The Brain, a User's Manual:
Enough Necessity
By Gabriel Blanchard
Sufficiency and necessity are twins, but not identical twins—though their younger sibling does mysteriously resemble both.
On Three Conditions
As we have said, Stoic logic deals with conditionals and disjuncts. We’ll deal with the former first.
Conditionals take one of three forms: they may be sufficient, necessary, or necessary and sufficient. All conditionals involve an antecedent, which we’re representing with A, and a consequent, B. The antecedent is the condition (which conditionals are named for, of course), while the consequent is what results or follows from that antecedent.
Sufficient: If A, then B (or, rephrased, B if A).
Necessary: B only if A.
Necessary & Sufficient: B if and only if A.
Like E statements on the square of opposition, the phrasing of the second and third conditionals is a little out of step from what we expect. This is just to ensure clarity according to the conventions of English grammar and syntax; natural language isn’t usually as rigorous as logicians would like.
Now: what do they mean?
The Sufficient Conditional
The sufficient conditional claims that if antecedent A, is true, then that fact by itself is enough to ensure that consequent B is also true, without any extra antecedents needed. “If you put a paper airplane in the fireplace, then it will burn up.” “If President Kennedy is assassinated, then it will trigger a third world war.” “If there is an all-powerful God, then evil cannot exist.” All of these—whether true or false or unknowable—are assertions that the first statement is sufficient to make the second true.
Establishing that this relationship between A and B really is the case is more difficult, and (as we went over last week) generally has to be based on observation, which is a distinct procedure from logic. However, if this is achieved, two arguments are thus opened.
One is called modus ponens, or “the method of setting something up.”1 The modus ponens syllogism goes like this:
If A, then B.
A.
∴, B.2
Pretty straightforward! But there’s also way to use the sufficient conditional backwards, so to speak. This is what’s called the modus tollens, “the method of taking something down.”
If A, then B.
Not B.
∴, not A.
You can prove anything you want by coldly logical reason—if you pick the proper postulates.
Modus tollens is often part of a broader form of argument called reductio ad absurdum, “reduction to absurdity.” This is sometimes confused with the slippery slope fallacy; the difference is that, in the case of the fallacy, the slope is not really as slippery as the arguer is maintaining, while in the case of modus tollens, the untruth of B really does logically involve the untruth of A.
Insufficiencies
There are a couple of typical fallacies based on the sufficient conditional, however: these are denying the antecedent and affirming the consequent. They take these forms (shown in red, because they are fallacious):
If A, then B.
Not A.
∴, Not B.
If A, then B.
B.
∴, A.
The reason that denying a sufficient conditional’s antecedent or affirming its consequent don’t work lies in the distinction between different kinds of conditionals. Here, remember, we are working with sufficient conditions for a result, not necessary ones. “If A, then B”—but some other antecedent (call it X) might also be enough to bring B about: “If A, then B, but also if X, then B,” and disproving A does nothing to disprove X. Affirming the consequent fails for exactly the same reason. (The fact that these names are used for these fallacies, without directly mentioning that they are fallacies specifically in the context of sufficient conditionals, is only a convention.)
The Necessary Conditional
The necessary conditional flips the details of the sufficient conditional around. That conditional makes antecedent A enough to guarantee consequent B; the necessary conditional states that, while the antecedent may not be enough to guarantee the consequent, the consequent cannot be true without the antecedent.
This too gives us two valid forms and two fallacies. The valid forms are:
B only if A.
B.
∴, A.
B only if A.
Not A.
∴, not B.3
And the fallacious forms:
B only if A.
Not B.
∴, not A.
B only if A.
A.
∴, B.
For the fallacies, it’s again important to remember that the terms of the argument are what is necessary to bring about some consequent, not what is enough to bring it about: You cannot make bread without flour, but all the same, flour by itself is not sufficient to make bread.
We’ll address the both-necessary-and-sufficient conditional next week.
1This actually predates the Stoics: Theophrastus, Aristotle‘s successor as head of the Lyceum, first described the modus ponens in detail.
2If you’re new to the series or it’s slipped your mind, the symbol ∴ means “therefore.”
3At a glance, this may seem wrong. The first looks like affirming the antecedent and the second looks like denying the consequent—and in a sense, they are, but this is the difference between the sufficient and necessary conditionals. They mirror, and thus reverse, each other’s requirements, a little like contraposition from the square of opposition.
Gabriel Blanchard has worked for CLT since 2019, and serves as the company’s editor at large. He lives in Baltimore, MD.
Thank you for reading the Journal. If you’re just discovering the Brain User’s Manual, you can find its full “table of contents” here, or go here for its predecessor series on identifying informal fallacies. You might also enjoy our series on “the Great Conversation,” the grand intellectual exchange that has been going on since the Bronze Age: we have posts on beauty, definition, law, the four loves, signs and symbols, and many more topics. Happy reading.
Published on 16th January, 2025. Page image of John Tenniel’s illustration of Tweedledum and Tweedledee for Lewis Carroll’s Through the Looking-glass: And What Alice Found There. Author thumbnail of the manticore from Edward Topsell’s 1607 book The Historie of Foure-Footed Beastes.