rule is called modus tollens, Latin for “method for denying.” Another way to look at the validity of modus tollens is by noticing that the following compound proposition is a tautology.

$$\left(\neg q_2 \land \left(q_1 \Rightarrow q_2\right)\right)\Rightarrow \neg q_1.$$

The above conditional proposition is true, and its hypothesis is true. Therefore, its conclusion has to be true.

We can verify this by the following truth table:

$q_1$	$q_2$	$\neg q_1$	$\neg q_2$	$q_1 \Rightarrow q_2$	$\neg q_2 \land \left(q_1 \Rightarrow q_2\right)$	$\neg q_2 \land \newline \left(q_1 \Rightarrow q_2\right) \Rightarrow \neg q_1$
T	T	F	F	T	F	T
T	F	F	T	F	F	T
F	T	T	F	T	F	T
F	F	T	T	T	T	T

Examples

Let’s look at a few examples to understand modus tollens further.

---

Consider the following propositions:

• $N_P$: Nemo is a pet fish.
• $N_L$: Nemo lives in a fishbowl.

---

Let’s assume that the following two propositions are true:

• $N_P \Rightarrow N_L$: If Nemo is a pet fish, then she lives in a fishbowl.
• $\neg N_L$: Nemo does not live in a fishbowl.

From these two facts, we can conclude $\neg N_P$ by applying modus tollens.

• $\neg N_P$: Nemo is not a pet fish.

If Nemo is a pet fish, then she lives in a fishbowl, and she does not live in a fishbowl, it means Nemo is not a pet fish.

Let’s look at another example:

By using modus tollens, we conclude the following fact.

• $\neg W$: It is not winter.

We must be cautious that we can not conclude $\neg W_{J}$ from the facts $W\Rightarrow W_{J}$, and $\neg W$.