**Deductive** inferences, which are inferences arrived at through deduction (deductive reasoning), can guarantee truth because they focus on the structure of arguments. Here is an example:

- Either you can go to the movies tonight, or you can go to the party tomorrow.
- You cannot go to the movies tonight.
- So, you can go to the party tomorrow.

This argument is good, and you probably knew it was good even without thinking too much about it. The argument uses “or,” which means that at least one of the two statements joined by the “or” must be true. If you find out that one of the two statements joined by “or” is false, you *know* that the other statement is true by using deduction. Notice that this inference works no matter what the statements are. Take a look at the structure of this form of reasoning:

- X or Y is true.
- X is not true.
- Therefore, Y is true.

By replacing the statements with variables, we get to the *form* of the initial argument above. No matter what statements you replace X and Y with, if those statements are true, then the conclusion must be true as well. This common argument form is called a disjunctive syllogism.

### Valid Deductive Inferences

A good deductive inference is called a **valid inference**, meaning its structure guarantees the truth of its conclusion given the truth of the premises. Pay attention to this definition. The definition does not say that valid arguments *have* true conclusions. Validity is a property of the logical forms of arguments, and remember that logic and truth are distinct. The definition states that valid arguments have a form such that *if* the premises are true, then the conclusion must be true. You can test a deductive inference’s validity by testing whether the premises lead to the conclusion. If it is impossible for the conclusion to be false when the premises are assumed to be true, then the argument is valid.

Deductive reasoning can use a number of valid argument structures:

Disjunctive Syllogism:

- X or Y.
- Not Y.
- Therefore X.

Modus Ponens:

- If X, then Y.
- X.
- Therefore Y.

Modus Tollens:

- If X, then Y.
- Not Y.
- Therefore, not X.

You saw the first form, disjunctive syllogism, in the previous example. The second form, modus ponens, uses a conditional, and if you think about necessary and sufficient conditions already discussed, then the validity of this inference becomes apparent. The conditional in premise 1 expresses that X is sufficient for Y. So if X is true, then Y must be true. And premise 2 states that X is true. So the conclusion (the truth of Y) necessarily follows. You can also use your knowledge of necessary and sufficient conditions to understand the last form, modus tollens. Remember, in a conditional, the consequent is the necessary condition. So Y is necessary for X. But premise 2 states that Y is *not* true. Because Y must be the case if X is the case, and we are told that Y is false, then we know that X is also false. These three examples are only a few of the numerous possible valid inferences.

### Invalid Deductive Inferences

A bad deductive inference is called an **invalid inference**. In invalid inferences, their structure does not guarantee the truth of the conclusion—that is to say, even if the premises are true, the conclusion *may* be false. This does not mean that the conclusion *must* be false, but that we simply cannot know whether the conclusion is true or false. Here is an example of an invalid inference:

- If it snows more than three inches, the schools are mandated to close.
- The schools closed.
- Therefore, it snowed more than three inches.

If the premises of this argument are true (and we assume they are), it may or may not have snowed more than three inches. Schools close for many reasons besides snow. Perhaps the school district experienced a power outage or a hurricane warning was issued for the area. Again, you can use your knowledge of necessary and sufficient conditions to understand why this form is invalid. Premise 2 claims that the necessary condition is the case. But the truth of the necessary condition does not guarantee that the sufficient condition is true. The conditional states that the closing of schools is guaranteed when it has snowed more than 3 inches, *not* that snow of more than 3 inches is guaranteed if the schools are closed.

Invalid deductive inferences can also take general forms. Here are two common invalid inference forms:

Affirming the Consequent:

- If X, then Y.
- Y.
- Therefore, X.

Denying the Antecedent:

- If X, then Y.
- Not X.
- Therefore, not Y.

You saw the first form, affirming the consequent, in the previous example concerning school closures. The fallacy is so called because the truth of the consequent (the necessary condition) is affirmed to infer the truth of the antecedent statement. The second form, denying the antecedent, occurs when the truth of the antecedent statement is denied to infer that the consequent is false. Your knowledge of sufficiency will help you understand why this inference is invalid. The truth of the antecedent (the sufficient condition) is only enough to know the truth of the consequent. But there may be more than one way for the consequent to be true, which means that the falsity of the sufficient condition does not guarantee that the consequent is false. Going back to an earlier example, that a creature is not a dog does not let you infer that it is not a mammal, even though being a dog is sufficient for being a mammal. Watch the video below for further examples of conditional reasoning. See if you can figure out which incorrect selection is structurally identical to affirming the consequent or denying the antecedent.

## Video

### The Wason Selection Task

### Testing Deductive Inferences

Earlier it was explained that logical analysis involves assuming the premises of an argument are true and then determining whether the conclusion logically follows, given the truth of those premises. For deductive arguments, if you can come up with a scenario where the premises are true but the conclusion is false, you have proven that the argument is invalid. An instance of a deductive argument where the premises are all true but the conclusion false is called a **counterexample**. As with counterexamples to statements, counterexamples to arguments are simply instances that *run counter to* the argument. Counterexamples to statements show that the statement is false, while counterexamples to deductive arguments show that the argument is invalid. Complete the exercise below to get a better understanding of coming up with counterexamples to prove invalidity.

## Think Like a Philosopher

Using the sample arguments given, come up with a counterexample to prove that the argument is invalid. A counterexample is a scenario in which the premises are true but the conclusion is false. Solutions are provided below.

**Argument 1:**

- If an animal is a dog, then it is a mammal.
- Charlie is not a dog.
- Therefore, Charlie is not a mammal.

**Argument 2:**

- All desserts are sweet foods.
- Some sweet foods are low fat.
- So all desserts are low fat.

**Argument 3:**

- If Jad doesn’t finish his homework on time, he won’t go to the party.
- Jad doesn’t go to the party.
- Jad didn’t finish his homework on time.

When you have completed your work on the three arguments, check your answers against the solutions below.

**Solution 1:** Invalid. If you imagine that Charlie is a cat (or other animal that is not a dog but is a mammal), then both the premises are true, while the conclusion is false. Charlie is not a dog, but Charlie *is* a mammal.

**Solution 2:** Invalid. Buttercream cake is a counterexample. Buttercream cake is a dessert and is sweet, which shows that not all desserts are low fat.

**Solution3:** Invalid. Assuming the first two premises are true, you can still imagine that Jad is too tired after finishing his homework and decides not to go to the party, thus making the conclusion false.

The content of this course has been taken from the free Philosophy textbook by Openstax