T. Parent

An Overview of Arguments in Logic

 

An argument is a set of statements one of which (the conclusion) is taken to be supported by the remaining statements (the premises). [Note that a “statement” can either be a whole sentence, or an independent clause within a sentence.]

Five types of Arguments: Inductive, Deductive, Abductive,  Practical, and Other.

An Inductive argument is an argument where the premises register the known cases of a certain phenomenon, and the conclusion suggests that unknown cases will be like the known cases.

Examples :

(P1) The sun rose today.                   (P1) Everyone in my family has been stung by a bee.
(P2) The sun rose yesterday.             (C) So, absolutely everyone has been stung by a bee.
(P3) The sun rose the day before

        yesterday.
(P4) The sun rose the day before

        the day before yesterday.
[etc.]
(C1) So, the sun will rise tomorrow.

 

Of course, the premises in each argument do not guarantee the truth of the conclusion. Still, an argument can be a good inductive argument to the degree that the conclusion is likely given the premise(s). (In assessing its likelihood, sometimes people talk of the “inductive strength” of the argument.)

 

A Deductive argument, on the other hand, is an argument where (roughly) the truth of the premises would guarantee the truth of the conclusion.


Official Definition: An argument is deductive if and only if [abbreviation: “iff”] it is not possible for the premise(s) to be true and the conclusion false.

Example of a deductive argument:

 (P1) Jim likes either Coke or Pepsi.
 (P2) Jim does not like Pepsi.
 (C) So, Jim likes Coke.

So with a deductive argument, if we get you to accept the premises, then you must accept the conclusion too. Why? ‘Cause in a deductive argument there’s no way for both the premises to be true and the conclusion false.

Unfortunately, most of the time a deductive argument is called (misleadingly) a ‘valid argument’. The label is misleading, since you can have a “valid” argument which is nonetheless a bad argument, all things considered. That’s because the premises might be totally implausible. Yet the argument still counts as “valid” if it is the kind of argument where if you granted the premises, the conclusion would be guaranteed.

So if you hear a logician call an argument “valid,” that does not mean that it is ultimately a good argument. Conversely, if an argument is “invalid,” that also does not mean it is ultimately a bad argument. Consider for instance that all inductive arguments are invalid, technically speaking, i.e., they are non-deductive. Still, as we saw, there can be good inductive arguments. Thus, if you say that an argument is “invalid,” you’re saying that the premises do not guarantee the conclusion, though the premises may still make the conclusion very likely for all that.

The term ‘valid’ is also misleading in that “validity” concerns a relationship between premise(s) and conclusion. It is not directly concerned with whether the statements in the argument are actually true. This is contrary to how we use the word ‘valid’ outside the logic classroom: Ordinarily, we sometimes say that someone has made a “valid point” or that someone’s perspective is “valid” when we mean that s/he  made a true statement. But this is NOT how logicians use ‘valid’—they only say that arguments are “valid.” (Consequently, logicians do not speak of a point or a perspective as “valid,” though they can say instead that someone has a good point or has a legitimate perspective, etc.)

Of course, not every argument is deductive (= valid). Here’s one example:

 (P1) Jim likes either Coke or Pepsi.
 (P2) Jim does not like Mountain Dew.
 (C) So, Jim likes Coke.

In this, it is possible for the premises to be true, and the conclusion false. That’s not to say the premises are actually true or the conclusion is actually false. Rather, it’s just to say that this combination of truth and falsity is possible. N.B., A non-deductive (= invalid) argument is also sometimes called a non-sequitur—it is an argument where the conclusion “does not follow” from the premise(s).

Some deductive arguments are also SOUND: An argument is sound iff it is deductive AND every premise is true. Thus, an argument is unsound iff it is some premise is false or is not deductive.

So, to check that an argument is sound, you have to verify that the argument is deductive and that every premise is true.

Example of a sound argument:

 (P1) If a thing is a rectangle, then it’s not a circle.
 (P2) This page is a rectangle.
 (C) So, this page is not a circle.

This argument is sound, since it is deductive, and all of its premises are true.

Example of an unsound argument:

(P1) If Bill Gates is poor, then I’m a monkey’s uncle.
(P2) Bill Gates is poor.
(C) So I’m a monkey’s uncle.

This argument is unsound: Although it is deductive, it is not true that Bill Gates is poor.

NOTE: Truth and Falsity are NOT properties of arguments, but of statements. Thus, we do not say that a deductive argument is “true;” rather, we say that it is valid or sound. Or, if we want to talk of “true” and “false,” we can evaluate the statements in the argument as true or false.

An Abductive argument is an argument that is neither deductive nor inductive, where the conclusion stands as an explanation of facts given in the premises.

Examples:

(P1) I can’t get online from my computer.                               (P1) I have a headache.
(P2) There’s nothing wrong with my hardware or                   (C) So, my head is shrinking

        software.
(C) So, the University network must be down.             

Note that in the first example, the conclusion does not explain (P2) in isolation. (The network being down wouldn’t explain why there’s nothing wrong with my hardware/software.) So the conclusion of an abductive argument is not one that explains why each premise is true individually; rather, it explains why the premises are jointly true, true all at once.

Consequently, in the first example, the conclusion is best seen NOT as an explanation of why I can’t get online per se. (That would just be an explanation of the first premise.) Rather, it’s best seen as an explanation of why I can’t get online despite my functioning hardware/software.

Confusingly, some inductive and deductive arguments also have conclusions which (in some sense) explain the premise(s). The second example I gave of an inductive argument is one where the conclusion (in some sense) explains the premise. Moreover, the conclusion is explanatory in the following deductive argument:

(P1) This figure is a triangle.

