Deductive Arguments and Inductive Arguments:
How to Tell the Difference
Inference: Drawing a Conclusion
Two Types of Inference: Certain versus Probable
Examples of the Two Types of Inference
Are Invalid Arguments Inductive?
Evaluating a Deductive Argument
Evaluating an Inductive Argument
Are the Premises in an Inductive Argument Unrelated to the Conclusion?
Can Inductive Arguments be Made Valid?
Inference: Drawing a Conclusion
We make inferences all the time. When the sky is full of dark gray clouds, we infer that it’s going to rain and that the wind will be cold. When it’s 5 p.m. on Friday we infer that Route 10 will be jammed. When there’s condensation on our car windows in the morning we infer that the temperature dropped overnight. That is, we draw conclusions from the conditions (evidence) we observe.
Some inferences are well-supported by the available evidence, even to the point of being so customary that we take them for granted like when we place a tray of water in the freezer and infer (assume, really) that in a few hours we’ll have ice, or when we place a slice of bread in the toaster and infer that in a few moments we’ll have toast.
Some inferences are poorly supported by the available evidence, or are at least quite speculative like when someone spends $100 on lottery tickets because they feel really lucky and infers that they are going to win, or when someone begins to notice clicking sounds whenever they use their telephone and infers that the CIA is spying on them.
Of course, well-supported inferences can sometimes turn out to be wrong, and poorly supported inferences can sometimes turn out to be right. Maybe the CIA is spying on you. Maybe the toaster has shorted out, and it’ll never toast another slice of bread.
Two Types of Inference: Certain versus Probable
Before we get to questions about the quality or accuracy or reliability of the supporting evidence, however, let’s begin by distinguishing between two broad types of inference. Some inferences are certain. That is to say, these inferences are such that if the supporting evidence is accurate, the inference could not possibly go wrong. For instance, if I have five apples, and I buy seven more apples, I now have twelve apples, and I can’t be wrong about that, unless I miscounted the first five apples, or miscounted the additional purchase of seven apples.
As another example, if I note that all doctors are humans and that all humans are fallible, then I cannot be wrong in thinking that all doctors are fallible, unless some humans are infallible, or some doctors are not human.
On the other hand, some inferences are uncertain but probable. The examples above involving our inferences about the chances of rain or the traffic at rush hour are of this sort. Very probably, if the sky has been gray all day, it will rain, and very probably if it’s rush hour there will be traffic. But while these inferences are quite probable indeed, they’re not certain. The sky could be gray all day and yet it might not rain. It might be rush hour, and yet the roads could be relatively clear on a given day. The inference about the rain is generally more chancy than the inference about the traffic (the first might be 60/40, whereas the second is more like 90/10), but both are merely probable inferences; neither is certain.
If an inference is certain, the probability of its being true when its supporting evidence is true is 100%. If an inference is only probable, then the chance of its being true when its supporting evidence is true might be very high – even 99.999 percent – but it will not be 100%, because there’s always a chance, however slim, that we could be wrong. The example above involving the toaster is like this. 99 times out of 100 times (or maybe even 99.999 out of 100 times), when we put a slice of bread in the toaster, it toasts. If, however, the toaster has shorted out, our next slice of bread will not toast, and we’d be wrong in our belief that it will.
Examples of the Two Types of Inference
The range of inferences that are certain is fairly small, and generally uninteresting, though this is not always the case. Consider the following:
The person who uses a tool is distinct from the tool they use
Our hands and arms are tools we can use to carry out certain tasks
Our feet and legs are tools we can use to get us from place to place
In fact, our entire body is a tool we use to interact with the world
Therefore, we must be distinct from our bodies
This inference is certain. If the supporting evidence is accurate, then the inference (the conclusion the statement that’s being supported or defended) must be true. The conclusion cannot be false while its supporting evidence is (all) true.
Inferences that are merely probable are much more common, and are usually more interesting, because they involve subjects where the evidence is less than conclusive and thus where we don’t know for sure (and maybe can’t know for sure) what is the truth of the matter. Consider the following:
Lee Harvey Oswald was a Marine, and was a good marksman
If Oswald shot Kennedy from the Texas School Book Depository building, however, he would have been above, to the right of, and behind Kennedy
The Zapruder film clearly shows Kennedy’s head jerking back and to the left when he is struck by the fatal shot
(Probably) Oswald did not act alone, and Kennedy was killed as part of a conspiracy
This inference is only probable because even if all of the supporting evidence is accurate, the conclusion might still be false. We just can’t be certain that Kennedy was killed as part of a conspiracy, even if the fatal shot seems to suggest more than one gunman. It has been suggested that even a shot from behind and to the right could “pull” a person’s head back and to the left, depending upon the force of the shot, and the reaction of the person’s nervous system and muscles to being hit. The evidence, then, might not point to what it seems to point to, even if it seems really, really likely that it’s pointing to the truth.
