1.1 Validity and Soundness
- Deductive arguments are those in which the premises are intended to guarantee the conclusion. However, intending to guarantee the conclusion does not always mean the argument actually does so.
- An argument is valid if it is necessary that, if the premises are true, then the conclusion is true. In a valid argument, there is a necessary connection between the premises and the conclusion: it is impossible for the premises to be true and the conclusion false.
- The definition of validity does not require that the premises or the conclusion are actually true—only that if the premises were true, the conclusion would have to be true as well.
- Example of a valid argument:
- All biologists are scientists.
- John is not a scientist.
- So, John is not a biologist.
- An argument can be valid even if its premises are false, as long as the conclusion would follow if the premises were true.
- Example of a valid argument with false premises:
- All sharks are birds.
- All birds are politicians.
- So, all sharks are politicians.
- An argument can have all true premises and a true conclusion and still be invalid if the conclusion does not necessarily follow from the premises.
- Example of an invalid argument with true premises and a true conclusion:
- Some Americans work in the movie industry.
- Angelina Jolie is an American.
- Hence, Angelina Jolie works in the movie industry.
- Invalid arguments are those in which it is possible for the premises to be true and the conclusion false. In other words, the premises do not guarantee the conclusion.
- Example of an invalid argument:
- All dogs are animals.
- All cats are animals.
- Hence, all dogs are cats.
- Truth and validity are distinct:
- An argument can have true premises and a true conclusion and still be invalid.
- A valid argument can have false premises and a false conclusion, or false premises and a true conclusion.
- Validity is about the logical connection between premises and conclusion, not about their actual truth.
- Soundness:
- A sound argument is a valid argument in which all the premises are true. Because validity preserves truth, a sound argument will always have a true conclusion.
- An unsound argument is either invalid or has at least one false premise.
- Unsound arguments fall into three categories:
- Valid, but with at least one false premise.
- Invalid, but with all true premises.
- Invalid and with at least one false premise.
- Example of a sound argument:
- All collies are dogs.
- All dogs are animals.
- So, all collies are animals.
- Example of an unsound argument (valid but with a false premise):
- All birds are animals.
- Some grizzly bears are not animals.
- Therefore, some grizzly bears are not birds.
- Example of an unsound argument (invalid but with all true premises):
- All birds are animals.
- All grizzly bears are animals.
- So, all grizzly bears are birds.
- Key points about validity and soundness:
- Arguments themselves are not true or false; only statements (premises and conclusions) are true or false.
- Arguments can be valid, invalid, sound, or unsound, but statements cannot.
- A valid argument with a false conclusion must have at least one false premise.
- Validity preserves truth: if the premises are true and the argument is valid, the conclusion must be true.
- Validity does not preserve falsehood: valid arguments with false premises can have either true or false conclusions.
- You can determine whether an argument is valid or invalid even if you do not know the actual truth value of the premises or conclusion.
Summary of Definitions (from 1.1):
- Logic: The study of methods for evaluating whether the premises of an argument adequately support its conclusion.
- Argument: A set of statements where some (premises) are intended to support another (the conclusion).
- Statement: A sentence that is either true or false.
- Deductive argument: An argument in which the premises are intended to guarantee the conclusion.
- Inductive argument: An argument in which the premises are intended to make the conclusion more probable, without guaranteeing it.
- Valid argument: An argument in which it is necessary that, if the premises are true, then the conclusion is true.
- Invalid argument: An argument in which it is not necessary that, if the premises are true, then the conclusion is true.
- Sound argument: A valid argument in which all of the premises are true.
- Unsound argument: An argument that is invalid or has at least one false premise.
1.4 Strength and Cogency
- Inductive arguments are those in which the premises are intended to make the conclusion probable, but not guarantee it.
- The goal of an inductive argument is for the premises to make the conclusion likely or probable, not certain.
- A strong argument is an inductive argument in which it is probable (but not necessary) that, if the premises are true, then the conclusion is true.
- In a strong argument, it is possible (but unlikely) that the conclusion is false, given the premises.
- Example of a strong argument:
- 98 percent of Star Wars fans hate Jar Jar Binks.
