Validity, Soundness, Sufficiency, and Inductive vs Deductive Arguments

Overview

We evaluate arguments for in terms of two features: (a) the contents of the argument (i.e., the premises) and the (b) the logical relationship between the premises and the conclusion. In the last few lessons, we focused on evaluating the content. We did this by evaluating premises for acceptability. In this next section we’re going to focus on the second criteria, i.e., the logical relationship between the premises and conclusion. We’re also going to look at two different categories of argument: deductive and inductive. It’s important to know the difference because each requires a different method of evaluation.

Validity and Soundness

Very generally, logical force is the strength of the logical relationship between the premises and the conclusion. To understand different kinds of logical force let’s begin by looking at on example of the strongest kind: validity.

Argument 1:
(P1). All philosophers like beer.
(P2). Ami is a philosopher.
(C). Therefore, Ami likes beer.

Ask yourself: If we assume that all the premises are true, is it possible for the conclusion to be false? The answer should be, no. If we assume that the premises are true, then the conclusion must be true. When premises are assumed to be true and logically “force” the conclusion to be true we say the argument is valid.

It’s important to notice that in critical thinking and logic, “valid” is a technical term. In everyday use, when people say “valid” they typically mean something like “true”. However, if you use valid this way around a philosopher they will give you a dirty look and maybe even secretly think bad things about you. Philosophers are very picky about how words are used. In this course, we are going to use ‘valid’ like philosophers. ‘Valid’ refers to the maximum possible logical force between premises and conclusion.

Here’s our technical definition:

An argument is valid if and only if if the premises are assumed to be true it’s logically impossible for the conclusion to be false.

We can state the same idea another way:

An argument is valid if and only if if the premises are assumed to be true then the conclusion must also be true.

It doesn’t matter which you use since they mean the same thing. Pick the one that you remember most easily. Also, you’ll notice one important qualification: When assessing validity we assume premises to be true whether they really are or not. This point is sometimes hard to grasp at first so let’s illustrate with an example:

Argument 2:
(P1). All philosophers can fly.
(P2). Ami is a philosopher.
(C). Therefore, Ami can fly.

Is this argument valid or invalid?

The answer is that it’s valid. Buy why? It has a false premise: It’s not true that all philosophers can fly (only some can!). This is where that little qualification “assumed to be true” comes in. Logical force/validity has absolutely nothing to do with the truth or acceptability of the premises. It’s only about the logical relationship between premises and conclusion. So, when we evaluate an argument for validity, we don’t care whether its premises are true or false. We first assume that they are true (regardless of whether they actually are) and then evaluate for validity.

If all of an argument’s premises turn out to also be true (not just assumed true) and the argument is valid, we say the argument is sound. Validity only concerns logical structure of arguments, soundness concerns premise acceptability of valid arguments. We don’t care about soundness for invalid arguments. Arguments can’t be invalid but sound. There’s another name for that feature of arguments which we’ll explore in the next section.

Here’s another way to think about it. Think of validity as the shape of the argument and soundness (i.e., premise acceptability) as the color. Whether something is a triangle or a square has absolutely nothing to do with its color. They are entirely different qualities. Similarly, when you evaluate an argument for validity don’t worry about whether the premises are true. To do so would be like deciding whether something is a square or a triangle based on its color. That would make no sense. Just assume that the premises are true and evaluate whether the premises ‘force’ the conclusion to be true. After you’ve determined validity then you can examine the argument for soundness.

Here’s a chart to show you the possible combinations:

Logical force Premise Acceptability Assessment
Valid All acceptable Sound
Valid One or more not acceptable Valid but not sound
Invalid All acceptable Invalid
Invalid One or more not acceptable Invalid

Let’s do a quick test to see how we’re doing. Decide whether the argument is valid or invalid and sound or not sound:

Argument 3:
(P1). If a ninja judo chops you, you’ll die.
(P2). If you die you’ll turn into a unicorn.
(P3). You’re not a unicorn.
(C). Therefore, no ninjas judo-chopped you.

Argument 4:
(P1). Yesterday every raven I saw was black.
(P2). The day before that every raven I saw was black.
(P3). Two days ago every raven I saw was black.
.
.
.
(Pn). Every raven I’ve ever observed is black
(C). Therefore, the next raven I observe will be black.

Argument 3 is valid but not sound. Why? First of all the argument isn’t sound because some of the premises aren’t acceptable. However, even though you might think all the premises are in the real world false, if we assume them to be true the conclusion must also be true. If you’re not a unicorn then it also means you didn’t die. And if you didn’t die it also means you didn’t get judo-chopped by a ninja. The important lesson here is that an argument can be valid even though all its premises are false. Such argument are valid but not sound.

