III Deductive Arguments: Validity and Soundness

When evaluating arguments, i.e. determining whether they are good or bad, strong or weak, persuasive or not persuasive, there are two questions we should ask (1) whether the premises provided appropriate support for the conclusion; (2) whether the premises are, in fact, true. These are the steps taken when evaluating a single argument. When evaluating a complex argument each of the single arguments of which it is composed must be evaluated and then an overall evaluation of how the single arguments fit together must be made.

1. Logical Correctness

The first question is a matter of "logical correctness."

An argument is considered to be "logically correct" when it satisfies the following condition:

If the premises were true, this fact would constitute good grounds for accepting the conclusion as true.

Notice that this condition presupposes that one is dealing with statements that are capable of being true or false. Nevertheless this condition is not concerned with whether the premises are in fact true. In evaluating arguments for logical correctness one is concerned with the relation between the premises and the conclusion not with the question of whether the premises are in fact true.

2. Deductive Validity

To make this condition more specific we have to specify what we take to be "good grounds". Certainly the truth of the premises guaranteeing the truth of the conclusion would mean the truth of the premises provided good grounds for accepting the conclusion as true. The criterion of logical correctness that requires the guarantee is called "the deductive criterion" of logical correctness

An argument form is deductively valid if and only if it is impossible that its conclusion is false given its premises are true.

Notice that this criterion for deductive validity does not require that the premises are true, nor that the conclusion is true, rather it says that IF the premises are true, the conclusion must be true. Deductive validity is a function of the form, or structure, of the statements in the argument and not a function of whether the statements are in fact true.

Consider the following two examples:

Argument 1

Argument 2

P1. All humans are mortal

P1. All mammals are four-legged

P2. You (the reader) are human

P2. You (the reader) are a mammal

C. You (the reader) are mortal

C. You (the reader) are four-legged

In Argument 1, both premises and the conclusion are true. In Argument 2, P1 and the conclusion are false. Notice that the arguments have the same form or structure:

P1. All A are B

P2. x is an A

C. x is a B

It is because of this form that we can say that the truth of the premises guarantees the truth of the conclusion. IF it were true that all mammals are four-legged, then it must be true that you, as a mammal, are four-legged. The argument form in these examples is one of many deductively valid argument forms. Other deductively valid arguments will be presented later. If an argument in ordinary discourse fits into a deductively valid argument form, then we can say that if the premises are true the conclusion must be true even though we don't know whether the premises are true. We can know that an argument is valid and not know the meaning of the terms in the premises and conclusion. For example:

P1. All pirons are elactical.

P2. All elacticals are verdish.

C. All pirons are verdish.

The terms in this argument may be from a highly specialized science in which they can be determined true or false or they may be nonsense. But that makes no difference to the validity of the argument. It is a deductively valid argument because of the form. IF the premises turn out to be true, they guarantee the truth of the conclusion.

3. Some Deductively Valid Argument Forms

It is useful for understanding and evaluating arguments to have knowledge of a relatively small number of deductively valid argument forms. Much of what you read in philosophy can be analyzed with them.

3a.  Universal Syllogism

Form

Example

P1. All A are B

P1. All dogs are mammals

P2. All B are C

P2. All mammals are warm-blooded

C. All A are C

C. All dogs are warm-blooded

 

3b. Predicate Instantiation

Form

Example

P1. All A are B

P1. All dogs are warm-blooded

P2. x is an A

P2. Fido is a dog

C. x is a B

C. Fido is warm-blooded

These two argument forms are deductively valid. This means that whatever is substituted for A, B, C and x, the truth of the premises guarantees the truth of the conclusion, provided the substitution is uniform, e.g. whatever is substituted for "A" in one premises must be substituted for "A" in all occurrences of "A" in other premises or conclusion. These two argument forms are part of predicate logic. " is a dog" and "is warm-blooded" are predicates, i.e. the properties of being a dog or being warm-blooded can be applied to individuals, e.g. Fido. Also, it can be asserted that everything that has one property also has an additional property, e.g. all things that are dogs are also things that are warm-blooded. These two argument forms are only a small part of predicate logic, still they are useful when critically reading a text.

