7 Chapter 7 Deductive Arguments
Deductive Arguments, Validity, and Soundness
Deductive arguments are a fundamental concept in logic and critical thinking. These arguments are structured to draw a conclusion that necessarily follows from a set of premises. In other words, if the premises of a deductive argument are true, the conclusion must also be true. This characteristic distinguishes deductive arguments from inductive arguments, which draw probable conclusions based on evidence.
When evaluating deductive arguments, two key terms come into play: validity and soundness. An argument is considered valid if its conclusion logically follows from its premises, regardless of whether those premises are actually true. In other words, validity is concerned with the logical structure of the argument, not the factual accuracy of its components. On the other hand, an argument is deemed sound if it is both valid and all of its premises are true. Soundness, therefore, combines logical validity with factual correctness, making it a stronger criterion for evaluating arguments.
Examples:
-
Valid but not sound:
Premise 1: All cats are reptiles.
Premise 2: Fluffy is a cat.
Conclusion: Therefore, Fluffy is a reptile. -
Both valid and sound:
Premise 1: All squares are rectangles.
Premise 2: All rectangles have four sides.
Conclusion: Therefore, all squares have four sides.
Exercise 7.1: Identifying valid arguments.
Determine whether each of the following arguments is valid or invalid.
1. No trees grow at the North Pole.
All oaks are trees.
Thus, no oaks grow at the North Pole.
2. The earth is flat.
Nothing flat is round.
Thus, the earth is not round.
3. All salamanders eat rust.
Nothing that eats rust likes fresh iron.
Thus, no salamanders like fresh iron.
4. 100% of Americans like hamburgers.
Jessica is an American.
Thus, Jessica likes hamburgers.
5. Geologists believe that the earth’s core is iron.
Iron is a metal.
Thus, there is metal at the earth’s core.
6. Millions of people have tried Bubble Up Soda.
Everyone who has tried it likes it.
So, if you try it, you will like it too.
7. My doctor tells me that if I take this medicine I will be cured.
I will take this medicine.
Thus, I will be cured.
8. Most Americans who can read are at least ten years old.
Thus, most Americans who cannot read are under ten years old.
9. All mice are rodents.
All rodents can fly.
Thus, all mice can fly.
10. Most mice can fly.
Most mice can sing.
Thus, something can fly and sing.
11. It rained last night.
Thus, the ground is wet.
12. The vast majority of frogs croak but do not speak.
Kermit is a frog.
Thus, Kermit croaks but doesn’t speak.
13. All trees bear fruit.
The oak is a tree.
Thus, the oak bears fruit.
14. Most walls are vertical.
Many walls are made of brick.
Thus, some brick walls are vertical.
15. Many who smoke cigarettes are polite.
Many who smoke cigarettes drink coffee.
Thus, some people who are polite drink coffee.
16. Only women can have babies.
Pat cannot have babies.
Thus, Pat is not a woman.
17. Anyone who could recite the Iliad by heart had an excellent memory.
Homer could not recite the Iliad by heart.
So, Homer didn’t have an excellent memory.
18. Only those who like to read dense prose like to read Husserl.
Marvin likes to read dense prose.
So, Marvin likes to read Husserl.
19. If Susan was at the party then Jack was there too.
If Jack was at the party, then Amanda was there.
So, if Susan was at the party then Amanda was there.
20. If Susan was at the party then Jack was there too.
If Jack was at the party then Amanda was there.
So, if Amanda was at the party then Susan was there.
21. If elephants can climb trees, then grasshoppers wear boots.
Elephants can climb trees.
So, grasshoppers wear boots.
22. If Sally went to the store, then she got ice cream.
Sally did not get ice cream.
So, Sally did not go to the store.
Exercise 7.2
1. For each of the problems (a) through (h) below, either give an example of an argument which fits the description, or else explain why there cannot be such an argument. Construct the arguments for your examples by taking your premises and conclusions from the following sentences.
All birds have bones. All dogs have bones.
All crows are birds. All worms are dogs.
All crows have bones. All worms have bones.
All birds are dogs.
A sample answer has been provided for the first problem.
(a) A valid argument with all true premises and a true conclusion.
Answer: All birds have bones.
All crows are birds.
Thus, all crows have bones.
(b) A valid argument with all true premises and a false conclusion.
(c) A valid argument with at least one false premise and a true conclusion.
(d) A valid argument with at least one false premise and a false conclusion.
(e) An invalid argument with all true premises and a true conclusion.
(f) An invalid argument with all true premises and a false conclusion.
(g) An invalid argument with at least one false premise and a true conclusion.
(h) An invalid argument with at least one false premise and a false conclusion.
Exercise 7.3
Determine whether each of the following arguments is valid or invalid.
1. Most phenomenologists like to read Husserl.
Most American philosophers do not like to read Husserl.
Thus, most phenomenologists are not American philosophers.
2. Either Ben or Carl is at home.
Carl and Danielle are not both at home.
Thus, if Danielle is at home then so is Ben.
3. Belinda believes that Benjamin Franklin lived in Boston when he was young.
Benjamin Franklin was the first Postmaster General.
Thus, Belinda believes that the first Postmaster General lived in Boston when he was young.
4. Erin does not know whether Paul was at the game.
Erin does not know whether Amelia was at the game.
Erin does not know whether Douglas was at the game.
Erin knows that Paul, Amelia and Douglas are the only member of the Pistol Pete Club.
Thus, Erin does not know whether anyone from the Pistol Pete Club was at the game.
If the conclusion of an argument logically follows from its premises.