- Kris is a Star Wars fan.
- So, Kris hates Jar Jar Binks.
- The argument is not valid (Kris could be among the 2 percent), but it is strong because the conclusion is probably true if the premises are true.
- A weak argument is an inductive argument in which it is not probable that, if the premises are true, then the conclusion is true.
- Example of a weak argument:
- 14 percent of Star Wars fans prefer Return of the Jedi to The Empire Strikes Back.
- Nina is a Star Wars fan.
- So, Nina prefers Return of the Jedi to The Empire Strikes Back.
- The conclusion is not likely, given the premises.
- Strength and weakness come in degrees:
- Unlike validity (which is all-or-nothing), inductive strength can vary. The higher the probability that the conclusion is true given the premises, the stronger the argument.
- For example, an argument is stronger if 99 percent of a group has a property than if 93 percent does.
- Cogency:
- A cogent argument is a strong argument in which all of the premises are true. The conclusion of a cogent argument is probably true.
- Example of a cogent argument:
- All or nearly all lemons that have been tasted were sour.
- So, the next lemon you taste will be sour.
- The argument is strong and the premise is true, so it is cogent.
- A cogent argument can have a false conclusion, since the premises do not guarantee the conclusion.
- Uncogent argument:
- An uncogent argument is one that is either weak or strong with at least one false premise.
- Uncogent arguments fall into three categories:
- Strong, but with at least one false premise.
- Weak, but all premises are true.
- Weak and with at least one false premise.
- Example of an uncogent argument (strong but with a false premise):
- Most wizards are muggles.
- Hermione is a wizard.
- So, Hermione is a muggle.
- Example of an uncogent argument (weak but with all true premises):
- According to a 2002 article in the National Enquirer, male members of Elizabeth Smart’s family were involved in a gay sex ring. So, they probably were.
- Example of an uncogent argument (weak and with a false premise):
- Huds Ford Bronco 4x4 will get good gas mileage. After all, Dennis’s Toyota Corolla gets good gas mileage, and like Dennis’s Corolla, Huds Bronco has velvet upholstery, tinted glass windows, and a six-disk CD player.
- Types of inductive arguments:
- Arguments from authority: The strength depends on the reliability of the authority.
- Arguments from analogy: The strength depends on the relevance and number of similarities.
- Statistical syllogisms: The strength depends on the proportion and how representative the sample is.
- Strength and cogency vs. validity and soundness:
- Validity and invalidity do not come in degrees; an argument is either valid or not.
- Strength and weakness do come in degrees; an argument can be more or less strong.
- A sound argument cannot have a false conclusion, but a cogent argument can.
- Terminology:
- Arguments can be strong, weak, cogent, or uncogent.
- Arguments are never true or false; only statements (premises and conclusions) are true or false.
- Premises and conclusions are not strong, weak, cogent, or uncogent—only arguments as a whole are.
Summary of Definitions (from 1.4):
- Strong argument: An argument in which it is probable (but not necessary) that, if the premises are true, then the conclusion is true.
- Weak argument: An argument in which it is not probable that, if the premises are true, then the conclusion is true.
- Cogent argument: A strong argument in which all of the premises are true.
- Uncogent argument: An argument that is either weak or strong with at least one false premise.
- Deductive logic: The part of logic concerned with the study of methods for evaluating arguments for validity and invalidity.
- Inductive logic: The part of logic concerned with the study of methods for evaluating arguments for strength and weakness.
Key Takeaways from Sections 1.1 and 1.4
- Validity is about the logical connection between premises and conclusion in deductive arguments; soundness adds the requirement that all premises are true.
- Strength is about the probability that the conclusion is true given the premises in inductive arguments; cogency adds the requirement that all premises are true.
- Validity and soundness are central to deductive logic; strength and cogency are central to inductive logic.
- Validity does not require true premises or a true conclusion; soundness does.
- Strength comes in degrees; validity does not.
- A sound argument cannot have a false conclusion; a cogent argument can.
- Arguments are evaluated for validity/invalidity (deductive) or strength/weakness (inductive), and for soundness/cogency based on the truth of their premises.