BA Chapter 5 Questions

counterexample

A counterexample to prove the invalidity of a given argument is another argument exemplifying the same form but with true premises and a false conclusion. ¦ To find a counterexample may require imagining a scenario that is "possible" in the sense that it involves no internal contradiction. The actual world is only one among many such possi¬ ble scenarios called "possible worlds."

What is an argument form? And how does an argument differ from an argument form?

An argument form is the type of logical mold or pattern that each argument exemplifies. Often the same argument form is the underlying pattern of many actual arguments. To show the form of an argument, it is customary to replace some words in it by "place holders" or symbols such as capital letters, keeping only the words that have a logical function.

What's the cash value of invalidity?

As discussed above, in all cases of invalidity, arguments fail to be truth-preserving—so that then any arrangement of truth values in premises and conclusion would be logically possible.

Sometimes people use 'valid' to mean 'true' or 'reasonable' and 'invalid' to mean 'false' or 'unreasonable.'

But these are not what 'valid' and 'invalid' mean in logical thinking.

The upshot of all this is:

In a valid argument, it makes no logical sense to accept the premises and reject the conclusion.

What matters is

whether the premises could be true and the conclusion false at once, because that would determine the invalidity of the argument.

Arguments are

¦ neither true nor false, ¦ but either valid or invalid.

At the same time,

it makes no sense to say that an argument is "true" or "false." Statements and beliefs can be true or false. But arguments can't have a truth value!

Recall that a proposition is

the content of a belief or statement, which has a truth value: it is either true or false.

To prove that an argument form is invalid, logical thinkers use the method of counterexample:

they try think of an argument with true premises and a false conclusion that has exactly that same argument form. Thus (9) is a counterexample to (8) above. That is, (9) is an example that proves (8)'s invalidity, because it shows that it is possible to have an argument with exactly the same argument form but with true premises and a false conclusion.

What does the expression 'cash value,' as used above, mean?

Money related

Arguments of the former type

are propositional, those of the latter categorical.

Statements are

either true or false, ¦ but neither valid nor invalid.

Such an argument is, by defi¬ nition,

invalid: its premises do not entail its conclusion. Note that we're introducing here some different expressions that all mean the same thing. To say that an argument is valid is equivalent to saying that its premises entail its conclusion. And both of these are equivalent to saying that the argument is truth-preserving, and that its conclusion follows necessarily from its premise or premises.

modus ponens

not an argument but an argument form showing a certain relation between premises and conclusion

modus tollens

offers a short list of some valid propositional argument forms,

What's the convention for representing categorical argument forms?

such as 'All,' 'No,' and 'Some,' the argument is probably better reconstructed as categorical.

'Validity' as a Technical Word

As we have been using the words 'valid' and 'invalid' here, there is no such thing as a "valid statement" or an "invalid statement." Though these expressions are sometimes heard in everyday language, 'valid' and 'invalid' are technical words in logic that cannot be applied to a single statement, but only to a relation between statements—namely, the relation called argument. 'Valid' can apply only to an argument whose premises necessitate or entail its conclusion, 'invalid' only to one whose premises fail to do this.

Is finding that an argument is invalid a reason to reject it? If yes, why? If not, why not?

No. Yet finding an argument invalid is not a conclusive reason to reject it, since it could still be a good inductive argument (more on this in Chapter 6). Once an argument is found valid, logical thinkers should then check whether its premises are true,

What does it mean to say that an argument's form is categorical?

On the other hand, when you see in the premises certain words indicating quantity, such as 'All,' 'No,' and 'Some,' the argument is probably better reconstructed as categorical.

What does it mean to say that a valid argument is 'truth-preserving'?

Only valid arguments are truth-preserving: If their premises are true, then it is not possible for their conclusion to be false.

Define validity and invalidity in terms of argument form.

Validity consists in this relationship, and nothing more. The fact that an argument might have one or more false premises is of no importance for its validity, which is entirely a matter of argument form. Invalidity is also a matter of argument form: an argument form is invalid if and only if an argument with that form could have true premises and a false conclusion. But 'could' here means 'logically possible,' which leaves open the possibility that a given invalid argument may have true premises and a true conclusion.

What does it mean to say that an argument's form is propositional?

When you see certain connections between propositions, such as 'Either ... or ... ' and 'If ... then ... ,' the argument is probably better reconstructed as propositional.

Could different arguments be instances of the same argument form? If so, how? If not, why not?

Yes, because often the same argument form is the underlying pattern of many actual arguments. To show the form of an argument, it is customary to replace some words in it by "place holders" or symbols such as capital letters, keeping only the words that have a logical function.

Could an argument be an instance of more than one argument form? If so, how? If not, why not?

