Reference Manual
Chapter One:
Formal and Informal Reasoning Basics
.pdf Version for Printing
This reference provides some of the basic points made in Chapter One. But it doesn't include everything of importance! Please spend the time working through all the tutorials. Often details for working homework problems -- the only good preparation for exams -- is available in the tutorials. Even if you can do this weeks homework without doing them all, there may be material in later units that will be very hard without a clear understanding of all that going on in the tutorials.
Contents: Section 1: Arguments and Form: Basics; Section 2: Determining Arguments and their Components; Section 3: Distinguishing and Judging Arguments: Validity and Soundness; Section 4:Distinguishing and Judging Arguments: Inductive Reasoning
In this chapter we introduce the general idea of logic, the study of correct reasoning. We start with the general notion of an argument and develop the concepts needed for argument analysis and evaluation.
1. Arguments and Form: Basics
Let's keep our first example in mind about Chris as we go over the basic ideas of logic. The primary notion is that of an argument.
An argument is a collection of statements including some (the premises) that are given as reasons for another (a conclusion).
Here's the familiar example ...
...and here are some of the basic ideas:
Usually formal logic can also be called deductive logic because the form of thinking allows one to deduce it's conclusion from its premises (as in the Chris process of elimination example argument described just above).
Informal logic is usually called inductive logic. Reasoning based on informal, inductive logic moves from statements of evidence (the premises) to a conclusion that extrapolates from, amplifies, or generalizes the evidence.
The process of elimination argument form we've been seeing will henceforth be called DS. The Chris "A" or "B" argument is an example:
Chris will get an "A" or a "B" in logic class.
Chris (it turns out) does not get an "A".
So, Chris will get a "B".
That is to say:
Any argument with the form: "Either A or B, but not-A, so B" is called DS.
Also, suppose Chris does better, he doesn't get a 'B', then still assuming that Chris will get an 'A' or a 'B', it follows that Chris will receive an 'A'. This is also DS.
Thus, there is a second version of the form DS: "Either A or B, but not-B, so A". [Note the slight difference: the second premise is "not-B" rather than "not-A".
Here's another
example.
If Chris gets an A, then he will be very happy. And (as it turns out) he does get an 'A'. So, it follows that Chris will be very happy.
This is an argument of a form we'll name as follows.
Any argument of the form "If A then B, and A so B" is called MP or "Modus Ponens".
We'll get lots of practice with this. So spend a moment to make some sense of it and then
move on...
If Chris gets an A, then he'll be very happy. But he turns out to be unhappy. So, it follows that Chris did not get an 'A'.
This is a different form:
Any argument of the form "If A then B, but not-B, so not-A" is called MT or "Modus Tolens".
These forms will turn out to have great significance. In this chapter, we will just begin to see why that is so.
"Picturing" Arguments
We will often need to display arguments in a way that makes its premises- conclusion structure as clear as possible. The first method we've seen for doing this is the Tree Diagram (from tutorials 1 and 4). Here is an example.
or |
Tree diagrams are especially useful for complex argument stuctures with more than one premise. The merging arrow used in this diagram, , indicate that the premises, (1) and (2), work together or "collaborate" to support the conclusion. Simple arrows, as in this diagram, , indicate that the premises each supports the conclusion independently of the other.
For simpler arguments, it's often better to give Standard Diagrams (see tutorial 2):
Chris will get an "A" or a "B" in logic class.
He does not get an "A".
Chris will get a "B".
To give a standard diagram, we write the premises first, draw a line, then write the conclusion. Here's another examle.
Chris has done well at college.
He has high LSAT scores.
Chris will likely be admitted to law school.
2. Determining Arguments and their Components
Distinguishing arguments From Non-ArgumentsWe need to keep in mind that there are many types of thinking in language that do not give arguments.
Here are five types of passage that you'll need to be able to disentangle.
Premise and Conclusion Indicators
Suppose we do have an argument.
All living beings deserve respect because life is sacred and the sacred deserves the greatest respect.
Now, how do we distinguish premises from conclusions or from other sentences which are not parts of an argument? For instance, in the example above arguing that life is sacred, how do we tell the conclusion from the premises? It's not always easy, but in this example there is a good hint. The word "because" is a premise indicator; it signifies that a premise follows. There are a number of roughly equivalent words or phrases in English; we'll call them all premise indicators. Several of the most common can be found in this table.
Premise Indicators: | |
---|---|
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.
Conclusion Indicators: | ||
---|---|---|
so | therefore | as a result |
thus | it follows that | consequently |
hence | in conclusion | so one can conclude |
3. Distinguishing and Judging Arguments: Validity and Soundness
One of the main points of logic is to be able to distinguish good reasoning from bad. There are two main parts to this process: (1) the judgment of the force or support of premises for conclusion and (2) the judgment of the correctness of the premises. The strongest sort of force or support is associated with valid arguments. The idea is that so long as the premises are assumed to be true, the conclusion is inescapable. We make this a bit more precise in the following terms:
An argument is valid just in case it is not possible that its conclusion be false while its premises are all true.
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.
It's worth noting that when we are concerned with validity, actual truth or falsity of statements need not matter. Instead, validity is only concerned with what happens IF premises are true.
Only when we are concerned with soundness (or cogency in inductive logic) do we need to think about whether the premises and conclusion are in fact true or false.
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 premises. 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.
4. Distinguishing and Judging Arguments: Inductive Reasoning
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.