The Confusing, Irritating, and Downright Annoying
Counterfactual Conditional
A Temporary Resolution to a Logical Conundrum
The Conditions That Prevail
"If"
- The Spartans reply to Phillip of Macedon's statement, "If I win this war, you will be slaves forever."
If you ever cracked a book on philosophy, you'll probably find discussions about the logical construction called the conditional. And you'll learn the conditional is 1) absolutely necessary for proper philosophical reasoning and 2) infuriating.
And what, you ask, is the conditional?
A conditional is nothing more than an IF-THEN statement. As anyone who has studied logic knows, you encounter IF-THEN statements all the time.
One of the reasons for their ubiquity is that they play an important role in deductive reasoning. For instance, let's say you have a premise.
If I think, then I exist.
(Which is a conditional.)
... and you also know that:
I think.
Therefore you can conclude:
I exist.
This type of construction can be succinctly put as an argument:
| Premise 1 | A → B | If I think then I exist. | |
| Premise 2 | A | I think. | |
| ---------- | |||
| Conclusion | B | I exist. |
This particular construct is called modus ponens. If you know that A → B is TRUE and A is TRUE, then we conclude B is TRUE.
Simple, no?
Well, no.
"Just Counter the Facts, "Ma'am"
Consider the statement:
If Pat Garrett did not shoot Billy the Kid, then someone else did.
This type of IF-THEN statement is called an indicative conditional. "Indicative" means there is nothing here that goes against accepted facts.
Now most people accept this statement to be TRUE. In fact, it seems obvious. So what's to debate?
Well, then, let's look at this variant:
If Pat Garrett had not shot Billy the Kid, then someone else would have.
Note that this statement, even if it recognizes that Pat Garrett really did shoot Billy the Kid, it nevertheless postulates for the sake of the argument that he did not. Because the IF-part of the sentence postulates something contrary to known facts, this type of conditional is called the counterfactual conditional.
And what, you ask, is the beef?
The beef is that to a philosopher, counterfactual conditionals are a pain. That's because a good philosophical statement should be TRUE or FALSE. And it's hard to see that this statement is one or the other.
But most of all, philosophers - at least traditional philosophers - like their sentences to be TRUTH FUNCTIONAL. This means that the truth or falsity of a complex sentence - and "IF A THEN B" is considered complex - depends only on the truth or falsity of the individual parts.
So if "IF A THEN B" is truth functional then it is TRUE or FALSE depending only on whether A and B are TRUE or FALSE. But again it's hard to see how the counterfactual conditional is truth functional.
However, without meaning to contradict thousands of philosophers, it may be that they are worrying needlessly. Instead let's see what happens if our two conditionals - the indicative and the counterfactual - are analyzed by classic philosophical logic.
So if worrying about the TRUTH of the conditionals has you wringin' your hands and cryin' so that the blues come fallin' like midnight rain, read on!
The Easy Part
Pat Garrett and Billy the Kid
First, let's take a relook at our first statement:
If Pat Garrett did not shoot Billy the Kid, then someone else did.
Now if you were to ask Joe or Josephine Blow on the street if this statement is TRUE, they'd probably snort and of course, it's TRUE. It's obvious.
But a philosopher will wag a finger and ask, can you prove the statement is TRUE?
And if you can prove the statement, what profound conclusions can you draw?
First it is a big mistake - and an error sometimes made even by university level teachers - to ask if the statement is TRUE. Instead, there is more to it than that.
An isolated statement can only be classified as TRUE or FALSE if:
- It is a PREMISE of an argument.
- It is a CONCLUSION of an argument and PROVEN to be TRUE.
Of course, if the statement is a PREMISE, that's that - at least within the context of the argument.
But if the statement is a CONCLUSION, you now have to ask two more questions:
- Is the ARGUMENT valid?
- Is the ARGUMENT sound?
As to what we mean by validity and soundness we'll get to that in a minute.
But if the statement:
If Pat Garrett did not shoot Billy the Kid, then someone else did.