 Propositional logic is the logic of propositions, or statements. In this logic, the variables in the valid argument forms are place holders for complete statements. In propositional logic statements are connected by logical connectives: "and", "or", "if ... then," and "not." The following are a few of the useful deductively valid argument forms in propositional logic.

3c. Affirming the Antecedent (also called Modus Ponens)

Form

Example

P1. If p then q.

P1. If John is a freshman then he can't enroll in Physics

P2. p

P2. John is a freshman.

C. q

C. John can't enroll in Physics

3d. Denying the Consequent (also called Modus Tollens)

Form

Example

P1. If p then q

P1. If Mary is a freshman then she lives on campus

P2. not q

P2. Mary does not live on campus

C. not p

C. Mary is not a freshman

 

3e. Disjunctive Argument

Form

Example

P1. p or q

P1. Either John loves Mary or he loves Susan

P2. not p

P2. John does not love Mary

C. q

C. John loves Susan

 

3f. Hypothetical Argument

Form

Example

P1. If p then q

P1. If Mary loves John then she loves a loser.

P2. If q then r

P2. If Mary loves a loser then she will be unhappy

C. If p then r

C. If Mary loves John then she will be unhappy

3g. Chain Argument

Form

Example

P1. If p then q

P1. If John is a loser then he will make Mary unhappy

P2. If q then r

P2. If John makes Mary unhappy Susan will hate him

P3. p

P3. John is a loser

C. r

C. Susan will hate John

The order of the premises in the above argument forms is the order for which most people intuitively see the validity of the arguments. In everyday discourse, the premises and conclusions won't always be presented in this order. Consider a "real life" version of the example of Chain Argument.

You know, Susan will wind up hating John. I'll tell you why. He's a loser, and if so, he will make Mary unhappy. And if that makes Mary unhappy, then Susan will hate him.

The premises and conclusion can be labeled as follows:

You know, (C) Susan will wind up hating John. I'll tell you why. (P3) He's a loser, and (P1) if so, he will make Mary unhappy. And (P2) if that makes Mary unhappy, then Susan will hate him.

3h. Reductio Ad Absurdum

Deductive arguments can be used to refute a view, as well as to prove a view. A form of refutation commonly used in Philosophy and other fields of inquiry is "Reductio ad absurdum "(literally "reducing to absurdity".) The reductio method identifies a premise that is not obviously false, combines it with other premises that are clearly true and then deduces by a valid argument a conclusion that is a contradiction or absurd (clearly known to be false.)

The basic structure of a reductio argument is:

P1. Q (the premise in doubt)

P2. (known to be true)

P3. (known to be true)

C. R (absurd, clearly known to be false)

For a successful reductio argument the argument form must be valid. For it if is, the premises cannot all be true and the conclusion false. Given the false conclusion, P1 must be false, since P2 and P3 are known to be true.

 Suppose someone argues that we ought to have the death penalty for first degree murder on the ground that the alternative - life in prison without parole - is a more severe penalty than death. This argument for the death penalty has been rejected by the following reductio ad absurdum argument. It reduces the premise that life in prison without parole is more sever than the death penalty to an absurdity. Abbreviating somewhat, the argument is as follows:

P1. Life is a more severe penalty than death

In doubt

P2. Lesser crimes should receive less severe penalties

Obviously true

P3. 2nd degree murder is a lesser crime than 1st degree murder.

Obviously true

C. Life for 1st degree murder & death for 2nd degree murder.

Absurd

P1 is the key premise in the argument for the death penalty. By showing that it leads to an absurdity in a valid argument, it is shown that the premise must be rejected and so also the argument for the death penalty.

A reductio argument is evaluated by asking: 1) does the premise in doubt really imply the absurdity, i.e. is the reductio argument valid; 2) is the conclusion really absurd; 3) can the premise in doubt be altered in a minor way so it does not imply the absurdity? Which of these approaches would be the best response by an advocate of the death penalty to the reductio argument?

