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Because How to Analyze and Evaluate Ordinary Reasoning Section 2 : Implication |
G. Randolph Mayes Department of Philosophy Sacramento State University |
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2.1 The concept of logical implication
In the previous section we introduced the idea of logical implication to capture the idea of support between premises and conclusion. But what exactly does it mean for premises to imply a conclusion? We'll define the term in a moment, but let's first look at an example that will make the definition easier to understand. Consider the following reasoning:
You'll notice that it in this example neither the word "because" nor any other reasoning indicator exists to help you distinguish the premises from the conclusion. Even so, it's pretty clear that its author intends this as reasoning in support of the conclusion that Chihuaua's are vicious dogs.
Premise 1: There is at least one Chihuahua that attacks people without provocation. Conclusion: Chihuahuas are vicious.
Notice that in our reconstruction we did not use all of the same words that occur in the example. The point of the reconstruction is to make the logical relationships clear. We will be working on this quite a bit later.
Now, if you were to be critical of this reasoning, you might say something like this:
The important thing to notice about this criticism is that in making it you do not deny the truth of the premises. You essentially took it for granted that Zeppo is a Chihuahua and that Zeppo is vicious. What you have pointed out is that that this information does not logically imply the conclusion.
This example brings out the essential feature of the concept of logical implication, which is that logical implication has nothing whatsoever to do with the actual truth of the premises or the conclusion. In fact, whenever we try to determine whether premises imply a conclusion, we simply assume that the premises are true, even when they are clearly false. On the basis of this assumption we then ask whether the conclusion must be true, or (what amounts to the same thing) whether it is still possible for the conclusion to be false. If it is not possible for the premises to be true and the conclusion to be false, then we say that the premises logically imply the conclusion.
Now here are three equivalent definitions of logical implication.
You can use any of these definitions, but it is important to see that they mean exactly the same thing.
Notice that when we say that logical implication has nothing to do with the actual truth of the premises, the term "actual" is key. Logical implication is a truth-preserving relation between premises and conclusion: if you put truth in (the premises), you get truth out (of the conclusion). But whether or not the relation exists has nothing to do with the quality of information you actually put in. Logical implication is like a perfect recipe that depends on good ingredients. If you use good ingredients, then a good result is guaranteed. If you use inferior ingredients, then the dish may turn out poorly. But that's not the recipe's fault.
2.2 Validity
Just as it is can be helpful to have more than one equivalent definition, it can also be helpful to have more than one equivalent term for saying the same thing. In logic the term "logically valid" is used to describe reasoning in which the premises succeed in logically implying the intended conclusion. When premises fail to imply the intended conclusion, we say that the reasoning is logically invalid.
2.3 Some notes on usage
For the sake of convenience we will often drop the term "logically" and just speak of implication and validity. However, It is important to note that the terms "imply" and "valid" have many different meanings in ordinary language. For example, we often use the term "imply" as a synonym for the word "suggest." In ordinary language you might say:
There is nothing fundamentally incorrect about this way of using the term "imply", but it's important to remember that this is not how we use it in logic. The term "validity" also has many different meanings. For example, we often use the term "valid" as a synonym for "true". In ordinary language we sometimes say that a person has made a "valid point," meaning that she has said something true or relevant, but this is not the same thing as saying that she has actually produced premises that logically imply a conclusion..
We should also note that the terms "valid" and "implication" are sometimes accompanied by the term "deductive". What we are calling "valid reasoning" is often called "deductively valid reasoning," and what we call "implication" is often called "deductive implication." Traditionally, the reason for this is to distinguish deductive relationships from inductive ones. We will speak of this difference in the next section, but for now all you need to know is that the term "deductive" adds nothing to the meaning of the terms "implication" and "validity". It is a redundant expression.
