The Logic Café
|because||for the reason that|
|since||for the following reason|
|for||on account of|
Now, on the other hand, we sometimes write things like
I've worked hard all morning so I deserve a good break this afternoon.
Here, the word "so" indicates that the conclusion is about to be given. We call it a "conclusion indicator". Again, there are many ways of indicating a conclusion. A number of them are given in the following table.
|so||therefore||as a result|
|thus||it follows that||consequently|
|hence||in conclusion||so one can conclude|
An argument is valid just in case
it is not possible that its conclusion be false while its premises are
An argument is invalid if and only if it is not valid.
So the definition of validity (the property of being valid) has to do with (1). Our second definition combines judgments (1) and (2):
An argument is sound if and only
if it is both (a) valid and (b) has only true premises.
An argument is unsound if and only if it is not sound.
But it can be a bit disconcerting to decide on soundness (the property of being sound)! That takes us rather far from the province of logic. So, it's good to point out that an argument's soundness is something that we won't often be able to decide as a matter of logic. When you are examined on soundness, you can expect matters that are fairly uncontroversial.
Think about the following argument. It's very uncontroversial and really rather uninteresting. But that makes it easier to judge.
All whales are mammals.
The animal who played Free Willy is a whale.
The animal who played Free Willy is a mammal.
Notice first that this argument is valid. Even if you don't know anything about whales or Free Willy, it's clear that the conclusion is inescapable given that the two premises (the statements above the line) are true. Second, the premises are true. So, the argument meets the two conditions required for it to be sound.
Now, consider another argument.
All whales live in the Southern Hemisphere.
Shamu (of San Diego, CA) is a whale.
Shamu lives in the Southern Hemisphere.
This argument too is valid. How can you tell? A test is to imagine the premises being true. Here you might have to imagine herding all the whales south of the equator! But imagine it anyway. Then notice that you are automatically imagining the conclusion being true as well. It's impossible for the conclusion to be false while the premises too are true. So, the argument is valid. But, of course, it's not sound. It has a false premise -- imagining that all whales live south of the equator does not make it so.
Now, not all arguments are meant to be valid or sound. We can only give valid and sound arguments when we have the most forceful evidence. When we do argue in this way, the reasoning is deductive; we'll say the study of such reasoning is "deductive logic".
An argument is deductive if and only if its premises are intended to lead to the conclusion in a valid way.
Note the word "intended" that is part of this definition. Whether or not an argument is deductive depends on how it is meant. Often we intend to give a valid argument but fail. (Didn't you ever give a "proof" in geometry class that was meant to validly imply some theorem, only to find you were wrong?) In any case, an argument may count as deductive even when it is not valid; judging an argument as deductive is a matter of interpretation not just logic.
Frequently we need to give arguments even when our evidence only makes a conclusion likely, but not inescapable. Then our thinking is often called "inductive". For example,
I have surveyed hundreds of students here at ITU and found that less than 10% say they are happy with the new course fees. My sample was selected at random. So, I conclude with confidence that the vast majority of ITU students do not find the course fees acceptable.
Here, the argument's author is clearly claiming that the evidence cited makes the conclusion likely to be true but not a certainty (surveys sometimes do go badly awry, for instance when the participants have some reason to lie.) So, this argument is a clear case of an inductive argument.
An argument is inductive if and only if its premises are intended to lead to its conclusion with high probability.
We do not say that an inductive argument is valid when it succeeds at supporting its premises as intended. This because an inductive argument does not intend to be valid, does not intend that its conclusion is inescapable. Rather, an inductive argument whose premises do support its conclusion as intended (i.e., they make the conclusion likely) is called "inductively strong":
An argument is inductively strong if and only if its conclusion is highly probable to be true given its premises.
Inductive strength is a counterpart to validity: by definition, deductive arguments are intended to be valid, inductive arguments are intended to be inductively strong. Of course, people often give arguments falling short of what was intended. That's why we have logic classes! But the point is that "valid" and "inductively strong" play similar roles for deductive and inductive arguments respectively: they support their conclusions as intended.
Finally, we need to define a counterpart to "sound" for inductive arguments. Remember, that an argument is sound if and only if it's both valid and has all and only true premises. For an inductive argument we just substitute "inductively strong" for "valid" to get the notion of cogency:
An argument is cogent if and only if it is both inductively strong and all its premises are true.
However, the Logic Café is primarily about deductive logic not inductive. We need to have these definitions about inductive logic, though, to set it off from what we will be studying. However, it's worth noting that even inductive thinking typically has parts (i.e., some of the inferences component to the thinking) that are deductive. So, what we study in this course will have wide ranging application. For more, see the tutorial.
Possibility is essential to the definition of a valid argument. In fact, it's essential to most concepts in deductive logic. We need to get straight on the appropriate sense of "possible".
There are at least two ways to think about possibility. First, we may be concerned with what may actually be the case. We could say, for example,
Well I don't know for sure, but it's possible that Jan Thomas is the best snow skier to ever live in Arkansas.
Here possibility is an indicator of one's doubt. Our limited knowledge leaves open the question of just who is the best Arkansan snow skier of all time.
The second meaning of "possible" is concerned with what might have been the case, perhaps in a completely different situation from what in fact occurs. We might say
Bill Clinton is the 42nd president of the U.S. but he might have lost the election of 1992 and never become president.
Here we talk about a possibility that might have occurred but did not.
The second meaning, possible as what might have occurred had things been different, is logical possibility. Roughly, at least, what is logically possible depends on what language allows. It allows, for instance, that someone other than Clinton is the 42nd US president. The only type of possibility of interest in this course will be logical possibility. So, hereafter, we'll just write "possible" instead of "logically possible" and ignore the first sense of the word "possibility".
Next, we consider properties of individual sentences, of pairs of sentences, and of sets of sentences.
First consider the sentence:
All tall women are female.
Contrast this obvious truth with the quite different:
All US presidents in the Twentieth Century are male.
The first example about tall women is true because of the language. It couldn't possibly be false. While the second example about the sex of US presidents is true, it's not true just because of logic. So we will call the first logically true while the second is true but not logically true. This second is true, but had things gone differently in US politics, it would have been false. Its truth value is not determined by logical considerations alone, so we call it logically indeterminate.
A sentence is logically true if and only if it could not possibly be false.
A sentence is logically false if and only if it could not possibly be true.
A sentence is logically indeterminate if and only if it is possibly true and possibly false.
Turn next to pairs of sentences. For example,
All whales are mammals.
No whales are non-mammals.
Intuitively these two sentences mean the same thing. We will call them logically equivalent. To give a definition that is a little more precise than talk about "meaning" allows, we utilize the notion of possibility yet again:
The two members of a pair of sentences are logically equivalent if and only if it is not possible for one of the pair to be true while the other is false.
Also, we say that one sentence logically entails another when the argument from the first as premise to the second as conclusion is valid. In other words,
One sentence logically entails a second sentence just in case the first could not possibly be true while the second is false.Finally, sets of sentences can tell coherent stories or, on the other hand, their members can conflict with one another. Roughly, a set of sentences is logically consistent when there is no contradiction between its members.
A set of sentences is logically consistent
if and only if it is possible for all members of the set to be true together.
A set of sentences is logically inconsistent if and only it it is not logically consistent, i.e., it is not possible for all members of the set to be true together.