5. Arrow Diagramming
Our goal now is to do argument analysis by considering three things: 1) structure, 2) validity, and 3) soundness. We will analyze structure by the use of arrow diagramming, validity by a close look at the support connection between premises and conclusions, and soundness by means of a discussion of truth in the context of argument.
Various diagramming methods are useful in revealing structural features of arguments. Among them, arrow diagramming gives the greatest insight into an entire argument as a skeletal whole so we can see how all the parts connect one to another. Continuing this analogy a little further, the bones of our skeleton refer to premises and conclusions; that is, they refer to statements that function as premises and conclusions. Ligaments refer to the various connections between argument parts; they are symbols on the diagram that represent these connections (ligaments are not intended here to represent reasoning indicators).
What we want to see by means of arrow diagramming is the degree of complexity that obtains in a given argument. Thus, we will center our attention on the techniques that are used for the visualization of argument complexity.
1A. An Overview of Argument Complexity
All arguments can be classified according to their complexity. Any argument will be either simple or compound. The degree of complexity in the structure of an argument may vary from the very simple to the extremely complicated. Key terms here are simple argument, compound argument, dependent premises, independent premises, and chain argument.
A simple argument is one that has singularity on both sides. The sides of an argument are premise side and conclusion side. This means that the entire argument leads to a single conclusion and that the premise side has either one premise or if it has more than one premise these premises will be dependent. Dependent premises are premises that work together in such a way that if one were false then the argument would be weakened or canceled. For example, note what happens when the second premise is omitted from the following argument:
P1. Speeding violates the civil law.
P2. Civil violations are morally wrong.
C. Therefore, do not speed.
Canceling the second premise does not cancel the support connection between the first premise and the conclusion. However, the argument is weakened and it is this fact that signals the dependency of the premises.
Compound arguments have multiplicity on at least one side. On the premise side, more than one premise is given as support for the conclusion and these premises are independent. Independent premises are premises that can be false or omitted without weakening or canceling the impact of the other premise as in the following example:
P1. Speeding violates the law.
P2. Speeding is dangerous.
C. Therefore, do not speed.
On the conclusion side, a compound argument may have more than one conclusion. Sometimes multiple conclusions may form a chain argument in which one conclusion becomes the premise for another (He's drunk, so he won't work, and therefore, he won't pay his bills; cf. Yanal, Basic, 47).
2A. A Sketch of Visualization Techniques
1B. The Vertical Arrow
Simple arguments have oneness on both sides, premise side and conclusion side. These are joined diagrammatically by a vertical arrow.
In a simple argument, the premise is a tightly knit unit, which forms one line of reasoning to the conclusion. The line of reasoning is represented on the diagram by the vertical arrow:
Since we always move down the argument in standard logical order, then our premises will be located at the top of the arrow and conclusions will be located at the bottom of the arrow, regardless of their order grammatically.
2B. Numbering
There are two things to keep in mind when numbering arguments. First, we number each entire sentence in the order we find them. This standardizes classroom work keeping us all on the same track. Second, since our interest is not so much with sentences but with statements, we may need to sub-divide sentences using capital letters. Thus, more than one statement in a given sentence will be sub-divided like this: 1ABeing a man, 1BSocrates is mortal. Now we are prepared to put an argument together in skeletal form.
3B. The simple argument diagram
The simplest simple argument has but one premise and one conclusion. If these are two different sentences, then the diagram will have the number one at the top of the arrow and a number two at the bottom: Idolatry is a great evil. Therefore, avoid it. This is simply sentence 1 that leads to sentence 2.
1
2
But note how "1ABeing a man, 1BSocrates is mortal" is diagrammed:
1A
1B
There is only one sentence, which is represented by the number 1. In this case, the existence of more than one statement within the same sentence necessitates the use of letters.
3A. Dependency
When a group of premises are dependent on each other, they form a unit and the argument of which they are a part is therefore considered simple. Although a simple argument may be very complicated, it is this principle of singularity that results in the classification as simple. This singularity is indicated on the diagram by a plus sign.
We now have ligaments for our skeletal look at an argument: the plus sign and the vertical arrow. The plus sign represents the connection which exists between premises. The vertical arrow represents the connection of premises to conclusions. What we lack is a way of representing the bones of the skeleton, which are, of course, the premise and conclusion statements. These are represented by numbers.
For an example of diagramming a simple argument with dependent premises let's walk through the speeding argument cited earlier: "Speeding violates the civil law. Civil violations are morally wrong. Therefore do not speed." First, number the sentences as they occur: "1Speeding violates the civil law. 2Civil violations are morally wrong. 3Therefore, do not speed."