Examples of reductio ad absurdum arguments can be found in the dialogues of Plato. Socrates asks a question and the proceeds to refute the answer by showing that it leads to a clearly false conclusion.

"Well said Cephalus, I replied, but as concerns your answer that justice is speaking the truth and keeping promises, are there not exceptions? Suppose that a friend when in his right mind has deposited weapons with me and he asks for them when he is not in his right mind, ought I to give them back to him? No one would say that I should or that I should be right in doing so, no more than they would say that I ought to always speak the truth to one in his condition."

"You are quite right he replied."

"But then, speaking the truth and keeping promises is not a correct account of justice."

The structure of the argument is as follows:

P1. It is just to tell the truth and keep promises

In Doubt

P2. A madman asks for the return of weapons I have promised to return.

True

C. I should return the weapons to him or tell him where they are.

Absurd

Since the conclusion is absurd, P1 cannot be a correct account of justice.

Philosophers, and other thinkers, frequently use the method of reductio ad absurdum. A student of philosophy can use them to assess the views of the philosopher he or she is reading. There is no mechanical way to generate reductio arguments. You must be imaginative and sometimes have knowledge about the subject matter of the view you wish to challenge. If you are not able to think of a reductio argument, that does not entail that the premises of the argument under consideration are true; it may be that you are not knowledgeable enough or clever enough.

 

4. Examples

Example a.

Many of the examples above are simple and not what you would encounter in real discourse of everyday and in the arguments of philosophers. They were used to most clearly show the validity of the argument forms. Here's an example from philosophy.

If we can cause animals to suffer, then what we do to them not only hurts them, it can harm them: and if it can harm them, then it can detract from the experiential quality of their life, considered over time; and if it can do that, then we must view these animals as retaining their identify over time and as having a good or ill of their own.

Tom Regan, The Case for Animal Rights

The first step in reconstructing this argument into its logical form is to label or pull-out the statements that can be the premises and conclusion of the argument. In this process we can remove what is redundant and those words not necessary to the structure of the argument.

(1) If we cause suffering to animals then we harm them.

(2) If we harm animals then we detract from the experiential quality of their life

(3) If we detract from the experiential quality of their life, then animals must be viewed as having an identity and a good or ill of their own.

Now that we have simplified the argument into these three statements, we can see that this is an extended version of Hypothetical Argument, without the obvious conclusion . It is an extended version of Hypothetical Argument because it has three premises rather than two. Abbreviating in order to see the structure:

Standard Form

Logical Form

P1. If cause suffering then cause harm

P1. If p then q

P2. If cause harm then detract from quality of life

P2. If q then r

P3. If detract from quality of life then an identity

P3. If r then s

The obvious conclusion from these three premises is:

C. If cause suffering then an identity

C. If p then s

It may be that Regan only means to present this hypothetical argument, but since we know that the title of the book is The Case for Animal Rights, it is reasonable to draw the conclusion that Regan will reach, namely that animals have an identity over time and a good or ill of their own. This is "s" in the formal reconstruction. To reach this conclusion we only need add the premise that we can cause animals suffering. The logical form of the argument is:

P1. If p then q

P2. If q then r

P3. If r then s

P4. p

C. s

This is an extended version of Chain Argument; it has three "if... then..." premises rather than two.

 

Example b.

Ralph will become a better student. He is studying logic and anyone who studies logic will be a better student.

Standard From

Predicate Instantiation

P1. Anyone who studies logic will become a better student

P1. All A are B

P2. Ralph is studying logic

P2. x is an A

C. Ralph will become a better student.

C. x is a B

Notice that "anyone" is being used in the same way as "all." The premise could be written, "All those who study logic will become a better student." The word "every" can function in the same way, e.g. "Every student who studies logic will become a better student."

Example c.

Bloogs does not wish to be an accountant for if he wished to be an accountant he would be enrolled in the Business College, and he is not.

Standard Form

Denying the Consequent

P1. If Bloogs wished to be an accountant, then he would be enrolled in the Business College.

