IV Non-Deductive Arguments

1. Distinguishing Deductive and Non-deductive Argument Forms

In the section on deductive arguments, the general concept of "logical correctness" was defined as follows:

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

Deductive logical correctness (validity) was defined as:

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

Not all arguments encountered in philosophy and other areas of inquiry can be formulated as deductive arguments. Non-deductive arguments are those argument forms in which the truth of the premises does not guarantee the truth of the conclusion, and yet they can be strong arguments. Compare the following two arguments:

P1. All freshmen are between 18 and 22 years of age

P2. John is a freshman

C1. John is between 18 and 22 years of age

P3 90% of freshmen surveyed have been between 18 and 22 years of age

P4. John is a freshman

C2. John is between 18 and 22 years of age.

The first argument is deductively valid, but we would certainly question the truth of P1. So we would be reluctant to consider it sound. The second argument is a more plausible way of constructing what a speaker would report. It is a strong argument, but the truth of the premises does not guarantee the truth of the conclusion. We are aware of freshman who are younger than 18 and older than 22.

Deductive validity is only one criterion for the logical correctness of arguments. When an argument does not fit a deductively valid form then the criterion for logical correctness is:

If the premises were true, the conclusion is likely to be true.

This is a matter of degree. In the second argument if only 10 freshmen at a large college were surveyed and 9 of them are 18 -22 the argument is not strong. But, if 500 freshmen were surveyed and 450 were 18 -22, this is a stronger argument. Yet it is possible that the premise is true and the conclusion false.

Common types of non-deductive arguments are inductive arguments (both general to specific and specific to general), analogical arguments, and explanations. NOTE. Do not make the mistake that deductive arguments are general to specific and inductive arguments specific to general. The arguments examined above are inductive arguments and they go from general premises to specific conclusions. Universal Generalization is a deductively valid argument form and it goes from general to general. A deductively valid argument can go from specific premises to a specific conclusion. For example:

The man who shot the duke in 1923 was killed later that year. Kraznakov was alive in 1924. Therefore, Kraznakov is not the man who shot the duke.

2. Specific to General Arguments.

Arguments of this sort are commonly called empirical generalizations because they start with premises reporting specific observations of the world and infer a general statement about the world, i.e. from a sample of a population to the entire population. The schematic form of empirical generalization is:

P1. X% of a sample of F are G

C. Most likely, X% of all F are G


P1. 33% of Midvale College students surveyed said they are Republicans

C. 33% of Midvale College students are Republicans


P1. 51% of a sample of registered voters said that they will vote for Senator Bloogs.

C1. 51% of all registered voters will vote for Bloogs.

C2. Bloogs will most likely win the election.

These arguments do not guarantee the truth of their conclusions when the premises are true. So, the criteria for logical correctness must be different than that for deductive arguments. The strong making characteristics of empirical generalizations are: 1) the evidence in the premises are true; 2) the sample size is large enough; 3) the sample is representative; 4) no counter evidence to the conclusion .

A person reading or hearing the argument about Republicans at Midvale College may doubt the truth of the premises because she does a quick survey of students and finds that 20% say they are Republicans. This would also be counter evidence to the conclusion. Neither of the examples above state how many individuals out of the total population were sampled. If Midvale has 5000 students and only 9 were surveyed, that would not be as strong evidence for the conclusion as sampling 900 students. Nor does either example tell us how the sample was selected. If the students selected were only those living on campus and excluded those living off campus, the sample would not be representative of the entire Midvale College population. If the voters surveyed were those living in one part of the state, then they may not be representative of the entire voting population. In order to avoid a non-representative sample, survey researchers commonly select the sample at random so that each member of the total population has an equal probability of being selected.


3. General to Specific Arguments

These arguments will have statistical premises (statements with "most", "many" "few", "a certain percentage.") As in the case of specific to general arguments, counter evidence to the premises or conclusion will weaken the argument. Another method of evaluating the strength of these arguments depends on background knowledge and knowledge of the specifics in the argument. Two general to specific arguments can have true premises and yield incompatible conclusions.


P1. Men who eat a diet high in fat are a high risk for a heart attack.

P2. Jones eats a diet high in fat.