... is indeed the CONCLUSION of an argument, there must be PREMISES. And as before we can construct the argument as follows:
| Premise(s): | ? |
| ---------- | |
| Conclusion: | If Pat Garrett did not shoot Billy the Kid, then someone else did. |
Fortunately, in our argument, there's only one premise and an obvious one at that. And that's why it's often left out.
| Premise 1 | Someone shot Billy the Kid. |
| ---------- | |
| Conclusion | If Pat Garrett did not shoot Billy the Kid, then someone else did. |
At this point, we will now lapse into what is known as formal logic. Formal logic was developed by the "analytic" philosophers of the 19th and early 20th centuries. Although George Boole - of "Boolean Algebra" fame - began the work in the 1850's, it was Bertrand Russell and his buddies who gave us the formal logic analysis we have today.
"Formal logic" does not mean you hold philosophical discussions wearing tuxedos or evening gowns. It means you represent natural language statements as formulas. And although there are tons of textbooks on formal logic, we'll just explain things as we go along.
First of all, we need to define some logical symbols, or as they are often called, the logical connectives.
| Logical Connectives | ||
| Symbol | Official Name | Plain English |
| ¬ | Negation | "Not" "It is false that" |
| ∧ | Conjunction | "And" |
| ∨ | Disjunction | "Or" |
| ∃ | Existential Quantifer |
"There is", "For some" |
| ∀ | Universal Quantifer |
"For all", "For every" |
| → | Material Implication |
"If-Then" |
Next - at least for our purposes here - we need to create a two part relation, S. This relation maps an assailant - the shooter - to the victim - the shootee.
S: assailant → victim
... which we will write as:
S(assailant, victim)
Now if we want, we can be general and not mention anyone in particular:
S(x, y)
... or we can be specific:
S("Pat", "Billy")
Now this relation on its own has no real tense. It can mean "Pat shot Billy", "Pat shoots Billy", or "Pat will shoot Billy". And as we'll show below there are ways of modifying this relation if you need to specify a tense more precisely.
OK. This is the IF-part of the statement. But what about the second part - the THEN-part?
If Pat Garrett did not shoot Billy the Kid, then someone else did.
Well, we're simply saying that someone shot Billy the Kid (even if Pat didn't). This is most simply written as:
∃x:S(x, "Billy")
...which if you look at the table of symbols you read as:
There exists an x such that x shot Billy.
... which means someone shot Billy the Kid.
The rather stilted rendering of the formula in ordinary language "There exists an x such that x shot Billy" is sometimes referred to as "Loglish".
[Note: At this point you may wonder why we dropped the "else" in the formal representation. We will explain that shortly.]
Now there's some things to know about this existential quantifier. "There exists an x" or "There exists some x" means that there is at least one x. But there can be more.
Also strictly speaking, on it's own the formula ∃x:S(x, "Billy") does not preclude the x being Pat. This possibility, though, is automatically ruled out since Pat is specifically excluded in the IF-part. Hence we don't really need to say "else" in our formal representation.
However, if you do want to restate that Pat did not shoot Billy in the THEN-part - in effect, include the "else" - you can do this. However, you can prove that it is unnecessary and leads to our formal statement without the "else". [Note: To see this proof, click here.]
So using our other symbols, we can write out the following formula:
| ¬S("Pat", "Billy") | → | ∃x:S(x, "Billy") |
From the meaning of the symbols and the relation, S, you can readily put this formula into Loglish as:
If Pat did not shoot Billy, then there exists an x such that x shot Billy.
which simply means:
If Pat Garrett did not shoot Billy the Kid, then someone else did.
... which is our original IF-THEN statement.
OK. But how do we write our premise?
Someone shot Billy the Kid.
Well, that's easy enough. We see that in Loglish this is:
There exists an x such that x shot Billy.
... and the corresponding formula is:
∃x:S(x, "Billy")
Now we can write the ENTIRE ARGUMENT in symbolic form:
| Premise | ∃x:S(x, "Billy") |
| ---------- | |
| Conclusion | ¬S("Pat", "Billy") → ∃x:S(x, "Billy") |
And yes, the premise is the same as the consequent of the conclusion of the argument.
All right. Remember that to prove our statement is TRUE we have to 1) prove our argument is VALID and 2) prove that our argument is SOUND.
And now we can ask what we mean by a VALID and SOUND argument.