Inferences that are certain, where the truth of the supporting evidence (the premises) guarantees the truth of what we’ve inferred (the conclusion), are called Deductive Inferences or Deductive Arguments. The relationship between the evidence or the grounds for the inference and the inference itself is such that the conclusion cannot be false if the premises are true. Of course, the premises might not actually be true, but even if they’re not, the relationship between the evidence and the inference is the same. The inference is Valid, even if the evidence is faulty. For instance, if we observe that:
All dolphins are fish
No fish are mammals
then we would be right to infer that
No dolphins are mammals
even though we’d be wrong about the facts (it’s true that no fish are mammals, but it’s not true that dolphins are fish). Arguments or inferences that are certain or intended to be certain are Deductive Arguments or Inferences. The inference is Valid when it is certain and is Invalid when it is meant to be certain but is incomplete. For instance:
All politicians are dirty rotten liars.
Clearly something is missing, but equally clearly the missing connection can be supplied if we were to add the missing (or implicit) premise
So, all politicians are dangerous.
All dirty rotten liars are dangerous.
Before we added this premise, the argument was Invalid (because incomplete), but we can see that the (completed) inference was intended to be certain because we can see quite clearly how to complete it, and we treat the additional premise as “missing” (or “implicit” that is, implied) because it’s clear how the argument is supposed to go, inasmuch as it’s clear what is the intended relation between politicians being dirty rotten liars, and politicians being dangerous: namely, anyone in power who’s a dirty rotten liar is going to be dangerous. Whether this conceptual relation actually holds (is actually true) is another thing; but the inference is both clear and certain.
Are Invalid Arguments Inductive? No.
It is for the above reasons that arguments that can be completed in this way are taken to be simply unsuccessful Deductive arguments, rather than Inductive arguments: the inference is not merely probable, but is certain once it is completed. Moreover, probable inferences generally cannot be made certain, as we shall see later on.
If an inference is certain and we know that all of the supporting reasons or evidence (premises) are true, or if they’re obviously true, then the argument is Deductive, and Valid, and Sound. For instance:
Water is a chemical compound
If an inference is Invalid (that is, Deductive but incomplete) it obviously cannot be Sound, since it is not (yet) Valid, and a Valid argument that has even one clearly false premise is Unsound. For instance:
Human beings drink water
Human beings drink chemical compounds
All reptiles are mammals
An inference can, however, be certain even when all the evidence is inaccurate:
All snakes are reptiles
All snakes are mammals
All humans are reptiles
Notice here that the inference is certain (if the evidence were accurate, the truth of the conclusion would be guaranteed), despite the fact that the evidence is not accurate. No humans are reptiles, and no reptiles have hair, but all humans have (at least some) hair. Thus, an inference can be certain, some (or all) of the premises can be false, and the conclusion can still be true.
All reptiles have hair
All humans have hair
Evaluating a Deductive Argument
To evaluate and critique a Deductive Argument, then, we should begin by determining whether the argument is Valid. If it’s Invalid, the first thing we need to do is to figure out how to make it Valid, and thus to fill in any missing (or “implicit”) premises. We do this because we want to learn as much as we can from the argument, and fixing it up requires us to think through the evidence, to think through the relationship between the evidence and the conclusion, and to think over what we know about the issue the argument involves. Simply rejecting the argument as Invalid does not require us to do any of these things, and an opportunity to test our beliefs against the ones in the argument, and to improve our own beliefs, will be wasted.
Once the argument is Valid (or if it’s already Valid), we identify which premises (pieces of evidence) seem to be correct, which seem to be incorrect, and which seem to be controversial or indeterminate. Obviously, we can’t criticize the inference itself; if the argument’s Valid, the inference is certain. So we must either criticize one or more of the premises or accept the conclusion. If the premises are true, so is the conclusion; if one or more of the premises is false, then the inference is unwarranted. If we can’t find anything wrong with the premises but we still think the conclusion is false, then there must be something wrong with the evidence, and we ultimately need to figure out where the evidence goes wrong, or is misleading.
Inferences that are merely probable are called Inductive Inferences or Inductive Arguments. As we already noted, the relationship between the evidence or the grounds for the inference and the inference itself is such that the conclusion might be false even if all the evidence is accurate. Of course, the premises might not be true, but even if they are, the relationship between the evidence and the inference is the same: the inference is only probable, so there will be some chance that even if the evidence is accurate, the conclusion is nevertheless mistaken. For instance:
The sun appears on the horizon in the East every morning, moves across the sky, and disappears below the Western horizon every evening.
From the standpoint of an observer standing on the surface of the Earth, it certainly looks like the sun is moving and we are not. Indeed, even though we now believe that the Earth is moving through space and that the Earth revolves around the sun, we cannot feel the Earth moving, however hard we try.
The Earth feels as though it is not in motion.
Thus, the sun revolves around the Earth.
Here’s another example:
Sound generated at a distance is not perceived immediately, but is heard after its cause is observed.
The transmission of light is, however, not instantaneous; light travels so fast (186,000 miles per second!) that its transmission looks instantaneous to us, just as it looks as though the sun is moving while we’re standing still. So here we've got two inferences (conclusions) plausibly supported by accurate and circumstantially compelling observational evidence, both of which turned out to be mistaken.
