Assignment 40 for practical work in media lab: Proving Invalidity For the Departments of Media Studies by Prof Dr Sohail Ansari

Proving Invalidity

It is relatively simple to prove an argument to be invalid. It is difficult to prove one valid, and that will be the thrust of most of the rest of the course. But, that comes later. For now, luckily, we're learning a simple way to demonstrate invalidity--the counterexample method. This is useful because if you can prove a particular argument to be invalid you can ignore it and don't have to worry about checking whether it's premises are true or not, which can be difficult.
To use the counterexample method it is necessary to be able to abstract the form from the content of a given argument. That done, you then substitute back into the argument a different, simple, familiar content. Your goal in doing this will be to try to force the argument into having all clearly true premises, but a false conclusion. If it is possible to do this to an argument, the argument is obviously invalid. If you cannot substitute words that give true premises and false conclusion it does not necessarily mean that the argument is valid. The argument may be valid or you may simply have failed to prove it invalid.
We extract the form by using capital letters to symbolize terms in the statements. (Capital letters are nothing more than a convenient convention. Any symbol would do as well.) One letter stands for each term.
A term is a simple component of a statement.

e.g. All hamsters are furry animals.

There are two terms in this statement,  hamsters and ‚ furry animals. Terms can be longer phrases such as "people who like to eat pasta"
Each term can be replaced by a single capital letter. Once again, common sense will help you out here.

e.g. All H are 

To prove invalidity of a deductive argument you can first abstract the form from the content of a given argument
STEPS:
1. Abstract the form from the content, listing the premises first and the conclusion last.
2. Beside the abstracted argument, set up the skeleton of the argument, noting that premises must be true and the conclusion false.
3. With a pencil, try substituting terms into the skeleton that result in all true premises and a false conclusion.
4. Be sure your substitution instances are consistent between the two arguments, i.e. if the letter "C" is replaced by the new term "mammals" in one part of the argument, it must do so everywhere.
Example:
All hamsters are furry animals.
All hamsters are creatures who like to eat carrots.
All furry animals are creatures who like to eat carrots.
All H are F.      T       All dogs are animals.
All H are C.      T      All dogs are mammals.
All F are C.      F       All animals are mammals.
Also note that "some" means "at least one" so "some dogs are animals" is true. It is also true that "all dogs are animals" and if all are then certainly some are as well. There is a tendency to think that "some dogs are animals" implies that "some dogs are not animals" and then assume that both statements are false. This is not the case. In logic, as a general rule, do not infer any more than what is given.
****You will only be asked to be able to give counterexamples to categorical syllogisms (made up of "all," "some," "no" statements) so you can ignore the parts of your reading where Hurley explains about substituting into conditional statements.******
ASSIGNMENT: (20 points each)
Construct counterexamples to each of the following arguments to prove that they are invalid. I suggest that you use terms from the following list: cats, dogs, fish, mammals, animals, because everyone agrees about them. (Understand that a fish is not a mammal, but it is an animal; cats and dogs are both mammals and animals.) But, if you want to be creative and use your own words keep in mind the sentences you construct must be clearly true or false, otherwise you haven't accomplished anything. Be consistent with your terms. Show your intermediate steps.*Be careful in determining which statement is the conclusion. Use indicator words as a guide (Hurley p. 3). Picking the wrong statement as a conclusion will cause the entire problem to be wrong!
1. No Coptics are Shinto. No Shinto are Hasidic. It follows that no Hasidim are Coptics.
2. No corporate directors are investigators since all auditors are investigators and no auditors are corporate directors.
3. No airline flights that allow smoking are flights that are safe for nonsmokers. Therefore, some airline fights that allow smoking are not international flights, since some flights that are safe for nonsmokers are not international flights.
4. All college professors are teachers, so all teachers are educators, since all college professors are educators.
5. Some persons who regret their crimes are convicted murderers. So some convicted murders are persons capable of being reformed, since all persons capable of being reformed are persons who regret their crimes.