First, the definition of a VALID argument is that if the PREMISES are TRUE, then the CONCLUSION must be TRUE.
Next, a SOUND argument, is a VALID argument where the PREMISES are TRUE.
And finally, the CONCLUSION of a SOUND argument is a TRUE statement.
At this point we need to review our classical logic. In logic an IF-THEN statement is called MATERIAL IMPLICATION. And Material Implication has the following TRUTH TABLE.
| Truth Table: If-Then Statements | ||
| A | B | A → B |
| TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE |
| FALSE | TRUE | TRUE |
| FALSE | FALSE | TRUE |
Note that the IF-THEN statement is ONLY FALSE if the IF-part - called the antecedent - is TRUE and the THEN-part - called the consequent - is FALSE. Otherwise, an IF-THEN statement is TRUE.
[Note: For an elaboration of how to derive this table and its sometime quirks, just click here].
First, we'll look at the question of VALIDITY.
Remember a VALID argument is an argument where if the PREMISES are TRUE then the CONCLUSION must be TRUE.
And if you look at the TRUTH table, our argument is certainly VALID.
Why? Well, as we noted, the PREMISE of the argument, ∃x:S(x, "Billy"), is the same as the CONSEQUENT of the CONCLUSION:
| Premise | ∃x:S(x, "Billy") |
| ---------- | |
| Conclusion | ¬S("Pat", "Billy") → ∃x:S(x, "Billy") |
And from the TRUTH TABLE, we know that if the consequent is TRUE, then the whole IF-THEN statement is TRUE.
So we see that whenever our particular PREMISE is TRUE, our CONCLUSION must always be TRUE.
The ARGUMENT is VALID.
That's well and good. But now we ask if our argument is SOUND and therefore our CONCLUSION TRUE?
Well, we have already proven the argument is VALID. So the only question remaining is whether our premise is TRUE.
"What is truth?" is a question that was once posed by an Italian politician. And there is a whole field of philosophy called epistemology that's dedicated to determining how a statement is really TRUE rather than just believed to be TRUE. Needless to say, people are still arguing over the matter.
But for any given argument things aren't quite so glum. As long as the disputants agree that the premises are TRUE, then from their point of view a valid argument is SOUND. And for all practical purposes that's good enough.
And there are three general ways to get people to agree premises are TRUE. These are:
- You believe the premises are TRUE because you want them to be TRUE. That's actually a pretty rotten way to decide if something is TRUE or FALSE but it's still a quite popular way to do it.
- A premise can be true by defintion. These type of premises are common when talking about mathematical proofs.
- Finally and what is probably the best way, a premise is accepted to be TRUE by gathering and evaluating the empirical evidence.
And it's the last method that we use for our argument. By analyzing all the evidence, testimony, and documentation, historians of the Old West agree that yes, someone did indeed shoot Billy the Kid. So our argument is indeed SOUND and the conclusion of the argument - which is our indicative conditional:
If Pat didn't shoot Billy, then someone else did.
... is indeed TRUE.
So much for our proof. But now the question is, what further conclusions can we draw from our now PROVEN and TRUE statement?
To answer that question we return, not to the thrilling days of yesteryear, but to back to the Truth Table for the IF-THEN statements. Since we've proven our statement is not just TRUE, but also has a TRUE consequent, we can grey out the rows of the Truth Table where the consequent is FALSE.
| Truth Table: If-Then Statements | ||
| A | B | A → B |
| TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE |
| FALSE | TRUE | TRUE |
| FALSE | FALSE | TRUE |
And we see that we are left with two acceptable possibilities for the antecedent, ¬S("Pat", "Billy"). It can be TRUE or FALSE.
¬S("Pat", "Billy") = TRUE or FALSE
And we can express this in formal logic as:
S("Pat", "Billy") ∨ ¬S("Pat", "Billy")
... which in plain English is:
Pat did shoot Billy or Pat did not shoot Billy.
And what, as Flakey Foont asked Mr. Natural, does it all mean?
First, you may know that this type of sentence - using a conjunction to link a statement with its negation:
A or ¬A
...is one type of logical tautology. A tautology is a statement that is always TRUE whether the individual components are TRUE or FALSE.
