Basics of argument structure and assessment

TIES542 Principles of Programming Languages, Spring 2017
Antti-Juhani Kaijanaho

The concept of an argument

The word argument means (among other things) a message conveyed in some (usually natural) language, by which someone attempts to convince someone else that a particular proposition (the conclusion of the argument, in Finnish johtopäätös) is true. A clear argument has recognizable premises (propositions which the argument relies on; in Finnish premissit or oletukset). Some of the premises are silent: they are not specifically mentioned but they are still evident from the context.

When analyzing an argument, we sometimes rewrite them in a special way: we list premises, each on its own line, and below them, separated from the premises by a horizontal line, we write the conclusion.

Example. "Haskell is terrible. Even in function calls they do not use parentheses correctly."

The conclusion of this argument is "Haskell is terrible" and its openly stated premise is "Even in fuction calls they do not use parentheses correctly." A number of silent premises can be supposed: that correct use of parentheses in function calls means putting call arguments in parentheses, with separate arguments separated by commas, like in Java and C#; and that a language is terrible if its parenthesis use is incorrect.

When analyzing the argument, we could write it thus:

\[ \begin{array}{l} \text{Even in function calls they do not use parentheses correctly.} \\\hline \text{Haskell is terrible.} \end{array}\]

Example ends.

Example. "In an object-oriented language, the state of an object must be mutable over time. In Haskell, state is immutable. Hence Haskell is not an object-oriented language."

The first two sentences are premises and the last sentence (except for the word "Hence") is the conclusion. We might write this argument when analyzing it as follows:

\[\begin{array}{l} \text{In an object-oriented language, the state of an object must be mutable over time.}\\ \text{In Haskell, state is immutable.}\\\hline \text{Haskell is not an object-oriented language.} \end{array}\]

Example ends.

Example. "If, in an object-oriented language, the state of an object must be mutable over time, and if, in Haskell, state is immutable; then Haskell is not an object-oriented language."

This is not an argument but a proposition.

End Example.

In practice arguments are usually nested: one first uses some premises to make an intermediate conclusion as a sub-argument, and that intermediate conclusion functions as a premise in another sub-argument. In principle, this nesting can go as deep as one likes, the only limiting factor being the ability of a human being to comprehend it. For example, an academic article or a doctoral dissertation is usually a huge nested argument.

There are many ways that an argument may be fallacious, but I will mention here two:

Infinite regress (in Finnish ääretön regressio) occurs when an argument relies on a premise that is justified by another argument whose premises are further justified by an argument, ad infinitum. It is like the old joke: if the Earth rests on a giant turtle, then what does the turtle rest upon? Well, another turtle; it is turtles all the way down!
Petitio principii, circular argument, or begging the question (in Finnish kehäpäätelmä) occurs when an argument relies on its own conclusion as a premise. Note that begging the question is very commonly used in another meaning as well ("A begs the question" means that A is a circular argument, but "A begs the question B" means that A requires B as its premise).

Because infinite regress and petitio principii are considered fallacies, a nested argument must have premises that are not justified by another argument. A good argument uses premises that are easier to believe than the conclusion.

Deductive arguments

An important and traditional way of analyzing an argument is to remove from it all references to the reality and replace them with abstract variables; all that remains is the form of the argument. For example, the second example above may be analyzed as folllows:

\[\begin{array}{l} \text{For all $x$, if $A(x)$, then $B(x)$.}\\ \text{$B(h)$ is false.}\\\hline \text{$A(h)$ is false.} \end{array}\]

The example is obtained from this formula by defining that $A(x)$ means that $x$ is an object-oriented language, $B(x)$ says that in $x$, the state is mutable, and $h$ stands for Haskell.

If an argument is meant to be analyzed in this manner – that is, by abstracting away all references to reality – it is deductive (in Finnish deduktiivinen). Such an argument is valid (in Finnish validi), if, whatever real or imaginary (however fanciful) one puts in place of the variables, the argument's conclusion is true whenever all its premises are true. In other words, an argument is valid if it is impossible that the conclusion is false when the premises are all true. The theory of deductive arguments is called formal logic; there are courses on it in this university in the disciplines of mathematics and philosophy.

There are many forms of argument that produce valid arguments. The following two forms one should recognize by name:

\[\begin{array}{ll} \textbf{Modus ponens} & \textbf{Modus tollens} \\ \text{A is true} & \text{$B$ is false} \\ \text{If $A$ then $B$} & \text{If $A$ then $B$} \\\hline \text{$B$ is true} & \text{$A$ is false}\\ \end{array}\]

In assessing a deductive arguments, there are three major questions to ponder:

Is the argument valid?
Do I accept all its premises?
Do I accept its conclusion?