2.4 The nature of implication
The concept of implication is of fundamental importance in logic, but the fact that premises imply a conclusion does not by itself mean that the reasoning is good, nor does the fact that premises fail to imply the conclusion by itself mean that the reasoning is bad. In the next sections we'll see why that is this is the case. First, let's look at some garden variety examples of valid and invalid reasoning.
This example is properly reconstructed as follows:
Premise 1: Jake is Rafael's dog. Premise 2: All dog's love their owners. Conclusion: Jake loves Rafael.
In this example we correctly observe that if the premises are true, the conclusion must be true. Hence, it is valid reasoning. Of course we do not know whether the first premise is true, and we actually know that the second premise is false. But we also know that the actual truth of the premises is not relevant to assessing its validity.
Now let's look at an example of invalid reasoning
This example can be reconstructed as follows:
Premise 1: Jake is a dog. Premise 2: Everything that barks is a dog. Conclusion: Jake barks.
The invalidity of this reasoning may not be immediately apparent. To see it notice that there is a big difference between these two statements:
If premise 2 said "Every dog barks", then it, together with the premise "Jake is a dog" would imply that Jake barks. But as originally stated, it is possible for the premises to be true and the conclusion false. Hence, it is invalid. One way to show this is to simply add a premise that is compatible with the truth of the original premises, but which clearly makes the conclusion false. This is what we call a counterexample. For instance:
Premise 1: Jake is a dog. Premise 2: Everything that barks is a dog. Premise 3: Jake is a basenji. Conclusion: Jake barks.
(A basenji is a breed of dog that doesn't bark. It yodels.)
2.5 The limits of implication
Now let's learn the limits of the concept of implication for evaluating reasoning.
The most obvious limitation, of course, is that the actual truth of the premises does ultimately matter. Typically we do not simply want to know whether a conclusion would be true under certain circumstances, but whether it is in fact true. But other limitations are of greater concern to us now.
Consider the following reasoning:
Clearly, it is of no value at all to provide a premise that makes exactly the same statement as the conclusion it is supposed to support. Oddly, however, this reasoning is valid. To see this, just reconstruct it, and apply the definition of implication.
Premise: Rafael has got to have a dog. Conclusion: Rafael has got to have a dog.
Since the premise and the conclusion are identical, it follows that if the premise is true, then the conclusion must be true.
Here is an even more peculiar example.
This is lunacy, but let's reconstruct it and assess it's validity anyway.
Premise: Dogs have four legs. Conclusion: Squares have four corners.
Now we assume that the premise is true, and ask ourselves whether it is possible for the conclusion to be false. Surprisingly, the answer to this is no. The reason is that squares have four corners by definition. There is simply no such thing as a square with more or less than four corners. The conclusion is what we call, a necessary truth. Hence, this perfectly useless reasoning is valid as well.
You might think that this example is based on some kind of confusion. You might express yourself as follows:
You're right, of course, they don't have anything to do with each other. The problem is that the concepts of implication and validity don't require them to have anything to do with each other. There is, in other words, nothing in these definitions that captures your intuition that in good reasoning the premises should somehow make or cause the conclusion to be true.
Finally, consider this example:
Reconstructing this reasoning we get:
Premise 1: Rafael has a dog. Premise 2: Almost all dogs have four legs. Conclusion: Rafael's dog has four legs.
Now it is probably obvious to you that this quite reliable reasoning is not strictly speaking valid. If we apply our definition it is easy to see that the premises do not imply conclusion, for given the truth of the premises it is still possible that Rafael has a dog that is missing a leg. Still, this reasoning clearly should not be dismissed as faulty. The premises do not guarantee it's conclusion, but they do seem to make it highly likely.
The reasoning of this example has an inductive character, which simply means that while the premises do not guarantee the truth of the conclusion, they do raise the likelihood of the conclusion. Reasoning in which the premises make the conclusion highly likely is called inductively strong. Later we will capture the concept of inductive strength in a slightly different way. The important point to see here is that inductively strong reasoning is good, reliable reasoning that is not captured by the concept of implication.
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