C1. Jones is at high risk for a heart attack

P3 Men who are not overweight, who do not smoke and who exercise regularly are at low risk for a heart attack.

P4. Jones is not overweight, does not smoke and exercises regularly

C2. Jones is at low risk for a heart attack.


The problem with both arguments is that neither of them consider all of the factors relevant to a heart attack. A stronger argument would be one that considers all four of the above factors and compares the rate of heart attacks for those men who have none of the four factors with those who have one, two, three and four of the factors. If Jones has two of the risk factors, then the important information for Jones would be his comparison to those who have one and those who have none of the risk factors.

4. Analogical Arguments

 This is a common type of non-deductive argument. Two things are analogous if they share one or more properties, i.e. they are similar in some respects. Hockey and soccer are analogous because in both a player has to put the ball into a net to score. An argument by analogy is an argument that because two or more things share a specific set of properties they will also share a further property. The general schema for arguments by analogy is:

P1. x has properties A, B, and C

P2. y has properties A, B and C

P3. x has property D

C. y has property D

Arguments by analogy are not limited to comparing single entities.

P1. t, v and x have properties A, B and C

P2. y has properties A, B, and C

P3. t, v, and x have the property D

C. y has property D

Arguments by analogy, like empirical generalizations and general to specific arguments, do not guarantee the truth of the conclusion when the premises are true. They are strong or weak depending on features of the argument.


A weak analogical argument:

P1. Bob has red hair, is from San Francisco and majors in Philosophy

P2. Bill has red hair and is from San Francisco

C. Bill is a major in Philosophy

A criterion for a strong analogical argument is that the similarities are relevant to the inferred property. In this case hair color and city of origin are not relevant to what a student's major is. A stronger argument would be:

P1. Bob is reflective, likes to discuss deep subjects , reads a lot and is a Philosophy major

P2. Bill is reflective, likes to discuss deep subjects, and reads a lot

C. Bill is a major in Philosophy

The premises of this argument could be true and the conclusion false, but the premises provide more support for the conclusion than those in the previous argument because the properties Bob and Bill share are more relevant to being interested in Philosophy.

 Another criterion for strong analogical arguments is that there are no relevant dissimilarities. If Bob is tall and Bill is short, that dissimilarity does not weaken the analogical argument. However, if Bill's parents are physicians and Bill does volunteer work in hospitals and takes biological science courses, while Bob does not share any of these characteristics, then this dissimilarity weakens the argument. Note again, that to make these evaluations of the logical strength of analogical arguments we have to have knowledge of the subject matter of the argument. This is unlike deductive arguments where we can determine validity simply by the structure of the argument. Evaluating the above arguments we had to appeal to our beliefs about sorts of activities a Philosophy major is likely to engage in, and we assumed some particular facts about Bob and Bill to show a dissimilarity that weakened the argument. If the subject matter of an analogical argument is in a area about which we are ignorant, we will not be able to do much to evaluate the strength of the argument. If we are not familiar with the persons or objects specified in the argument, we will not be able to find specific dissimilarities. However, we will be able to say in some cases that if the objects in question were dissimilar in a particular relevant respect, that this would weaken the argument.


P1. Jones is a Republican, owns a small business, and supports tax cuts

P2. Smith is a Republican and owns a small business

C. Most likely, Smith will support a tax cut.

Based on our general knowledge we can say that the similarities between Jones and Smith are relevant to support of tax breaks. We don't know Jones and Smith so we can't identify a relevant dissimilarity. However, we may believe that a person's age and wealth are relevant to support for tax cuts, e.g. a person who is a senior citizen and not wealthy is less likely to support tax breaks, If that is so, and if we know that Jones is rich and young and Smith is old and not rich, then that would be a relevant dissimilarity.


In the nineteenth century it was frequently argued by analogy that there must be life on Mars.

The planet Mars posses an atmosphere with clouds and mists resembling our own; it has seas distinguished from the land by a greenish color, and polar regions covered with snow. the red color of the planet seems to be due to the atmosphere, like the red color of our sunrises and sunsets. So much is similar in the surface of Mars and the surface of the earth, that we readily agree that there must be inhabitants there as well.