If you accept all premises of a deductive argument but cannot accept its conclusion, then you must find a flaw in its form that makes it an invalid argument.

You are not compelled to accept the conclusion of a valid argument. You can always turn the argument around: since the argument is valid but its conclusion is not true, some of its premises must be false. You can thus respond to the argument by a counterargument, where your goal is to show that a particular premise is false. Philosophers describe this strategy by saying that one person's modus ponens is another person's modus tollens.

Inductive arguments

If an argument is not deductive, it is inductive (in Finnish induktiivinen). Such an argument is not meant to be assessed based on its form and it is a fallacy to accuse an inductive argument of being invalid. Most inductive arguments are not in fact valid arguments, but many of them are still very useful and instructive.

The argument "I have observed during my lifetime that the sun rises every morning and sets every evening. Therefore, the sun will rise tomorrow morning as well." is not deductive and thus it is not useful to ask whether it is valid. It is quite possible that the sun does not rise tomorrow, though I doubt this will be so here in Jyväskylä; it could happen if the rotation of the Earth suddenly changed overnight so that Jyväskylä is always away from the sun (though this event would probably kill us all instantly due to the accelerations involved).

There is much disagreement among researchers about the theory of inductive argument assessment. I like to use a theory where the strength of an argument is interpreted as the probability of its conclusion being true given the premises. This requires, of course, a subjective interpretation of probability: everyone may select probabilities as they like, so long as they follow the axioms of probability theory.

It is possible to give a practical interpretation to subjective probability through a betting metaphor: the probability you set for a particular proposition determines the betting odds you are willing to accept in favor of that proposition. Now, following the axioms of probability theory means that if you were to become a bookie, you could not suffer a certain loss regardless of how the bets turn out (in other words, you would not be vulnerable to a Dutch book). More dramatically: what sort of a proposition you are willing to stake your life on, that is, what sort of proposition you believe strongly enough that you are willing to die if it is false?

This theory entails the following interesting result. Let us consider the following kind of an inductive argument:

\[\begin{array}{l} \text{Procedure $A$ was found better than procedure $B$ in an experiment.}\\\hline \text{Procedure $A$ is better than procedure $B$.} \end{array}\]

Now the strength of the argument depends not only on the result of the experiment but also on how likely we considered the conclusion before the experiment. In other words: your posterior probability (in Finnish posterioritodennäköisyys) depends not only on the strength of the evidence (in Finnish näytön vahvuus) but also on your prior probability (in Finnish prioritodennäköisyys). Simplifying a lot: our opinion depends not only on the evidence but also on our prior opinion (which may be or may not be based on prejudice). Those of you who are familiar with probability theory may recognize this to be an instance of Bayes' theorem; hence, this sort of argument analysis is sometimes called Bayesian.

In practice, precise probability calculations are rarely done. Instead, an inductive argument is usually assessed intuitively based on how convincing it seems. Of course, in this situation the effectiveness of an argument depends not only on its content but also on how it is presented (hence, the study of rhetoric is important for people who want to become good argumenters). We usually use probability theory mainly as a check to insure that we deal with arguments in a rational manner at least to a degree.

You could assess an inductive argument by trying to come up with additional premises required to make the argument deductively valid. This technique allows you to identify reasons why you assess an inductive argument differently than your discussion partner, and may move the discussion forward. You should do this carefully, however, as it is very easy to make a strawman argument (in Finnish, olkiukkoargumentti), where you attack not the original argument itself but your own caricature of it. Strawman arguments are disrespectful and crude.

The challenge in inductive argumentation is that, unlike a deductive argument, it cannot compel the reader in any manner. It is perfectly rational to accept the premises of an inductive argument and reject its conclusion. Thus it is possible that a discussion does not result in a consensus result even if all parties are rational and argue in good faith. Sometimes it is necessary to agree to disagree.

Literature

Basic works on argument analysis are for example

Roy van den Brink-Budgen: Critical Thinking for Students. 4th ed, How-To Books, 2011.
Matthew McKeon: Argument. Internet Encyclopedia of Philosophy, accessed 2017-01-09.

The theory of deductive arguments is studied in introductory courses to formal logic. As to the theory of inductive arguments, see for example

Colin Howson & Peter Urbach: Scientific Reasoning: The Bayesian Approach. 3rd ed., Open Court, 2006.
Antti-Juhani Kaijanaho: Evidence-based programming language design: a philosophical and methodological exploration. Dissertation, University of Jyväskylä, Jyväskylä Studies in Computing 222, 2015; Chapter 4. Available in JYX.