Argument 4 is invalid. Even if it’s true that every raven you’ve ever observed was black, it’s not logically necessary that the next one you see will also be black. It might be pink or white. Of course it’s unlikely that the next one you see is pink or white but it’s logically possible. And so, we say the argument is invalid because even though it’s very likely that the next raven you see will be black, it’s not logically necessary. Don’t confuse “very likely” with “logically necessary” (i.e., validity). Regarding soundness, we don’t give an evaluation because we only care about soundness when we’re talking about valid arguments. Since we’ve determined the argument to be invalid, we don’t worry about soundness.

Argument 4 leads us to our next topic:

Inductive vs Deductive Arguments

We need to distinguish between two kinds of arguments. Let’s look at arguments 2 and 4 again:

Argument 2:
(P1). All philosophers can fly.
(P2). Ami is a philosopher.
(C). Therefore, Ami can fly.

Argument 4*:
(P1). Yesterday every raven I saw was black.
(P2). The day before that every raven I saw was black.
(P3). Two days ago every raven I saw was black.
.
.
.
(Pn). Every raven I or any other person has ever observed is black
(C). Therefore, the next raven I observe will be black.

Argument 2 is called a deductive argument. A deductive argument is one in which if we assume the premises to be true then the conclusion must also be true. Otherwise stated, if we assume the premises to be true, it’s impossible for the conclusion to be false. A good deductive argument is a valid argument.

Argument 4* is an inductive argument. An inductive argument is one in which if we assume the premises to be true it’s possible for the conclusion to be false. Inductive arguments are probabilistic arguments. Even though every raven I’ve ever observed has been black, it’s still logically possible that the next one I see is some other color. By definition, inductive arguments aren’t valid but we still assess their logical strength. Our method is similar to assessing validity for deductive arguments. A strong inductive argument will mean that, if we assume the premises to be true, it’s very probable that the conclusion is true (or improbable that it’s false). Argument 4* is a strong inductive argument. A weak inductive argument will mean that, even if we assume the premises to be true, the conclusion isn’t likely to be true.

As with deductive arguments we can also evaluate the acceptability of an inductive argument’s premises. With deductive arguments we used the concept of ‘soundness’ to describe premise acceptability of valid arguments. If we’ve determined that an inductive argument is strong and all its premises are acceptable we say that argument is cogent. An uncogent inductive argument is one that where one or more premises are false or the logical connection between premises and conclusion is weak.

How do I know if an argument is deductive or inductive?
To figure out whether an argument is deductive or inductive we use the concept of sufficiency. An argument’s premises are sufficient if, assuming the premises to be true, the conclusion is also guaranteed to be true. It this sounds a lot like validity, it’s because it is!

To determine whether an argument is deductive or inductive think of sufficiency as “enoughness”. Ask yourself: Are the premises (when assumed true) enough to logically guarantee that the conclusion is also true? An argument whose premises are sufficient is a valid deductive argument. That is, they are enough on their own to guarantee that the conclusion will also be true. If not, then the argument is either an invalid deductive argument or an inductive argument.

At this point, it’s important to note that just because an argument is inductive (i.e., not valid) it doesn’t follow that it’s not a good argument. For example argument 4* is a good argument. A good inductive argument will be cogent; that is, the premises will give strong logical support to the conclusion and the premises will be acceptable.

Some argument forms are by definition inductive and the entire second half of this course is dedicated to evaluating the various forms. For now, all you need to know is that inductive argument doesn’t mean bad argument. Inductive arguments are probabilistic arguments. Some will make the probability of the conclusion very high and others will make it very low.

Science and Inductive Arguments

Science makes generalizations about how the world is. Scientists draw conclusions about the world based on observations. For example, a scientist might conclude that medicine X causes Y effect based on observing a series of particular cases. This means that arguments for scientific generalizations about the world will never be valid. That is, they will always be inductive. You can never make a valid argument for All Xs cause Y or All Xs are Y. Can you figure out why?

Argument 4 is a common illustration of why scientific claims are always inductive. Let’s see why.

Suppose you’re an ornithologist (bird scientist). You want to prove that all ravens are black. So you go about your life observing ravens. Every single time you’ve ever observed a raven it was black. Can you conclude that necessarily all ravens are black? Think about all the places where ravens could exist. Have you been to all those places? No. If you’ve never left your country, it’s possible that there are pink ravens in some other country. You just didn’t see them because you never went there.

“Ah ha!” you say, “I have ornithologist friends in every country and they told me they’ve only seen black ravens. Therefore, I can conclude that all ravens are black.” Let’s assume that between you and your friends, no raven has escaped observation and that they are all black. Can you now conclude that it’s a scientific law that all ravens are black?

