Counterexamples. A simple logic proof method.

Transcription

1 Counterexamples A simple logic proof method.

2 Valid In ordinary language: often used as a synonym for Good. Daytime TV is full of people with valid opinions who make valid points. Don t use valid this way (for this class)! Try to never use it this way! Valid has a very precise technical meaning.

3 What Valid really means. A valid argument is one that guarantees the truth of the conclusion whenever the premises are true. Example Valid argument. All dogs are canines. No Cats are canines. No dogs are cats.

4 But, valid is a conditional concept Arguments whose premises are false, but which would guarantee the conclusion if the premises are true, are still valid. Eg: valid argument with false premises. All dogs are cats. Obviously false. (O.F.) All cats are whales. OF All dogs are whales. OF

5 Think of it this way If all dogs were cats, and If all cats were whales Then all dogs would be whales. Valid is a description of arguments, not parts of arguments (which can be either true or false)

6 We are not good at hearing validity. We are often focused on meaning and truthfulness. It takes a lot of practise to hear good arguments vs bad.

7 More odd Valid arguments. All dogs are cats. OF All cats are canines. OF All dogs are canines. Obviously true. If all dogs were cats and all cats were canines, Then it would be true that all dogs are canines. This example illustrates that the falsity of the premises doesn t imply the falsity of the conclusion.

8 This is a valid argument All X are Y All Y are Z Therefore all X are Z. Despite the fact that you don t know what X, Y or Z represent. Ignore Actual truth or falsity when considering validity, concentrate on the relationship between the premises and the conclusion.

9 Examples of invalid. All dogs are mammals. All cats are mammals. All dogs are cats. O.F. All dogs are mammals All cats are mammals No dogs are cats. (all true, but still invalid for reasons as follows )

10 How do you know that is invalid? Any argument that permits a counterexample is invalid. A counterexample is an argument with the same form as the original argument, but which has obviously true premises and an obviously false conclusion.

11 Counterexample Example Original argument All x are y All z are y No x are y. Counterexample: All fish are cold-blooded. All spiders are cold-blooded. Therefore no fish are spiders.

12 Using Counterexamples Counterexamples are a form of PROOF that an argument is invalid. They are also effective in contexts where people don t know their logic. Someone who has never studied reasoning can often be convinced by ear that their argument is flawed with the presentation of a counter example.

13 Providing counter examples. Take the original argument: Identify the form (the structure of the argument) Think of another argument with the same form but with obviously true premises and an obviously false conclusion. Ex next page.

14 Original argument All Premiers are charismatic men. Pallister is a charismatic man, Pallister will be Premier. Counterexample: Fill in the blanks. P1 All are P2. is C: will be

15 Fallacies and Counterexamples In the next section of the course we will study the fallacies We will use a lot of counter-examples to help understand why these arguments are fallacies.

16 Logical Fallacy Two senses: Any argument that fails to adequately support its conclusion. It is impossible to define all the ways you can be wrong. Any argument that fits into common patterns of error in reasoning. The fallacies.

17 To be a fallacy To be a fallacy a series of statements must first be an argument: You re a jerk, therefore you re wrong. Is a fallacy (ad hominem aka fallacy of abuse.) You re a jerk, is not a fallacy, it is mere abuse.

18 Example of a fallacy Childhood obesity has increased now that so many children are playing video games, so obviously video games cause obesity in children. This argument relies on the evident fact that video games and obesity have occurred together to conclude one is the cause of the other.

19 Cum hoc, ergo procter hoc A common causal fallacy, know by its traditional latin name: With this, therefore because of this. Counterexample: (to expose the fallacy) Video games have been gradually increasing in popularity since the 80 s, and my hair has been decreasing since then, so obviously video games have caused my baldness.

20 Formal versus Informal fallacies Formal fallacies are fallacies that violate some specific logical rule or law. Ex: All Geese can fly All ducks can fly All geese are ducks. This argument commits the formal rule regarding the distribution of terms in an argument.

21 Formal versus Informal fallacies Formal fallacies are fallacies that violate some specific logical rule or law. Ex: All Geese can fly All ducks can fly All geese are ducks. The category of things that can fly includes both ducks and geese. The way this argument refers to flying things, they can be either ducks or geese, not both. This argument commits the formal rule regarding the distribution of terms in an argument.

22 Formal vs informal fallacies Informal fallacies are not violations of specific logical rules Instead are errors of reasoning common enough to be named, recognized, and studied. Traditional education in law often focused heavily on the informal fallacies. Informal fallacies are often grouped by category in various ways. There are many different groupings in different texts.

23 Additional sources for Fallacies Extensive collection of fallacies. Collection of fallacies relevant to the website: rebutting holocaust deniers. This site indicates a proof condition for each fallacy given.