P1. If p then q

P2. Bloogs is not enrolled in the Business College

P2. not q

C. Bloogs does not wish to be an accountant

C. not p

Example d.

Look Bloogs, can't you see that this sample is verigated? Let me convince you. All the pirons we have found so far have been elactic. And all the elactic samples we have are verigated. This sample is a piron, so it has to be verigated.

The conclusion of this argument is "This sample is verigated." The argument uses "all," so the approach to reconstructing the argument is to use the two predicate logic argument forms.

Standard Form

Logical Form

P1. All pirons so far are elactic.

P1. All P are E

P2. All elactics are verigated

P2. All E are V

P3. This sample is a piron

P3. x is a P

C. This sample is verigated

C. x is a V

This argument does not fit one of the two valid argument forms in predicate logic. It is a combination of Universal Syllogism and Predicate Instantiation.

P1. All pirons are elactics

P2. All elactics are verigated

C. All pirons are verigated

P3. This sample is a piron

C. This sample is verigated

Only the most simple arguments in ordinary language can be reconstructed as one of the basic valid argument forms listed above. More often, arguments will have to reconstructed as a combination of basic valid argument forms.

Example e.

The universal right to health care will be enacted if those who have adequate health care vote for liberal democrats. They will vote for liberal democrats if they are concerned about the lack of access to health care of those who are poor and do not have adequate health care. Those who have adequate health care are concerned about those who do not. So, the universal right to health care will be enacted.

Standard From

P1. If those who have adequate health care vote for liberal democrats then universal right to health care will be enacted.

P2. If those who have adequate health care are concerned about the lack of access to health care of those who are poor and do not have adequate health care then they will vote for liberal democrats.

P3. Those who have adequate health care are concerned about those who are poor and do not have adequate health care.

C. Universal health care will be enacted

Notice that in the original statement of the argument, the "if" clause of the conditional statements came after the "then" clause. This is common in ordinary discourse. The order is reversed in the standard form. If we rewrite the standard from using the notation of propositional logic, we have the following:

P1. If p then q

P2. If r then p

P3. r

C. q

This is an instance of the valid argument form Chain Argument although the order of the first two premises is not the same. To clearly see this form we can rewrite the argument form as follows:

P2. If r then p

P1. If p then q

P3. r

C. q

 

5. Soundness

So far we have been concerned with the validity of arguments; we have been answering the first question about arguments, viz. whether the premises provide appropriate support for the conclusion. Seven deductively valid argument forms have been presented. The second question about arguments is whether the premises are true. This is just as important as whether an argument is logically correct. If an argument is valid, and its premises are false, then the argument for the conclusion is not persuasive, for validity of an argument only tells us that IF the premises are true the conclusion is guaranteed to be true. If the premises are false, the conclusion of the valid argument may in fact be true, but the valid argument hasn't shown this. Consider the following argument:

P1. If Detroit is on the east coast then is it in Michigan

P2. Detroit is on the east coast.

C. Detroit is in Michigan

This is a valid argument, an instance of Affirming the Antecedent. The conclusion is true, and both premises are false. If the geography of the U.S. were different than it is, then the premises could be true and would thereby guarantee the truth of the conclusion.

An argument that is valid and has true premises is a sound argument.

Validity + true premises = soundness

Both conditions are required for soundness. An argument with true premises that is not valid is not a sound argument. Assume that Bill lives in Michigan

P1. If Bill lives in Michigan then he lives in the mid west

P2. Bill lives in the mid west

C. Bill lives in Michigan

The premises and conclusion of this argument are true, but it is not a valid argument and hence it is not a sound argument. The argument form is not Affirming the Antecedent nor is it Denying the Consequent. Rather it is Affirming the Consequent.

P1. If p then q

P2. q

C. p

The invalidity of Affirming the Consequent can be shown as follows: in a valid argument the truth of the premises guarantees the truth of the conclusion; so if an argument form can be constructed with true premises and a false conclusion it cannot be valid.