The answer is no because it’s still possible that in the future, perhaps long after you and your friends are gone, someone will observe a pink raven. There’s always a logical possibility that a pink raven will exist no matter how many black ravens have been observed in the past or in the present. This is why scientific generalizations about the world are always probabilistic. The more observations you make, the more likely it is that the generalization is true. However, since it’s impossible to observe every single instance past, present, and future, and there’s always the possibility of counterexample. In the raven example, I can’t observe every possible raven, much less future ravens, and so there’s always the possibility that my conclusion (all ravens are black) could turn out to be false. I just didn’t encounter that particular raven.

Does this mean we reject scientific generalizations about the world? Of course not. This knowledge merely reminds us that scientific generalizations are probabilistic. And the more instances we observe, the more likely the generalization is to be true. That is, the more observations of black ravens (and no non-black ravens) the greater the likelihood that “all ravens are black” is true. But it’s never 100% because there’s always the logical possibility of a future counterexample.

This fact about science explains, in part, why scientific conclusions change sometimes. Scientists encountered more examples, some of which didn’t conform with the original generalization.

How to Fool and Be Fooled: Science Denialism, Pseudo-Science, and Conspiracy Theories: So You’re Tellin’ Me There’s a Chance Fallacy. 

Suppose I show you a jar with 100 hundred marbles. 99 are blue and 1 is red. I blindfold myself, reach into the jar and make the following argument:

Argument 5:
(P1). 99 of the 100 marbles are blue.
(P2. 1 of marbles is red.
(C). Therefore, I’m going to pick a blue marble.

Are my premises sufficient for the conclusion? That is, do they guarantee that it will be true that I pick a blue marble? No. However, it’s a cogent argument. The premise strongly supports the conclusion and the premises are acceptable/true.

Suppose instead of Argument 5, before reaching into the jar, I made this argument:

Argument 6:
(P1). 99 of the 100 marbles are blue.
(P2). 1 is red.
(C). Therefore, I’m going to pick a red marble.

Are my premises sufficient for the conclusion? No. Notice that Argument 6’s premises aren’t sufficient either. Does the fact that neither argument is sufficient mean it’s equally reasonable to believe both conclusions? Of course not. The reasonable thing to believe is the conclusion with the highest probability of being true even though picking the blue marble could possibly turn out false.

This is almost too obvious to state but this is exactly were science denialism, pseudo-science, and conspiracy thinking go wrong. They latch on to the fact that claims about the world are necessarily inductive, and therefore possibly false. Then they make a false equivalence between probabilities. In terms of the RRAR method, they fail the Relativity test. There is always a chance of a scientific consensus being false, however, the relative probability of it being true is much higher. I call this the “so your tellin’ me there’s a chance” fallacy or confusing possibility with probability.

Link

Examples

Anti-GMO proponents have for the last 25 years continued to make the claim that there’s a chance that GMOs could cause massive irreversible environmental damage. The claim is made despite no positive evidence. They justify their view by exploiting the nature of scientific claims. They’re right that it’s possible. But for something to be possible is not the same as it being probable. The probabilities favor the opposite argument; i.e., that any such threats, if they exist, are very small. There is over 25 years of positive evidence supporting this. Yes, I’m tellin’ you there’s a chance GMOs could pollute the gene pool but it isn’t probable relative to the what the positive evidence supports so far.

Conspiracy theorists use the same tactics. For example, 9-11 “truthers” are fond of saying since we don’t know all the evidence for what happened in 9-11 it’s possible that the government is covering up major facts. It’s not even clear what it would mean to know all the facts. Does that include knowing how many paper clips were in each person’s desk? Obviously not. We should be concerned with relevant known facts. There are mountains of positive evidence that conform with the view that 9-11 was perpetrated by a group of Saudi terrorists under the control of Bin Laden. There is no positive evidence to the contrary, only speculation, arguments from ignorance, and anomaly hunting.

So, are the conspiracy theorist right by tellin’ us there’s a chance that 9-11 was a plot by the US government? Sure. But the probability of the possibility has to be weighted relative to the probability of the direction of the wealth of positive evidence. Again, the mistake here is similar to thinking the argument for picking the red marble is just as reasonable as the argument for picking the blue marble because, hey, there’s a chance the marble might not be blue. I’m tellin’ you there’s a chance!

The Bottom Line

The bottom line is that when you evaluate an inductive argument you have to recognize that it is by definition a probabilistic argument. That means, if there are competing conclusions, a critical thinker should adopt the belief that is more likely to be true relative to others. Otherwise, they’re confusing possibility and probability. And, given what we’ve learned about burden of proof, if you want to argue for a positive conclusion, you need to provide positive evidence not just arguments from ignorance.

Let me add one caveat. The claim here is not that a dominant scientific view is never overturned or that conspiracies never happen. To claim this would be to ignore history. However, we’re interested in current available reasons and evidence for particular conclusions. And so, on any particular issue, be it a claim about conspiracy or opposition to a dominant scientific view, views change because new positive evidence on balance, supports one conclusion rather than another.