We see then that the conclusions we can draw from a statement like "If Pat Garrett did not shoot Billy the Kid, then someone else did" is automatically TRUE. In other words, all we can conclude from our indicative conditional argument is that either Pat shot Billy or he did not. But we knew that already.
Yes, after all that work, we come to the conclusion that the indicative conditional - at least the type we studied here - is pretty much useless.
And from this, you ask, philosophers make a living?
Well, not exactly. Remember, we have that other IF-THEN statement:
If Pat Garrett had not shot Billy the Kid, then someone else would have.
... to worry about.
Now as the $500 an hour psychiatrist said, we can begin, yes?
The Hard Way
So we've learned that the indicative conditional:
If Pat Garrett didn't shoot Billy the Kid, then someone else did.
... is something most people intuitively understand. And we've seen any conclusions from it are trivial and in fact useless.
On the other hand, we mentioned that the alternative statement:
If Pat Garrett had not shot Billy the Kid, then someone else would have.
... which is a counterfactual conditional, causes a lot of consternation.
What makes philosophers rend their garments in frustrations is that they can't tell if the counterfactual conditional is TRUE or FALSE. And you will hear some bewailing that such conditionals do not appear to be TRUTH FUNCTIONAL. Remember, a truth functional statement is one whose truth or falsity depends solely on the truth or falsity of the individual parts.
But although you hear loud lamentations about such counterfactual conditionals, you rarely see anyone actually attempt an analysis. So could it be our new sentence is truth functional after all? And so we can prove the statement is TRUE or FALSE as we did above - perhaps by uncovering hidden premises?
Certainly we can at least give it the old college try. And first of all we need to realize that our earlier treatment was too simple - particularly in defining our function, S.
Look at it this way. For the indicative conditional we were only referring to a particular instant in history. This was the night of July 14, 1881, in Fort Sumner, New Mexico, when Pat Garrett shot Billy the Kid.
But our counterfactual conditional makes no such restriction. It says that if Pat had not shot Billy, someone else would have although perhaps at some other place and time.
So our original two part relation, S, where we associated an assailant with a victim:
S(assailant, victim)
... should really be a multivariable relation where we include the time and place.
S(assailant, victim, time, place)
So instead of writing "Pat shot Billy" as:
S("Pat", "Billy")
... we should write:
S("Pat", "Billy", "July 14, 1881", "Fort Sumner")
And our original statement - the indicative conditional - should have been:
| ¬S("Pat", "Billy", "July 14, 1881", "Fort Sumner") | → | ∃x:("x", "Billy", "July 14, 1881", "Fort Sumner") |
But as the reader can easily verify, this expanded argument can be proven to be just as TRUE - and just as trivial - as is the simplified version.
But what about our new statement:
If Pat Garrett had not shot Billy the Kid, then someone else would have.
... which we now see should be restated as:
If Pat Garrett had not shot Billy the Kid in Fort Sumner, New Mexico, on July 14, 1881, then someone else would have shot Billy either then and there or at some other time and place.
So just how do we turn this into a logical formula? We'd really like to know how to do this.
I thought you would as Captain Mephisto said to Sidney Brand. It's very simple really.
Using the table of logical connectives, we quickly see that the rather verbose statement:
If Pat had not shot Billy at Fort Sumner on July 14, 1881, then there exists some x, y, and z, such that x would have shot Billy at y time, and at location z.
... can be cast as the formula.
| ¬S("Pat", "Billy", "July 14, 1881", "Fort Sumner") | → | ∃x∃y∃z: S (x, "Billy", y, z) |
But the more grammatical reader may object. The formula does not represent that our counterfactual conditional in English sentence was in the subjunctive mood. That seems to be contradicting the whole point of separating out indicative conditionals from the counterfactual.
But look what happens if we state the counterfactual formula using the indicative wording in Loglish:
If Pat did not shoot Billy at Fort Sumner on July 14, 1881, then there exists some x, y, and z, such that x shot Billy at time y and at location z.
So although this may not be exactly what you say in ordinary conversation, it still means the same thing as our original counterfactual conditional. In other words, using formal logic we can express a counterfactual conditional using the same basic symbols and formulas as the indicative conditional. We just don't specify an exact time or place in the consequent.
