I love logic. That is probably because, as my parents would tell you, I have always loved to argue. Some things never change. But, by the grace of God in me, as he has sanctified me over the years I have learned to love to argue about things that really matter; things which deserve to be argued for or against. What was prone to be more like sinful quarreling has been transformed into an attempt to use reasoned persuasion for what is true, good, and beautiful.1 Christians need to learn logic and put it to service for the kingdom of Christ (for our own good and for that of our neighbor).
There is much which I enjoy in the study of logic, from fallacy detection, to syllogism making, to analyzing the validity of arguments via truth tables, etc. Today I want to share with you a very powerful argument form known as the Constructive Dilemma. Dilemmas have at least two major contexts wherein it may be important to know their nature and workings. The first is the context of life itself. The second is debate, whether on a stage with a live opponent or in written word on the page.
A dilemma is a problem wherein a person finds themselves stuck between two opposing possibilities, neither of which are desirable.
For instance, there have been occasions in the past (whether this practice still occurs or not I do not know) where a person who has been apprehended for a crime was faced with either going to prison or being forced into military service. Presumably the preference of the individual is to be free but that option is not available to him. He has to pick between these two options. What might his thought process be?
Let’s suppose the prison sentence would be for 20 years. On the other hand let’s suppose the military service would only be for 8 years. The convicted might reason thus:
If I take the prison sentence then I will lose many more years of freedom than if I pick the other option. However, if I take the military service and am sent into combat then I might die and live only a very short life (and life in prison is better than no life at all). I will either pick prison or military service, therefore I will either lose many years of freedom or I may die early.
Obviously the individual facing such a circumstance would have to weigh carefully his options and decide what he thinks is his better of the two options. It is a difficult choice for him precisely because neither option is desirable to him.
Let’s take another instance that might occur for someone who is not a criminal (someone like you).2
Suppose that you are at work and you come to realize that one of your fellow employees has been stealing from the company. This person is your friend but what he is doing is obviously wrong and harmful to your company and the well being of all its employees. Should you turn them into your boss? You might reason thus:
If I turn this person in to our boss then my friend will be fired and maybe even go to prison. If I don’t turn them in then I will be complicit in something immoral and our company will suffer. I either will or wont turn them in. Therefore my friend will be fire and maybe go to prison or I will be complicit in something immoral and our company will suffer.
What you hopefully see about both examples is that the structure of each dilemma is essentially the same. In logic we talk about schematizing arguments. What this means is that we can examine the underlying structure of an argument without reference to the specific content. We might liken this to looking at the skeleton of a person and ignoring the flesh.
So let’s get morbid with our arguments and boil off the flesh.
The basic form of a Constructive Dilemma, when symbolize as a schema, looks like this:
(p⊃q) ᐧ (r⊃s)
p v r
∴ q v s
Now let’s break this down some more and explain a few things. If you have no experience with Propositional Logic (a.k.a. Symbolic Logic) then there are some things which will want explaining immediately, namely the symbols themselves.
To begin with, lower case letters are variables, just like in algebra. Let “p” stand for whatever you like. Upper case letters, on the other hand would be used to represent an actual and particular argument. The sideways “horseshoe” symbol indicates a hypothetical statement (an if/then statement). The symbol itself is called the conditional. So p⊃q is read as “if p then q”. Hence, “If it is raining outside then the driveway will be wet” could be symbolized as R⊃W (R=Rain, W=Wet) if I want to represent this particular statement or it could be symbolized as p⊃q if I want to represent any possible statement with the same form/structure. The letter which comes before the ⊃ symbol is called the antecedent and the letter which follows the ⊃ symbol is called the consequent.
The middle dot “ᐧ” represents a conjunction which joins two statements together. Words like and, but, or still are examples of conjuncts which would be symbolized with the middle dot. Example, “The dog is running and the cat is hissing.” Two simple sentences, “The dog is running.” and “The cat is hissing.” are joined into a compound statement with the word and. To symbolize this particular compound statement I can symbolize it as D ᐧ C or to represent any compound statement of this form I can symbolize it as p ᐧ q.