Prior to twentieth century astronomical instruments and exploration of Mars via satellites and landings, knowledge of Mars was limited to what could be seen by the naked eye and by telescope. So this argument was persuasive. Today we know that the atmosphere of Mars will not support inhabitants that are any thing like us. We have discovered relevant dissimilarities.

5. Non Argument Uses of Analogy.

The best way to explain something to a person who has yet to understand it is to make a comparison with something that the person is familiar with. Explanatory analogies are frequently used to explain science to lay persons.

Carbon is gregarious stuff; the carbon atom has an outer shell with four electrons available for making shared electron pairs - or covalent bonds - four hands so to speak, to clasp its neighbor - where oxygen, say, has but two and hydrogen only one.

It it not always easy to determine whether an analogy is an explanation or an argument.

For the average user, trying to understand the workings of a computer is like trying to understand what your nerves and muscles are doing while you run.

This could be seen as an explanation of why it is difficult, if not impossible, to use a computer and simultaneously try to understand its workings, or taken as an argument that just as it is ridiculous for us to try to understand what our nerves and muscles are doing while we are running, it is ridiculous for us to try to understand what a computer is doing while we use it.

 6. Explanation and Abduction

An argument is an attempt to justify  a statement, i.e. show that it is true. Suppose Jones says to Smith that Smith was late for work, and Smith protests that he was not. Jones might then say, "I was in my office at 8:00 and didn't see you in your office, further, when I looked out my window at 8:30 I saw you coming in the door." Jones is providing reasons why it is true that Smith was late for work. Jones is presenting premises to support the conclusion that Smith was late.

P1. Smith was not in his office at 8:00

P2. Smith came in the door at 8:30

P3. Starting time is 8:00

C. Smith was late

Now suppose that Smith says. "OK. I was late, but it wasn't my fault. There was an accident on Vine Street and it held up traffic for 20 minutes." Smith is not giving an argument to prove that he was not late, he admits that he was. He is now giving an an explanation of why he was late.

Smith's explanation has the following form:

P4. There was an accident on Vine Street

P5. Traffic was not moving

P6. I was held up for 20 minutes

C. I was late for work

Both inferences support the truth of the same conclusion, viz, that Smith was late for work. There is an important difference: Jones's argument is an inference from premises to conclusion. Jones's argument is designed to prove a matter on which he and Smith disagreed. When Smith's finally agrees that he was late he is providing an explanation of a matter not in dispute. An explanation begins with a statement known to be true, and provides statements to show why is it true. To give an explanation is to reason from the fact to be explained to some statements that provide the explanation. Jones' initial argument is a deduction; it reasons from premises to a conclusion. Smith's explanation is an abduction, it reasons from the conclusion to the premises. It is called an abduction because it is reasoning "up" to premises rather than "down" to a conclusion.

There can be alternative explanations for a particular fact. Suppose again that Smith is late for work. Jones wonders why, for Smith is nearly always on time. Jones could come up with several explanations: a traffic accident held him up; his car broke down; his alarm clock failed; he is sick, etc. Each one of these explanations can be constructed as an argument in which the premises support the conclusion. This shows that the logical strength of the argument from the statements that explain to the statement of the fact that needs explaining is not the only criterion for a good explanation.

Another criterion is that the statements offered as an explanation are true. Jones could call the police and find out if there was an accident on the streets Smith drives to work. If there was not, this explanation can be rejected. Jones might also cast doubt on the truth of this explanation by observing that other workers drive to work on the same route as Smith, and they weren't late. Another criterion for a good explanation is that is complete, i.e. that it explain all aspects of what needs explanation. Smith's explanation was that the accident held up traffic for 20 minutes and suppose that this confirmed by the police. This explains why Smith was late, but it doesn't explain why he was 30 minutes late.

Explaining why things in the world are the way they are is one of the tasks of science. Scientists collect facts about the world; in addition they formulate general laws and theories to explain why things are the way they are. Suppose we put an ice cube in a glass and then fill the glass to the brim with water; the ice cube floats on the water and part of it will stick above the water level. When the ice melts, will the water level rise and overflow, will it remain the same, or will it go down? Suppose we try this a few times and each time the water level stays the same. Why does this happen. We want an explanation. The explanation is given by the law of buoyancy.