We now have to set up our argument as we did before. We have to set up our PREMISES and CONCLUSION.
Here we come to the major difference between the indicative and counterfactual conditionals. Remember that the indicative conditional is non-commital on whether the antecedent is TRUE or FALSE. It is literally a "either-or" statement. Therefore when we proved the argument was valid (and sound), the antecedent did not need to be listed as a specific premise.
But what about the counterfactual conditional? Well, here things get a bit confusing.
First from the grammar, we know the counterfactual conditional accepts the fact that Pat shot Billy.
| If Pat HAD NOT shot Billy | ≡ | Pat really did shoot Billy. |
But on the other hand, for the sake of the argument, we are hypothesizing that he did not.
| If Pat HAD NOT shot Billy | ≡ | Pat really did shoot Billy but what if he didn't? |
So for our argument are we assuming he did or did not?
Remember the name. It's the counterfactual conditional.
The IF-THEN statement is so named because the well accepted fact that Pat did shoot Billy is known to be TRUE. But somewhat contradictorily this FACT is actually outside the framework of the present argument. That is, for our argument we are saying Pat did not shoot Billy
| Premise | ¬S("Pat", "Billy", "July 14, 1881", "Fort Sumner") |
So our entire - and yet partial - argument is:
| Premise 1 | ¬S("Pat", "Billy", "July 14, 1881", "Fort Sumner") |
| Premise 2 | ? |
| ---------- | |
| Conclusion | ¬S("Pat", "Billy", "July 14, 1881", "Fort Sumner") → ∃x∃y∃z: S (x, "Billy", y, z) |
So one of the premises is the same as the antecedent. And so we now look at the TRUTH TABLE for the IF-THEN statement with TRUE antecedents:
| Truth Table: If-Then Statements | ||
| A | B | A → B |
| TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE |
| FALSE | TRUE | TRUE |
| FALSE | FALSE | TRUE |
OK. But we notice now were are indeed at least one premise shy. That is, our counterfactual conditional also presupposes that it was inevitable that Billy the Kid would be shot. That is, it assumes someone, at someplace, and at sometime would shoot Billy.
∃x∃y∃z: S (x, "Billy", y, z)
... and the whole argument is:
| Premise 1 | S("Pat", "Billy", "July 14, 1881", "Fort Sumner") |
| Premise 2 | ∃x∃y∃z: S (x, "Billy", y, z) |
| ---------- | |
| Conclusion | ¬S("Pat", "Billy", "July 14, 1881", "Fort Sumner") → ∃x∃y∃z: S (x, "Billy", y, z) |
We see, then from the Truth Table, that we now have a conditional where both the antecedent and the consequent are TRUE. Therefore the whole IF-THEN statement is indeed TRUE.
TRUE → TRUE ≡ TRUE
And here - at long last - we find what the problem is with the counterfactual conditional. It's not a question if the conditionals are truth functional or not. They are. It's simply people can't agree if this second PREMISE:
∃x∃y∃z: S (x, "Billy", y, z)
... is TRUE or FALSE.
Remember that in the indicative conditional:
If Pat Garrett did not shoot Billy the Kid, then someone (else) shot Billy the Kid.
.. we have no problem accepting that the consequent (and premise):
Someone shot Billy the Kid
... is TRUE.
Not so with the counterfactual conditional.
Certainly some historians will deny it was inevitable that Billy would have been shot by Pat or anyone else. Pat's deputy, John Poe, heard that Billy was at Fort Sumner. Pat, though, thought this wasn't likely and John said it was only at his considerable prodding that Pat agreed to go and look. Had Pat been less accommodating to his deputy, Billy would have headed down to Mexico and disappeared into history. So by this scenario, if Pat had not shot Billy, then no one would have.
But others argue that Billy was going to stick around Fort Sumner. He believed he had friends who would protect him and he had no reason to head to Mexico or anywhere else.
And since he was an escaped fugitive wanted for murder (which now included the killing of two lawmen during his famous escape from the Lincoln County jail), he would never have surrendered. There were also a number of other posses after Billy, including one headed by famed Pinkerton detective Charles Siringo. So if we accept this premise, then Billy would have been shot by Pat, by Charles, or by someone else.