Finally the “v” is the symbol used for a disjunct, the most common of which is the word “or”. The compound statement “I will go to the movies or I will go to dinner.” can be symbolized as M v D. If I want to symbolize the universal form of this statement, rather than the particular case, I can symbolize it as p v q.
Finally the ∴ symbols stands for conclusion words like therefore, so, and consequently, etc.
Now you know what is needful to continue on in our discussion of dilemmas.3
The Constructive Dilemma is actually a combination of two instances of a more basic argument known as modus ponens. These arguments begin with a hypothetical statement for their first premise, such as, “If it is raining outside then the driveway will be wet.” and then they follow in the second premise with simple categorical statement which corresponds to the antecedent of the original hypothetical statement in the first premise, in this case, “It is raining outside.” The conclusion can then be drawn from the premises, namely, “Therefore, the driveway will be wet.” Symbolized to fit this particular modus ponens argument we get:
R⊃W
R
∴ W
The schema of any modus ponens is:
p⊃q
p
∴ q
Now if we go back to examine the basic form of a Constructive Dilemma we will notice something interesting.
(p⊃q) ᐧ (r⊃s)
p v r
∴ q v s
The Constructive dilemma is just two modus ponens smushed together. Premise 1 of our dilemma is just a conjunction of two hypothetical statements which each represent the opening of a modus ponens argument. Premise 2 then says we must affirm either the antecedent of the first hypothetical or the antecedent of the second hypothetical. The conclusion then tells us that we will necessarily end up with either the consequent of the first hypothetical or the consequent of the second.
So that’s how it works. Sometimes in life these dilemmas arise naturally and we must take them on as best we can (but I would not leave you unarmed). There are three ways we may attempt to rescue ourselves from what logicians call “the horns of the dilemma”. We can attempt to grasp the horns, or we may try to go between the horns, or we may seek to rebut the horns.
Let’s set up another situational dilemma that will serve to illustrate these three means of escaping a dilemma.
If I fail my math test then my parents will ground me but if I ace my math test then my friends will think I am a huge nerd. I will either fail my math test or I will ace it. Therefore my parents will either ground me or my friends will think I am a huge nerd.
(F⊃G) ᐧ (A⊃N)
F v A
∴ G v N
Grasping the Horns of a Dilemma
To grasp the horns of a dilemma means to take on one, or both, of the hypotheticals in the first premise by denying their truth and giving an explanation. In our sample case, for instance, I could deny the first hypothetical, “If I fail my math test then my parents will ground me.” I could say “This is not true, my parents are kind and generous people who know that I will do my best and that we can’t always succeed at everything we try. They will not ground me if I happen to fail this test as long as I try my best.”
Likewise I could grasp the other horn and say something like, “My friends won’t think I am a nerd for doing well, they appreciate intelligence and they strive to do well themselves.”
In either case, I can simply invalidate the dilemma by showing that some part of its premises are false.
Going Between the Horns of a Dilemma
To go between the horns of a dilemma means to call out what is known as the “either/or fallacy”. The second premise of our dilemma insists that I will either fail or ace my test. Clearly, however, there is a third option. I could do moderately well on my test without “acing it” or failing it. So this dilemma is faulty because it employs a common fallacy of trying to force a person into a “false dilemma”.
Rebutting the Horns of a Dilemma
This third approach is often the most humorous way to try to defeat a dilemma. What it does is simply rearranges the material of the dilemma to make it sound more positive. In this case we could restate the argument like this:
If I fail the test then my friends won’t think I am a huge nerd. If I ace the test then my parents won’t ground me. I will either fail the test or I will Ace it. Therefore I will either not be grounded or my friends won’t think I am a huge nerd.
I will lay out the argument and counter-argument in linear fashion so it is easier to see what I’ve done here.
Original Argument: (F⊃G) ᐧ (A⊃N) F v A ∴ G v N
Counter Argument: (F⊃~N) ᐧ (A⊃~G) F v A ∴ ~N v ~G
In the counter argument, in the first premise the antecedent stays the same in both hypothetical statements but the consequent of both hypotheticals is swapped and negated (~). The second premise remains the same as in the original argument. The conclusion now represents the swapped consequents in premise one.