(C) Hence, this figure is a closed, three-sided figure.

After all, if the figure is a closed three-sided figure, that “explains” why it is a triangle. But still, the argument is deductive, because the truth of the premise would guarantee that the conclusion is true.

Thus, in order to be certain that an argument is abductive, you must first show that the argument is neither deductive nor inductive.

Like an inductive argument, however, an argument is a good abductive argument to the degree that the conclusion is likely given the premise(s). (Since abduction and induction are both evaluated by the probability of the conclusion, oftentimes logic books will call both types of argument “induction.”) N.B., If the conclusion of an abductive argument is the most likely explanation out of all the explanations available, then the abductive argument is sometimes called an inference to the best explanation.

A Practical argument is an argument where the conclusion is a statement of what should or ought to be done, yet the argument is not deductive, not inductive, and not abductive.

Examples:

(P1) Stocks are low right now                           (P1) I need to make money.

(P2) The economy will recover soon.                (P2) Kidnapping children makes money.

(C) So, I should buy stocks right now.  (C) So, I should start kidnapping children.

                                               

As should be clear, these two arguments are not deductive. Re: the first argument, even if stocks are low and the economy is expected to recover, it is still possible that I should NOT buy stocks right now. After all, I might have barely enough money to feed my family.

 

Still, the first example can be a good practical argument if we’re talking about someone who has expendable income. But even in that case, it remains possible for the premise to be true and the conclusion false for different reasons. So the argument is still non-deductive.

 

When is a practical argument a good practical argument? NOBODY KNOWS. That is still debated vigorously among ethicists. However, there is a sub-type of practical argument, called a decision-theoretic argument, and it is known what makes these arguments good or bad (under certain assumptions). Very briefly, you have a good decision-theoretic argument when the conclusion recommends an action that is expected to “maximize profit” among the available options. (No need to go into more detail at this point...)

 

Note: Some arguments with a “should” or “ought to” conclusion are NOT practical arguments. Consider the following inductive and deductive arguments (respectively):

 

(P1) I shouldn’t have played the lottery today.                        (P1) Thou shalt not steal.

(P2) I shouldn’t have played the lottery yesterday.                        (C) I should not steal this ipod.

(P3) I shouldn’t have played the lottery the day before that.

[etc.]

(C) I shouldn’t play the lottery tomorrow.

 

(Arguably, there are also abductive arguments with “should” or “ought to” conclusions as well.) So remember that the term ‘practical argument’ is reserved for an argument that is NOT any of the previous three types of argument—AND has a “should” or “ought to” conclusion.

Other arguments exist besides the previous four types. Some arguments in the “other” category are “mixtures” of the previous types of arguments. Consider, for instance:

(P1) My car is usually out of gas.

(P2) My car currently isn’t running.

(C) So, my car is currently out of gas.

The conclusion here seems to be inductively and abductively inferred. Consider that if the argument just consisted of (P1) and (C), it would plausibly be inductive. But if the argument just consisted of (P2) and (C), then it would look abductive. Yet since you’ve got both premises, it looks like inductive and abductive reasoning is being used.

A different kind of “other” argument is an enthymeme: In these arguments, too much is left unsaid for us to classify the reasoning more precisely. For instance, consider:

 

(P1) The Democrats took control of the Congress and the White House.

(C) So, predictably, the economy stopped its downward slide.

 

How exactly is (C) supported by (P1) in this case? Are we making an induction based on past cases (which aren’t explicitly mentioned)? Or are we deducing the conclusion from a suppressed premise like “whenever the Democrats are in control, the economy improves”? It’s impossible to say. So when an argument is enthymematic to this degree, we put it in the “other” category.

Relatedly, some arguments can’t be classified more precisely, simply because they are just plain awful. Consider:

(P1) I have ten toes.

(P2) Penguins live in Antarctica.

(C) So, Obama’s economic plan will fail.

 

Observe that out of context, these three sentences would not seem to be an argument at all. But here, they indeed constitute an argument since one statement is marked as the conclusion, and other statements are marked as premises. So in this case, the three statements here are an argument; it’s just that it’s a really bad argument. Because of that, it’s not at all clear how the premises are meant to support the conclusion; hence, the argument goes in the “other” category.

Finally, some arguments in the “other” category are arguments by analogy. Here’s a famous example:

(P1) A watch has a designer.

(P2) The universe is like a watch.

(C) So, the universe has a designer.

 

Note that the truth of the premises would not guarantee the conclusion; hence, the argument is not deductive. Moreover, the conclusion is not meant to explain why the premises are jointly true. So it isn’t abductive either.

 

Some logic books, however, classify arguments by analogy a type of inductive argument. I myself think this is backwards: Inductive arguments are a type of argument by analogy, if induction assumes that the unknown cases will be like the known cases. But even ignoring that, it seems best not to classify arguments by analogy as inductive. That’s because normally when logicians speak of induction, they do not have analogical reasoning in mind. (And conversely, they are not normally thinking of induction when they talk of analogical reasoning.)

 

Thus, I’ve put arguments by analogy in the “other” category. But unlike the just-plain-awful arguments, it is not obvious whether the watch-argument (for example) is a bad argument. Its worth would depend on how appropriate the analogy is in (P2)—and specifically, whether the universe is similar in the right way to a watch. I’ll let the theologians among you decide that one. But generally, an argument by analogy is a good argument to the extent that the analogy is a “tight” one (to put it roughly).