Determining Validity and Invalidity in Deductive Arguments

PHIL 121: Methods of Reasoning

February 6, 2013

Instructor:Karin Howe

Binghamton University

Standard Form of a Statement• Certain kinds of statements need to be put

into standard form before we can analyze them.

• What do we mean by the "standard form" of a statement? What kinds of statements are we worried about here?– Conditionals– Categorical propositions

Standard Form of a Conditional• The statement following the word 'if' is the

antecedent; accordingly, the statement that follows the word 'if' is placed before the statement following the word 'then' (which is the consequent). The statement is then said to be in "standard form."

• The statement "If Marvin stays, then Nancy leaves" is in standard form; whereas the statement "Nancy leaves if Marvin stays" is not.

Standard Form of a Categorical Proposition• Categorical propositions are statements that

involve the quantifiers "all," "some," or "none." (we will be studying these kinds of statements in depth in the next section of the course)

• Categorical propositions also have a "standard form," which looks like this:– Quantifier (subject term) copula (predicate term)

– The copula is always some form of the verb "to be"

– Subject and predicate terms need to be able to be expressed as classes of things (nouns or noun phrases)

Examples• All cats are mammals.

– This statement is already in standard form. Yay!

• Some cats are furry.

– This statement is almost in standard form -- we've got quantifier (subject term) copula all set, but we run into problems with the predicate term, which is not a noun or noun phrase. How can we fix this?

– Like this: Some cats are furry things.

• All hamsters like donuts.

– This statement is also not in standard form - here we also need to get the copula part in there.

– We do so like this: All hamsters are things that like donuts.

Standard Form of an Argument

• We can also talk about the "standard form" of an argument.

• Basically, it's a convention for writing out arguments

1. List the premises of the argument, putting each premise on a separate line and number each line.

2. Draw a line under the premises.3. Write the conclusion under the line, identifying it as

the conclusion by using either a conclusion indicator, or by placing this special conclusion symbol () in front of the statement.

Example

1. Kangaroos can fly.

2. Karin is a kangaroo._____

Karin can fly. (Therefore Karin can fly.)

Plan for today:

• Walk through an example of each method (interactively)

• Work through some more examples as a class where I specify the method

• Finally, work through as many examples as we can as a class where the method is left open ended -- it is up to you to decide which method will work the best!

Identifying valid/invalid forms

• The gerbil is in the teapot if the cat is. The gerbil is in the teapot. Therefore, the cat is in the teapot.

• Standard form:1. If the cat is in the teapot, then the gerbil is in

the teapot.2. The gerbil is in the teapot._______

Therefore the cat is in the teapot.

Next…• Abstract away from content

– C = the cat is in the teapot

– G = the gerbil is in the teapot

1. If C, then G

2. G_________

Therefore C

• Identify the form of the argument– Affirming the consequent

• Valid or invalid?– INVALID

Another example (do via the same method)

• If there is a rabbit hole nearby then that means there is company around. If there is company around, then there will be food available. Therefore, if there is a rabbit hole nearby then there will be food available.

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Counter-example

• Either the cat or the gerbil have been in the teapot. The gerbil has been in the teapot. Therefore, the cat has not been in the teapot.

1. Either the cat has been in the teapot or the gerbil have been in the teapot.

2. The gerbil has been in the teapot.______

Therefore, the cat has not been in the teapot.

Next …

• Abstract away from content– C = the cat has been in the teapot– G = the gerbil has been in the teapot

1. Either C or G

2. G_________

Therefore not C

Then …

• Pick a domain– Good choices: the natural numbers, facts about

mammals, any other easily accessible domain

• Restate the premises in that domain

• Determine invalidity (why can't we determine validity?)

Another example (provide a counter-example)

Some hamsters like to get into teapots. Some things that like to get into teapots are not cats. Therefore, some hamsters are not cats.

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Applying the definition of validity/invalidity

• Case 1: the premises are all true in the real world

• Case 2: at least some of the premises are false in the real world, or their truth value is unknown or uncertain

• Special case: the premises are inconsistent

Example

• Milk is white, and so is snow. Snow is a frozen liquid, and milk is liquid, so snow is made of milk.

• Standard form:1. Milk is white.

2. Snow is white.

3. Snow is a frozen liquid.

4. Milk is liquid.________

So, snow is milk.

Next step …

1. Milk is white.

2. Snow is white.

3. Snow is a frozen liquid.

4. Milk is liquid.________

So, snow is milk.

• Which kind of case is this? – All the premise are true in the real world.

– Also, the conclusion is false.

– In effect, this argument is its own counter-example.

More examples

• All gerbils like to get into teapots. All gerbils are not cats. Therefore, nothing that likes to get into teapots is a cat.

• Invalid arguments do not establish their conclusions. There are proofs of God's existence that are not invalid. This shows that some proofs for God's existence establish their conclusions.

• No one who knowingly and needlessly endangers his or her health is rational. Thus, college students who smoke are not rational, because every college student who smokes is knowingly and unnecessarily endangering his or her health.

• If it rains for an hour a day a week, the drought will be broken. Thus, either it rains for an hour an hour a day for a week or the drought will not be broken.

• If God hadn't wanted there to be poor people, He would have made rich people more generous, Hence, he must want there to be poor people, since He did not make the rich more generous.

• All healthy diets contain significant amounts of protein. All diets that include meat contain significant amounts of protein. Hence, all healthy diets include meat.