Yes, because the fact that an argument might have one or more false premises is of no importance for its validity, which is entirely a matter of argument form.

Being truth-preserving is

a characteristic a valid argument has in virtue of the form or pattern it exemplifies. Some arguments have the characteristic of being truth¬ preserving because the statements that constitute their premises and conclusion are connected in certain ways, forming distinctive patterns of relationship that transfer the truth of the premises (if they are true) to the arguments' conclusions.

Recall inductive arguments:

although their premises might provide some reasons for the conclusion, they would never entail it. According to this definition, all inductive arguments fail to meet the standard of validity.

When is an argument invalid? When is an argument valid?

an argument form is invalid if and only if an argument with that form could have true premises and a false conclusion. In a valid argument, it makes no logical sense to accept the premises and reject the conclusion.

Other arguments have it

because within the statements that constitute their premises and conclusions there are some expressions, usually called terms, that bear certain relationships to each other that make the arguments' conclusions true if the premises are true.

Contradictory statements

cannot have the same truth value: if one is true, the other must be false.

Its premise 1

features two simple propositions connected by 'if ... then ... and its premise 2 asserts the first of those two simple propositions.

Since a valid argument's premises,

if true, determine that the conclusion is true, valid arguments can also be said to be truth-preserving.

Invalidity

is also a matter of argument form: an argument form is invalid if and only if an argument with that form could have true premises and a false conclusion.

Validity

is one of the standards used to evaluate deductive arguments. Whether an argument is valid or not is never a matter of degree, but instead one of all or nothing. An argument cannot be 'sort of valid.' It's either valid or it's not. Furthermore, there is a simple test to determine the validity of an argument. Validity is best thought of as a kind of relation between premises and conclusion in an argument, where the actual truth or falsity of the component statements is largely irrelevant. What matters is: do the premises necessitate the conclusion? If so, it's valid. If not, it's invalid.

An argument form

is the type of logical mold or pattern that each argument exemplifies. Often the same argument form is the underlying pattern of many actual arguments. To show the form of an argument, it is customary to replace some words in it by "place holders" or symbols such as capital letters, keeping only the words that have a logical function. (4* 1) is a valid argument form, because any argument with this underlying form would be valid: if its premises were true, its conclusion would have to be true.

For an argument to be valid,

it is of no importance whether it has all false premises, as in the case of (4), or a false conclusion with at least one false premises as in (7) or even all false statements as in (5).

What's the convention for representing propositional argument forms?

such as 'Either ... or ... ' and 'If ... then ... ,' the argument is probably better reconstructed as propositional.

A deductive argument is

valid if and only if its premises necessitate or entail its conclusion, where 'entail- ment' is defined as in Box 1.

In claiming that false conclusions are 'possible,'

we have in mind logical possibility. Whether (2) and (3) would be likely to have true premises and false conclusion in our actual world, with things being as they are, is beside the point. Rather, if there is some scenario, 'possible' in the sense that it implies no internal contradiction, in which these arguments' premises could be true and their conclusions false at once, then the arguments are invalid.

VALID VS. INVALID ARGUMENTS

1. Arguments may be divided into two groups: those that are valid and those that are invalid. 2. Only valid arguments are truth-preserving: If their premises are true, then it is not possible for their conclusion to be false. 3. Only in a valid argument do the premises entail the conclusion. 4. A logical thinker who accepts the premises of a valid argument cannot reject its conclusion without contradiction. But this doesn't happen in the case of an invalid argument.

Validity and Argument Form

In any argument exemplifying a valid form, there is a relationship of entailment between premises and conclusion. If the argument's premises are true, its conclusion cannot be false. Validity consists in this relationship, and nothing more. The fact that an argument might have one or more false premises is of no importance for its validity, which is entirely a matter of argument form.

What's the cash value of validity?

The cash value of standards for argument evaluation, such as validity, is simply their practical impact: knowing whether an argument meets them or not determines the attitude we should have about its conclusion on the basis of its premises.

What is entailment? How is entailment related to validity?

There is entailment in an argument if and only if the truth of the argument's premises guarantees the truth of its conclusion—in the sense that, if the premises are all true, the conclusion cannot be false. Such an argument is valid and truth-preserving.

entailment

There is entailment in an argument if and only if the truth of the argument's premises guarantees the truth of its conclusion—in the sense that, if the premises are all true, the conclusion cannot be false. Such an argument is valid and truth-preserving.

What does it mean to say that 'validity' is a technical term?

Though these expressions are sometimes heard in everyday language, 'valid' and 'invalid' are technical words in logic that cannot be applied to a single statement, but only to a relation between statements—namely, the relation called argument.