P1. If Brenda's large ring is a real diamond then it is valuable

P2. Brenda's ring is valuable

C. Brenda's ring is a large real diamond.

This argument is not valid because the premises can be true and the conclusion false. P1. is true; large real diamonds are valuable. Suppose Brenda's ring is a real ruby, then it is valuable and hence P2 is true. But if her ring is a real ruby, then the conclusion is false. So this is not a valid argument, and therefore it is not sound, even though the premises are true.

 6. Determining Truth of Premises.

In the examples above the premises used were known true or false by common knowledge or by stipulation. This will not be so in the case of arguments commonly encountered in the real world.

To determine the truth or falsity of premises, we have to know relevant information about the subject matter in the premises. This is unlike validity. To determine validity, we have to be competent speakers of the natural language used in the argument so we can sort out premises and conclusions, and we have to know deductively valid forms of argument. Whether the premises are true is irrelevant to validity.

The truth or falsity of premises is not a subject matter of logic, but of all the other areas of human knowledge and inquiry. Determining whether a valid argument about environmental hazards is also sound, i.e. its premises are true, depends on our knowledge of environmental science. However, there are some matters of logic that clarify inquiry into the truth of premises.

Propositions, i.e. declarative statements, can be divided into three types: empirical, normative and conceptual. Empirical statements are statements of fact; they say something about how the world is. The following are empirical statements:

1. The mean distance from the earth to the moon is 238,866 miles.

2. Humans are descended from non-human primates

3. The Amazon River is the largest river in the world.

4. The disagreement over the morality of slavery was a cause of the Civil War.

5. The rate of acceleration due to gravity on earth is 32 feet per second per second.

6. The costs of health care are increasing faster than the rate of inflation.

Normative statements are statements about how things ought to be in the world, or about how things in the world are good, bad, right, wrong, evil, our duty, our right. Examples are:

1. Killing an innocent human being is immoral.

2. Love thy neighbor.

3. Every American should have access to adequate health care.

4. We must stop polluting the environment with toxic waste from industrial plants.

5. We have a duty to tell the truth even when doing so would be to our disadvantage.

6. Physician assisted suicide ought to be legally permissible.

Conceptual statements are statements about what the concepts expressed by words mean. Examples are:

1. A bachelor is an unmarried male.

2. An electron is a negatively charged particle circling the nucleus of an atom.

3. A legal right is an enforceable claim that a person may do or not do some act without interference from others.

4. First degree murder in the law is the killing of one person by another with the premeditated intent to kill.

5. A bicep is the contractor muscle in the upper arm.

6. A touchdown is scored whenever an offensive player has possession of the football in the end zone.

When these kinds of statements appear in argument we determine their truth in different ways. The truth of conceptual statements can often be determined by our personal knowledge of how we use the terms. In the event that we are not sure of how terms are used, we can consult a dictionary. In most cases a standard collegiate dictionary will suffice, but sometimes the terms are from a technical field and do not appear in a standard dictionary, or the author is using the term in a technical way that does not match the definition in a dictionary. Or, the dictionary may give us more than one definition. In that case we will have to discern what an author means by a term by noting how it is used, unless of course the author provides a definition of the term. One of the things to watch for in an argument is clarity of use of terms and consistent use of terms in the premises and conclusion.

The truth or falsity of empirical statements is determined in the first place by observation. The distance of the moon from the earth has been accurately determined by the observations of astronomers. The increase in the cost of health care is determined by economists and others carefully tracking the records of cost of various health care procedures over a period of several years. We commonly know the truth of empirical statements without making our own observations. For example, we know how far it is from New York to Los Angeles not because we have made our own measurements, but because we can look it up somewhere or ask someone we believe has more knowledge then we do. Many of our most important beliefs about the world are like this. None of us can make direct observations to confirm all the beliefs we rely on in our life. We have to trust the expertise and honesty of others. If the premises in an argument are not common knowledge, we ought to be sure that the sources of the facts stated are experts, honest and not biased. If we cannot determine that immediately, we have to set aside accepting or rejecting the conclusion of a valid argument until we are able to ascertain this.