An object in water is buoyed up by a force equal to the weight of the water it displaces.

This implies that if we put an object in water it will sink until it displaces a volume of water that is equal in weight to the weight of the object. The ice cube is frozen water; when it melts it will fill in the volume it displaced and the level will remain the same because the volume of water in the glass does not increase.

The structure of a common type of scientific explanation is this:

General laws

Initial conditions

Fact explained

In the case of the ice cube the pattern is:

General law of buoyancy

Ice cube floating in a filled glass of water

The level of water remains the same when cube melts

Among the reasons this is a good explanation is that the general law of buoyancy can be used to predict other phenomena. It will explain why a one cubic foot block of Styrofoam floats higher in the water than a one cubic foot block of wood. One cubic foot of Styrofoam weighs one pound; one cubic foot of wood weighs, say, 25 pounds. The Styrofoam will sink until it displaces an amount of water equal to one pound. The wood block will sink until is displace an amount of water equal to 25 pounds. The volume of water for 25 pounds is greater than the volume for one pound, so the Styrofoam will float higher in the water.

Another criterion for a good explanation it that the statements that explain a fact do not imply something that is not true. For example, suppose a student receives a poor grade on an essay and the student knows that her view disagrees with that of the professor. The student explains the poor grade with the claim that the professor give poor grades to students who disagree with his view. The structure of the explanation is:

P1. If an essay disagrees with Professor Bloogs' views then he will give it a poor grade

P2. Susan's essay disagrees with Professor Bloogs' views

C. Susan received a poor grade

The explanation in P1 can be tested by predicting the grades of other students. If James' essay disagrees with Professor Bloogs views then it should have received a poor grade.

P1. If an essay disagrees with Professor Bloogs' views then he will give it a poor grade

P2. James' essay disagrees with Professor Bloogs' views

C. James' received a poor grade

Suppose James essay did disagree with Professor Bloogs' view and it received a good grade. In that case C, above is false, and so P1 must be false. Susan might also explain Professor Bloogs bias by claiming that if an essay agrees with his view it will receive a good grade. This predicts that if Bill received a good grade then his essay must have agreed with Professor Bloogs view. But if Bill has a poor grade and his essay agrees with Bloogs view, then the explanation of bias is not confirmed.

Suppose Simmons has a cold and explains this by stating that he went out of doors on a cold and rainy day without a coat. The general statement explaining Simmons' cold is that exposure to cold and rain causes colds. This explanation could be tested by seeing how well it predicts the occurrence of colds in those who go out in a cold rain without a coat. There is another way to evaluate the explanation - ask whether the general statement is consistent with other beliefs we have about colds. If we believe with modern medicine that a cold is an infection of the upper respiratory system, then we would say that a cold is the result of exposure to germs - to a bacteria or virus. The claim that colds are caused by exposure to cold rain is not consistent with the claim that colds are caused by exposure to germs. We have to give up one or the other of these claims, or we have to combine the two explanations in a consistent way. We might say that germs are necessary for the occurrence of an upper respiratory infection and add that exposure to a cold rain weakens the immune system and makes it more likely that exposure to germs will lead to a cold.

In summary, the criteria for good explanation are:

1) logical strength, i.e. the explanatory statements must strongly support the statement of the fact to be explained;

2) truth of the explanatory statements;

3) breadth of the explanation - it can explain other related phenomena

4) the explanatory statements predictions are not disconfirmed and there are confirming instances beyond the statement initially calling for explanation;

5) the explanatory statements are consistent with our well founded beliefs.


The examples of explanation used so far have been from everyday life and from science. Philosophers also use explanation. It is often claimed that a student of philosophy must identify and evaluate the arguments of philosophy, i.e. find the premises and conclusions and determine if the arguments hare logically strong and if the premises are true. The methods of deductive arguments are then the proper approach to philosophy texts. Many philosophical arguments are best understood as deductive arguments. But philosophers also use abduction. The philosophical theses they put forward are premises in an explanation of a phenomena.