The premises are a unit; they make the move to the conclusion stronger when taken together. There is one line of reasoning from premise to conclusion because the premises are a unit. This dependency and the move to the conclusion are both reflected on the diagram
1 + 2
3
4A. Compound Arguments
Compound arguments have multiplicity on at least one side. If the multiplicity occurs on the premise side we have independent premises and the structure may look like this:
1 2
3
In contrast to the simple argument, this argument has two lines of reasoning from distinct pieces of evidence to the same conclusion. Here is an argument with two lines of reasoning from separate premises to the same conclusion: "Speeding violates the civil law. Speeding is dangerous. Therefore, do not speed."
If we have two distinct lines of implication for different conclusions, the diagram will look like this:
4
3 8
The chain argument will have a statement at the bottom of one arrow and at the top of another arrow leading to a final conclusion. This statement is both a conclusion and a premise. It is a conclusion to the first argument and a premise for the second:
1
2
3
To understand this and similar arguments, it may be helpful to keep in mind that the relation of 1 to 3 is remote whereas the relation of 1 to 2 and of 2 to 3 is direct.
5A. Additional diagramming notes
To round off our diagramming procedures, we need to establish a method for representing weakened premises and missing elements.
1B. The Weakened Premise
When a premise is given but qualified or reduced in strength by another statement, as may be done by concession, this qualification is numbered and attached to that premise with a line (sometimes a dotted line).
3
1 + 2
5
2B. The Missing Element (enthymeme)
Missing elements are indicated by the star (*) instead of a number:
1 + * 4
3 + 5
6A
Thought questions: 1) does the star represent sentence 2? 2) Why is 2 not listed? 3) What is the difference between the connections 4 to 5 and 5 to 6A? 4) Make some more observations about this diagram that unpack its meaning.
Arrow Diagramming WS1
1. An arrow diagram is a structural, skeletal, analysis of an argument. In reference to the skeleton analogy, answer the following questions:
a. What are the bones of an argument?
b. What are the ligaments of an argument? What do the ligaments signify?
c. How are sentences indicated on the arrow diagram?
d. When more than one statement occurs within a sentence, how are they represented on the arrow diagram?
e. What 3 (or 10) observations can be made regarding the last example on the previous page (2B, the missing element)?
2. Arrow Diagram these examples. Only include matters that are structurally relevant to the argument on the diagram. Before doing so, number the argument.
a. Since all free men are citizens of Berlin and I am a free man, then my point is obvious. I am a citizen of Berlin.
b. The will of God is changeable. For the Lord says (Gen. 6:7): It repenteth me that I have made man. But whoever repents of what he has done has a changeable will.
c. Since speeding violates the law, then you should not speed. Besides, it is unsafe.
d. Speeding violates the civil law, civil law violations are immoral, and it is unsafe. So, you know I am telling the truth when I say, do not speed.
e. He is characteristically untrustworthy. Therefore, he will not make a good employee nor will he be a good husband.
f. Big Macs are better than Whoppers and Big Boy Classics are better than Big Macs. You should buy the better burger and children love only Classics. So, you should buy Big Boy Classics.
Arrow diagramming WS2
1. Diagram this argument by filling in the enthymatic conclusion:
The bigger the burger, the better the burger.
Burgers are bigger at Burger King.
Now make this filled out argument into a chain.
2. Diagram this pro-life argument:
Because a fetus is biologically human, it is a person, and so it should not be killed.
3. Diagram this portion of an abortion argument:
A fetus is so unlike a person that it has no significant right to life.
4. What is the relationship between the premises in this argument? Is it simple or compound? Diagram the argument.
As well as being unhealthy, working seven days a week violates God's law. So, one day in seven should be set aside for rest.
5. Arrow Diagram:
Voluntary euthanasia ought not to be legalized. In the first place, it is difficult to ascertain whether the consent of the patient is really voluntary. Thus, there is substantial danger that the law will be abused. Second, the judgment that a person is incurably ill may be wrong on two counts: the diagnostic judgment may be in error, and the judgment that no cure will be available within the life expectancy of the patient may be in error. Thus, there is substantial risk that some will die unnecessarily. Finally, to legalize voluntary euthanasia is to accept the "thin edge of the wedge," thus preparing the way for the legalization of involuntary euthanasia (Churchill, Logic, 554).
6. Arrow Diagram:
The death penalty by its very nature is a cruel and unusual punishment and is thus in violation of the Eighth Amendment of the Constitution. The death penalty is excessive and unnecessary. In addition, it is abhorrent to currently existing moral values. For these two reasons, the death penalty is within the meaning of "cruel and unusual punishment" as established by previous decisions of the Supreme Court.
Arrow Diagramming WS3
1. Arrow diagram Wesley’s last argument, which is his 29th paragraph (per my outline in the internet download it is argument 18). Once the diagram is complete, what lessons can be cited regarding diagramming?
2. Study the Sabbath Argument Arrow Diagram and answer the questions about it.