Students frequently hold the view that normative statements are not true or false but simply a matter of opinion. They sometimes add that this means that one person's view is just as "true" as any others'. There is a correct intuition behind this, namely that normative statements cannot be determined true or false by observation, i.e. they are not empirical statements. Many philosophers share the view that we cannot say that normative statements are true or false. But this does not imply that one normative statement is just as acceptable as its contrary. One example is the statement that "It is wrong to kill an innocent human being." Would anyone of us seriously entertain the view that it is morally permissible to kill an innocent human being? The view that all normative statements are equally acceptable may stem from an unbalanced diet of examples. The morality of abortion is controversial, so also the question of whether the death penalty is a just form of punishment. But these controversial cases should not lead us to regard all normative statements as controversial and the debates about them irreconcilable. If an argument has normative premises, we need to ask ourselves whether they are acceptable. If we find the normative premises acceptable and the empirical premises true, and the argument is valid, then we must accept the conclusion.

Arguments with normative conclusions frequently do not state a normative premise. An earlier example was:

Given that many persons are sentenced to death due to mistakes or careless work by police or prosecutors, the death penalty should be abolished.

This argument was constructed into premises and conclusion are follows:

P1. Many persons are sentenced to death due to mistake or careless work by police or prosecutors.

P2. It is wrong for persons who do not receive a proper trial to be found guilty and be sentenced to death.

C. The death penalty should be abolished

P2. is an implicit premise. Without it the argument is not valid. P1. is an empirical statement and C. is a normative statement. Validity is sometimes characterized as an argument where the conclusion is contained in the premises, or it is said that there can't be anything in the conclusion that is not already contained in the premises. So, for C to be a valid conclusion, there must be a normative statement in the premises. P1 states a matter of fact; assuming it is true, it does not alone support the conclusion. It is good reason for abolishing the death penalty, only when we also state that what it describes ought not to happen, that it is wrong that it happens. Philosophers summarize this by stating that you cannot infer a normative conclusion from only empirical premises.

7. Counter Examples

One way to cast doubt on the truth of a premises is to find a counter example. This technique is used to show that one or more of an arguments premises are false. It is most effective against premises that make universal statements.

P1. All poodles are white or black

P2. Sara's dog is a poodle

C. Sara's dog is white or black.

This is a valid argument; if the premises are true the conclusion must be true. But P1 can be shown false by pointing out a poodle that is brown. You may be able to do this because you or someone you know has a brown poodle. Whenever there is a universal claim in the premises of an argument, trying to think of a counter example is a good way to cast doubt on its truth.

An example from philosophy is St. Augustine's refutation of the claim of astrology that a person's future can be predicted by the position of the stars at the time of the person's birth. This view is committed to the following:

1. The positions of the stars at the time of a person's birth can be used to predict the person's future.

What follows from this is:

2. If two persons are born at the same time, then the predictions of their future will be the same.

Augustine observed that in the case of twins, they are born at the same time and under nearly identical circumstances. So, astrology would predict the same future for them. Augustine further noted that twins do not have the same future. The major events in their lives are not the same. The difference in the lives of twins is a counter example to the astrologers claim that the position of the stars at the time of birth can be used to predict a person's future.

The method of counter example can be used to show that the conclusion of a valid argument is false and therefore one or more premises must be false. For, if an argument is valid then the premises can't be true and the conclusion false. We can apply the astrologer's claim to a particular case.

P1. All persons born when the stars are in arrangement C will have futures F1

P2. Bill and Bob are twins both born when the stars were in arrangement C

C. Bill and Bob will both have futures F1.

Counter Example: Bill had future F1 and Bob had a different future, F2. This shows that C is false and therefore that either P1 or P2 is false. P2 is an easily confirmed fact, so P1 is the false premise.

Counter examples are effective against universal claims. So one way they can be avoided is to not make universal claims. An astrologer might modify P1 to something like, "It is highly likely that all persons born under the same arrangement of stars will have the same future, but they will not when their futures are influenced by things in the heavens we cannot know."