7. Abductive Reasoning in Ethics

In scientific explanation a premise is abduced to explain a known phenomena and to then predict other phenomena. Issues in ethics present themselves for justification. "Was it right for Jones to lie to Smith?" "Are experiments on non-human animals ethical?" "Should an American citizen have a goavernment protected right to health care?" Answers to these sorts of questions are frequently presented by arguing from cases where the answer is clear. This is sometimes interpreted as reasoning by analogy.

It is wrong for a doctor to lie to a person about a test result, even if the doctor thinks that lying is in the patient's best interest. We know this because even doctors would agree that it would be wrong for a financial adviser to lie to them about a potential investment, even if the financial adviser thinks that this lie is in the doctor's best interest.

The analogical interpretation would be:

P1. A financial adviser lies to a doctor when he think doing so is in the doctor's best interest.

P2. A doctor lies to a patient when she thinks doing so is in the patient's best interest .

P3. The financial adviser is morally wrong.

C. The doctor is morally wrong.


A better interpretation of the argument is that it abduces a general rule about lying from the example of the financial adviser and then uses this general rule to apply to the case of the doctor. The argument starts from the view that we would all agree that it is wrong for a financial adviser to lie to a doctor, or any customer, on the ground that the financial adviser believes lying is in the interest of the customer. There is an implicit general rule that supports this view - that professionals should tell clients the truth and allow them to determine what is in their best interest. Once this general rule is abduced from the case of the financial adviser, it can be applied to the case of the doctor and show that it is wrong for doctors to lie to patients.

C1. It is wrong for a financial adviser to lie to a client when he believes it is the best interest of the client

P1. Professionals should tell clients the truth and allow them to determine what is in their best interest.

C2. It is wrong for a doctor to lie to a patient when she believes it is in the best interest of the patient.

C1. is written above P1 to show that P1 is abduced from C1, and C2 is written below P1 to show that it is deduced from P1.

This is simple presentation of the issue of professionals lying to clients. Situations involving lying are usually more complicated. One can ask about the competence of the doctor's patient and of the adviser's client. If they are deluded in some significant way, they may choose to do something that they would not choose if they were competent. This does not totally undercut the general rule, rather it requires a more precise formulation.

Professionals should tell clients the truth and allow them to determine what is in their best interest, except when the client is not competent and because of this likely to choose a course of action that they would not choose if competent.

 There may be other circumstances where lying to a patient, or a client, is justifiable. Ethical analysis and argument frequently proceeds from the examination of many and divers cases in order to abduce the most plausible general rules of ethical conduct.

 Here is an abductive argument that is also a reductio ad absurdum argument.

A full grown cat is capable of much more than a human infant. The cat has a sense of its own interests, and is capable of receiving and displaying affection. Further, it engages in play and activities like hunting that show that its level of intelligence is as great, if not greater than, that of an infant. Hence, if we consider painful, destructive experiments performed on cats to be morally legitimate then we must also accept the legitimacy of similar experiments on human infants.


This argument starts from what the author believes is a widely accepted view - that it is morally permissible to do painful experiments on cats. He abduces a general rule from this and some facts about the behavior and capacities of cats, and then applies it to human infants.

P1. It is morally permissible to do painful experiments on beings who have a sense of their own interests, display affection, and have intelligence.

P2. An human infant is not more capable of these behaviors than a full grown cat.

C. It is morally permissible to do painful experiments on a human infant.

This conclusion is one that nearly everyone would reject as morally outrageous. Since the argument is valid, one or other of the premises must be rejected. The author believes that only P1 can be rejected since P2 is obviously true. But without P1, there is no justification for painful experiments on cats, and we should find it just as outrageous as painful experiments on human infants.

Note the similarity between the structure of this argument and of explanations in science. A particular statement of fact is explained by subsuming it under a general law which explains the statement. The explanation can be rejected if we can deduce from it a prediction that turns out false. In this ethical argument the practice of painful experiments on cats was subsumed under a general ethical rule which justifies the practice. The general rule was rejected by showing that one can deduce from it a statement of what is permissible that we would all reject.

Much more could be said about this argument. For example, there may be alternative general rules that justify painful experiments on cats that would not justify painful experiments on human infants.