The transmission of sound is not instantaneous.
Light generated at a distance, however great, is instantaneously perceived.
The transmission of light is instantaneous.
A probable inference is Strong when the premises are true and the truth of the conclusion is more probable than not (more than 50% likely to be true, but less than 100% likely) given the evidence. A probable inference is Weak when one or more of the premises is false, or when the conclusion is less probable than not (less than 50% likely to be true, but greater than 0% likely) given the evidence.
Here’s an example of a fairly unambiguously Strong Inductive argument:
Those who smoke cigarettes or are exposed to second-hand smoke over a period of many years develop lung cancer at a rate that is statistically significantly higher than the rate among the general population
Here’s an example of a fairly unambiguously Weak Inductive argument:
Smoking causes lung cancer
Louis Pasteur, who introduced the germ theory of disease, was ridiculed and ignored by other scientists and physicians
Louis Pasteur was, however, a scientific genius, and was right about germs, but was ahead of his time
Royal Raymond Rife, who proposed a germ theory of cancer, was ridiculed and ignored by other scientists and physicians
Therefore, Royal Raymond Rife was a scientific genius, and was right about cancer, but was ahead of his time
Evaluating an Inductive Argument
The only way we can evaluate an Inductive Argument or Inference is to consider the extent to which the evidence points to the conclusion given (or the extent to which it might just as well point to some other, even more likely conclusion), and to consider how strongly the evidence points to this conclusion, and how accurate the evidence is. Thus, we have to be able to defend our view of the matter that is, we have to be able to back up the beliefs and attitudes that lead us to say that the evidence points strongly to the conclusion, or only points weakly to the conclusion.
The relationship between the evidence and the inference where the inference is certain is, of course, very strong in fact, it’s as strong as it can be (100% strong). Thus, there is a very close and obvious connection between what the evidence says and what the conclusion says. This connection is in some sense less tight and will obviously seem less inevitable in the case of inferences that are only probable. This is because the evidence is being used to suggest or recommend that the conclusion is true, rather than to guarantee its truth. But this doesn't mean that in an Inductive inference the conclusion does not follow from the premises. It might seem that if the evidence involved in an inference is accurate but the conclusion turns out to be false then the inference didn't really follow from the evidence in the first place. But this isn't so. To say that a conclusion follows is not to say that it will have to be true if the evidence is accurate; it means that the evidence provides support for it, regardless of whether that support is decisive or somewhat less than decisive.
Are the Premises in an Inductive Argument Unrelated to the Conclusion? No.
The relationship between the premises and the conclusion of an Inductive Argument is less tight than the relationship in a Deductive Argument more of a suggestion than a guarantee but the premises and the conclusion are not unrelated. If they were not related at all, not even suggestively, then the inference wouldn’t just be weak, it’d be non-existent. Consider the following:
There's no proof that souls don't exist
The inference here is not just poor; there is no inference. The premise and the conclusion are not meaningfully related to one another. This is a case where the premise(s) are unrelated to the conclusion. The premise here is not really evidence for the conclusion at all, at least not by itself, and it does not even point suggestively in the direction of the conclusion. (The fact that something has not been disproven does nothing at all, not even by way of suggetion, to show that it's real.) This, then, is the difference between the premises being unrelated to the conclusion, and the premises being related to the conclusion suggestively, but not in such a way that it’s impossible to draw any other conclusion.
Therefore, souls exist
Can Inductive Arguments be Made Valid? No.
Inductive Arguments cannot (generally) be made Valid, no matter how many further plausible premises we add. To make an Inductive Argument Valid we would almost always have to add a false or highly implausible premise and thus make the argument worse. Certain kinds of inferences are thus better understood as being probable than as being certain but with faulty evidence. This is because inferences which are only probable deal with situations and matters about which certainty is impossible or unavailable. To make an argument Valid is to make the inference certain, and if certainty is out of the question, then we cannot make the argument Valid, no matter how hard we try.
Predictions illustrate this quite nicely: even if it looks really, really obvious that it’s going to rain tomorrow, it might not rain, contrary to all of our expectations and all of our best evidence. Generally speaking, there’s just no way we can know for sure (with certainty) what’s going to happen in the future, so inferences about the future generally won’t be certain.
There are some exceptions. I can know for certain that unless I make a mistake, I will never, no matter how many times I do it, get anything other than 12 when I add 7 and 5, even in future cases. Moreover, I know for certain that I am going to die (though I don’t know when, thankfully). Most of the time, though, we cannot be certain of the future even when we’re pretty darn sure of what’s going to happen. (In some sense, my belief that I will die someday isn't really a prediction at all but simply a specific instance of a general rule: everybody dies.) We have every reason to think that the moon will be visible in the sky tomorrow, but if an enormous asteroid comes along and obliterates the moon, it won’t ever be visible again (except as debris), and we know that this could happen, even though it seems really, really unlikely. The point here is that we can’t be completely sure of what’s going to happen, even in cases where we have little doubt, and this distinguishes cases where inferences are merely likely (even if they’re really, really likely) from cases where there’s no chance at all of the inference going wrong.