Summary

1. An argument is valid if and only if if its premises are assumed to be true then the conclusion must also be true OR if its premises are assumed to be true then it’s impossible for its conclusion to be false.

2. An argument is sound if and only if it is valid and its premises are all true.

3. Validity and soundness apply to deductive but not inductive arguments.

4. If an argument’s premises, when assumed to be true, logically force a conclusion to be necessarily true then it is a deductive argument.

5. If it’s possible for a conclusion to be false even though all the premises are assumed to be true, the argument is inductive.

6. Inductive arguments can be strong to weak depending on how well the premises support the conclusion. That is, depending on how relevantthey are.

7. An inductive argument is cogent if (when assumed to be true) its premises strongly support its conclusion and the premises are actually true.

8. An inductive argument is uncogent if either its premises weakly support its conclusion OR one or more of its premises are false.

9. Identifying an argument as deductive vs inductive: (a) Assume the premises to be true; (b) evaluate whether it’s possible for the conclusion to be false (no matter how improbable); (c) if yes, then the argument is inductive; if no, then the argument is deductive. Otherwise stated: if we assume the premises to be true, a deductive argument’s premises will always be sufficient; an inductive arguments premises might be relevant (to varying degrees) but never sufficient.

10. An argument is sufficient if, assuming the premises to be true, they are ‘enough’ to guarantee the conclusion to be true. If an argument’s premises are sufficient then it is a deductive argument. If not, then it is an inductive argument.

11. Inductive arguments are probabilistic. That means it’s always possible to for the conclusion to be false. In general, when there are two competing conclusions on an issue the reasonable thing is to believe the one with the highest probability of being true.

 

 

HOMEWORK

A. Is the argument Valid, Valid but not sound, Invalid and not sound? 

1.  All monkeys live underground. Koko is a monkey. Therefore, Koko lives underground.

2. Some people like watching TV and everyone likes judo-chopping. Therefore, some people like both watching TV and judo-chopping.

3. If you drink caffein you’ll feel energized. If you feel energized you’ll generally be more productive. Therefore, if you drink caffein you will be more productive.

4. All limimus eat pamumas. Frank is a limimu. Therefore, Frank eats pamumas.

5. Some people are drug addicts and some people are criminals. Therefore, some people are both drug addicts and criminals.

6. Some monkeys live underground. Koko is a monkey. Therefore, Koko lives underground.

 

B. Is the argument (a) valid or invalid (b) if it isn’t valid, explain why not (c) if the argument is inductive say whether it is a strong, medium, or weak inductive argument (your answer to ‘b’ will help you decide). You’re only required to do Q. 1-10 but you may do the others for extra practice if you want.

E.g., Everyone I know who has student loans also has a part-time job. Kelley has a student loan so she must also have a part-time job.

(a) invalid

(b) Invalid because it’s possible for someone to have student loans and not have a part-time job. You can’t move from a small set of individual observations about a class of things to a conclusion about everything in that class of things.

(c) This is a weak inductive argument (because your sample size is small and based only on your own personal experience rather than a large survey, thus the inference to the conclusion will be weak.)

 

1. If you drink a lot of coffee you’ll have to pee a lot. If you have to pee a lot you’ll miss important parts of the lecture. You drank a lot of coffee therefore you’ll miss important parts of the lecture.

2. 99% of students like pizza. Bob is a student, therefore Bob likes pizza.

3. All the single ladies put their hands up. Angelina Jolie is a single lady, therefore she put her hands up.

4. Either Canvas works properly or I lose my mind. Canvas doesn’t work properly. I lose my mind.

5. Our government has deceived us in the past therefore you can’t trust anything they say.

6. Our government has told the truth in the past, therefore everything they say now is likely to be true.

7. These diet pills helped my aunt lose 5 pounds, therefore they’ll work for you.

8. These diet pills didn’t help my aunt lose any weight, therefore they won’t work for you either.

9. 10 of the wine bottles tested positive for trace amounts of glyphosate, therefore all wine has trace amounts of glyphosate.

10. Some of my students like football. So, my student Mike probably also likes football.

11. Every time I come home, the downstairs neighbors’ dogs bark. When I come home tonight they’ll bark.

12. If you’re not sure how to do this homework assignment you should see your TA. If you plan on seeing your TA but can’t make their office hours, you should see if they can meet you at another time. Mary isn’t sure how to do this homework and can’t make office hours, therefore Marry should see if her TA can meet at another time.

13. Everyone likes ice cream. Some people like Donald Trump. Therefore, some people like ice cream and Donald Trump.

14. Some people like ice cream and some people like Donald Trump. Therefore, some people like both ice cream and Donald Trump.

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