It is important to understand that some dilemmas are strong and others are weak. The stronger a dilemma is the less it is open to these three kinds of responses. Some dilemmas are real and therefore cannot be denied as an either/or fallacy. Some hypothetical statements are true and therefore cannot be denied as false, and some counter arguments really don’t work to our advantage.
When it comes to situational dilemmas, those which arise in the course of life, sometimes we find ourselves simply stuck with two bad options and nothing we can do about it but ask God for mercy and grace to deal with it.
But not all dilemmas are situational, some of them are designed to trap us by an opponent (or perhaps the other way around). These kind of dilemmas are those we face, or force others to confront, when we are debating someone or some idea. Socrates provides us with a classic example during his trial in Plato’s Apology.
Plato: And now, Meletus, I will ask you another question—by Zeus I will: Which is better, to live among bad citizens, or among good ones? Answer, friend, I say; the question is one which may be easily answered. Do not the good do their neighbours good, and the bad do them evil?
Miletus: Certainly.
Socrates: And is there any one who would rather be injured than benefited by those who live with him? Answer, my good friend, the law requires you to answer—does any one like to be injured?
Miletus: Certainly not.
Socrates: And when you accuse me of corrupting and deteriorating the youth, do you allege that I corrupt them intentionally or unintentionally?
Miletus: Intentionally, I say.
Socrates: But you have just admitted that the good do their neighbours good, and evil do them evil. Now, is that a truth which your superior wisdom has recognized thus early in life, and am I, at my age, in such darkness and ignorance as not to know that if a man with whom I have to live is corrupted by me, I am very likely to be harmed by him; and yet I corrupt him, and intentionally, too—so you say, although neither I nor any other human being is ever likely to be convinced by you. But either I do not corrupt them, or I corrupt them unintentionally; and on either view of the case you lie. If my offense is unintentional, the law has no cognizance of unintentional offences: you ought to have taken me privately, and warned and admonished me; for if I had been better advised, I should have left off doing what I only did unintentionally—no doubt I should; but you would have nothing to say to me and refused to teach me. And now you bring me up in this court, which is a place not of instruction, but of punishment.
Extracting the relevant data, Socrates argument is thus:
If Socrates corrupts the youth of Athens intentionally then Miletus is a liar. If Socrates corrupts the youth of Athens unintentionally then Miletus is a liar. Socrates either corrupts the youth of Athens intentionally or unintentionally. Therefore Miletus is a liar.
(I⊃L) ᐧ (U⊃L) I v U ∴ L v L (or just L)
This is a very powerful dilemma that Socrates uses against his accuser. What options does Miletus have here? It has been established that either antecedent being true leaves Miletus in the place of being a liar (or at least wrong). There is no either/or fallacy because there is no third option between intentional and unintentional unless Miletus wishes simply to retract the original charge itself that Socrates is corrupting the youth of Athens. The counter-dilemma does absolutely nothing for Miletus here, it just doesn’t work. Check mate, Socrates for the win.
The power of the dilemma is formidable when wielded rightly. Dilemmas of various strength and weakness are to be found all over the place. In the news, in presidential debates, in great literature and history, and even in biblical literature. Once you master the form of the dilemma you can build your own and play with them. They can be quite comical at time or they can be devastatingly powerful weapons. There are other versions too. There is also something called a Deconstructive Dilemma which utilizes the Modus Tollens instead of the Modus Ponens as its base.
Christian’s would do well to know how to wield such arguments for good and also to know how to deconstruct faulty arguments used by wicked men to mislead or even oppress those who don’t know how to see through them. Good use of logic, in the hands of good people, is one way to practice chivalry in our modern world. This is but a small excursion into the realm of logic and I commend you to dive down the rabbit hole as far as you can go.4
Of course I sometimes still get drawn in to petty quarrels, God forgive me.
I mean, probably. Maybe you are reading this from prison, I don’t know.
There are some other symbols in propositional logic which are not here being considered. The ≡ symbol, known as the biconditional, is the sign for logical equivalence. The tilde ~ is the symbol for negation.
Lewis Carroll, by the way, was a pen name. His real name was Charles Dodgson and he was an Oxford logician and mathematician who wrote several books on logic. Even Alice in Wonderland and its accompanying stories is a way of demonstrating logic by intentionally creating humorous fallacies.