So the problems arises because we ask if the counterfactual conditional is TRUE. That is, does it arise from an argument that is not only valid, but the premises are also TRUE.
First the argument is indeed valid. The conclusion is simply an IF-THEN statement where both the antecedent and consequent are premises. Therefore if the premises are TRUE the conclusion is always TRUE. And so the argument deriving the counterfactual condition is VALID.
As far as the argument being SOUND, we are accepting for the sake of the argument that the premise forming the antecedent of the counterfactual conditional is TRUE. And so we also must accept that the premise forming consequent must really be TRUE, and not just for the sake of the argument.
And there's the problem. The premise forming the consequent is a matter of debate. No one can agree if it's TRUE or FALSE. And so you can't decide counterfactual conditional - at least the type of our example - is TRUE or FALSE.
To sum it all up, people cannot decide if the counterfactual conditional is TRUE or FALSE because the answer lies outside the realm of logical analysis. We'll just have to wait until historians agree on what Billy the Kid's ultimate fate would have been. But the counterfactual conditional itself is perfectly truth functional.
"You can prove anything you want by coldly logical reason - if you pick the proper postulates." - Isaac Asimov (I, Robot).
One thing that emerges from all the brouhaha is the indicative conditionals and the counterfactual conditionals - at least the type we looked at - are essentially the same IF-THEN statement:
¬A → B
However, the actual arguments differ.
| Premises | Indicative Conditional | Counterfactual Conditional |
| Premise 1 | ----- | ¬A |
| Premise 2 | B | B |
| Conclusion | ¬A → B | |
Both of these arguments are valid since if the premises are TRUE, then the conclusion must always be TRUE.
But to determine if the conditional - indicative or counterfactual - is TRUE, then we must know - or at least agree - that the premises themselves are really TRUE.
In the indicative conditional, the only premise, B, is something that most people accept as TRUE. Therefore, there's no problem accepting the argument as SOUND. The conclusion, the indicative conditional, is likewise TRUE.
For our counterfactual conditional, though, the argument hypothesizes that ¬A, is also TRUE. Since the argument is a "what-if" case, whether the premise is TRUE or FALSE in the "real world" isn't important.
But the other premise, B, is something that must be TRUE for the IF-THEN statement to be TRUE. But for the counterfactual conditional, there is considerable disagreement whether B really is TRUE or FALSE. So there is disagreement if the argument is SOUND. Hence it's not possible to determine if the conclusion - our counterfactual conditional - is TRUE or FALSE.
To belabor an already belabored point, we can legitimately analyze our indicative and counterfactual conditionals as truth functional statements with no problem. The problems on determining if the conditionals are TRUE or FALSE arise when people can't agree to accept or reject (often unstated) premises.
Lest we raise the ire of philosophers everywhere, we don't want to imply that the study of counterfactual conditionals is a waste of time. Heaven forfend! After all, we've only dealt with a single example of both the indicative and counterfactual conditionals. There are of course other variants.
But most of all we've also seen that analyzing the individual sentences case-by-case is extremely time consuming. So if philosophers work out general principles common to all counterfactuals, this would certainly be a boon to all humankind.
References
Introduction to Logic, Patrick Suppes, Van Nostrand, 1957.
Formal Logic: Its Scope and Limits, Richard Jeffrey, McGraw-Hill (1981).
"Some Remarks on Indicative Conditionals", Barbara Abbott, Proceedings from Semantics and Linguistic Theory, Vol. 14, pp. 1 - 19, (2004).
"On Conditionals", Dorothy Edgington, Mind, Vol. 104, pp. 235 - 329, (1995).
"Ancient Logic", Stanford Encyclopedia of Logic.
"Truth Tables and the Concepts of Logic", Jack Sanders, Symbolic Logic, Lecture, Rochester Institute of Technology.
The Death of Billy the Kid, John Poe, Houghton Mifflin, 1933.
The Authentic Life of Billy the Kid, the Noted Desperado of the Southwest, Whose Deeds of Daring and Blood Made His Name a Terror in New Mexico, Arizona and Northern Mexico, A Faithful and Interesting Narrative, Pat Garrett, McMillan, 1927, (Original Publication: New Mexico Printing and Publishing Company, 1882).