The Brain User's Manual:
Conjunction
Junction
By Gabriel Blanchard
The necessary-and-sufficient conditional and the disjunctive lie before us; and then ... well, we can burn that bridge when we come to it.
The Third Condition
In our last installment, we dealt piecemeal with sufficient conditionals and necessary conditionals, “ifs” and “only ifs.” We have now to discuss the “if and only ifs”: statements that the antecedent A is both enough by itself to make the consequent B true, and also that B cannot do without A. They take this form:
B if and only if A.1
Conditions which are both necessary and sufficient are uncommon compared with the other two, but in logical terms, they are more useful. This is because one can affirm or deny either the antecedent or the consequent, and in all four cases, it allows a logical deduction to be made—both pairs of fallacies are mutually eliminated from the other two types of conditionals.
B if and only if A.
B.
∴, A.
B if and only if A.
Not B.
∴, not A.
B if and only if A.
A.
∴, B.
B if and only if A.
Not A.
∴, not B.
Disjunction Junction, What Is or Isn’t Your Function?
Handily enough, the multi-directional deductions of the necessary-and-sufficient conditional suggest our final topic in deductive logic: disjunctives, which is just a fancy word for “‘or’ statements.” These come in two kinds: the inclusive and the exclusive. These work a little bit like the subcontrary and contrary relations on the square of opposition.
All disjunctives have the form “A or B”; the A and B in this statement are called the disjuncts. In principle, this logic can be extended to lists with more than two disjuncts; this is how we get not only dilemmas but also trilemmas and all the rest. However, a list of two is the smallest list we can meaningfully reason about—there is neither dis– (which means “put asunder”) nor –junct (which means “what hath been joined”) with a list of only one. In three-or-more item disjunctive statements, you can sometimes find blended statements that are partly inclusive and partly exclusive, but the nuts and bolts of the logic of those comes down to their purer components, expressed separately here.
These papers have afforded me an insight into the lives of two men, which has confirmed my hunch that the external is not the internal. This was especially true about one of them. His external mode of life has been in complete contradiction to his inner life. The same was true to a certain extent with the other also ...
Inclusive Disjunctives
In an inclusive disjunctive, both of the disjuncts can be true simultaneously,2 but a minimum of one disjunct has to be true (if we are any good at building our disjunctives, anyway). As a result, by a procedure a little bit like modus tollens, we can prove the truth of a disjunct by eliminating its rival. However, in an inclusive syllogism, affirming one of the disjuncts will get you no further information, since both can be true; it isn’t exactly a fallacy to affirm a disjunct, but it’s a fallacy to pretend you can draw any conclusions from doing so.
In natural language, logicians tend to use “or” for the inclusive disjunctive, and the default meaning of “or” in contexts like these is the inclusive version. In formal logic, you may see the symbol ∨ used to indicate an inclusive “or.”
Valid Inclusive Disjunctive Syllogisms
A or B.
Not A.
∴, B.
A or B.
Not B.
∴, A.
A or B.
A.
∴, ?
A or B.
B.
∴, ?
Their Semi-Symbolic Representations
A ∨ B.
Not A.
∴, B.
A ∨ B.
Not B.
∴, A.
A ∨ B.
A.
∴, ?
A ∨ B.
B.
∴, ?
Exclusive Disjunctives
These, as you may have guessed (you’re very smart), go the other way from inclusives, though they are not exact opposites: rather, exactly one of the disjuncts must be true, no more and no less; they are mutually, well, exclusive. This aligns a little more closely with the way we tend to use the word “or” in natural speech. This makes them more polyvalent than their inclusive cousins. To mark an exclusive or, logicians normally specify “either A or B.”
Valid Exclusive Disjunctive Syllogisms
Either A or B.
Not A.
∴, B.
Either A or B.
Not B.
∴, A.
Either A or B.
A.
∴, not B.
Either A or B.
B.
∴, not A.
Their Semi-Symbolic Representations
Either A ∨ B.
Not A.
∴, B.
Either A ∨ B.
Not B.
∴, A.
Either A ∨ B.
A.
∴, not B
Either A ∨ B.
B.
∴, not A.
Exclusive disjunctives are also known by another name: dilemmas (or “trilemmas” for mutually exclusive sets of three disjuncts, and so on). False dilemmas are sophistical, of course, but a genuine dilemma, well-expressed, can be a very powerful form of argument, in both the abstract and practical spheres.
In Un-clusion
And that’s it! If you began from the beginning, have read all the way to the end, and stop when this piece stops, then you have completed our introduction to informal logic. Congratulations!
Except … something does still seem lacking, doesn’t it? You’ve seen critical thinking textbooks, and they tend to run for more than a few dozen pages! And even if you combined this with our series on fallacies, both together would only cover tricks of speech used to cover bad arguments and the deductive side of logic. Isn’t there inductive logic, too? How does that work? And how do you go about constructing an argument that’s actually—you know—good?
In ancient Greek poetry and theater, the ode was a specific kind of poem with a three-part structure: A concluding, harmonizing epode followed two contrasting segments, which were called the strophe and the antistrophe. The word strophe meant “turn,” and in drama, a strophe might be sung while the chorus danced toward one side of the stage; then, as they turned their dance back to re-center themselves, they would sing a turning-back, an antistrophe, in the same meter but complementing the strophe by balanced contrast.
Discussing the subject of the book, Aristotle wrote that it was “an antistrophe to dialectic” (i.e., logic) in his slender volume, Rhetoric. We will tackle that next.
1On the whole, we’ve avoided the notation of formal logic in this series to keep things accessible, tossing in a symbol here and there (like ∴). However: a formal symbol for the necessary-and-sufficient conditional is iff with two f’s. This is not only relatively easy to recall, but also rather funny to look at (and was used to great effect here).
2Or, mutatis mutandis, some or all of the more-than-two-options can be true simultaneously. The point is that these disjuncts are not mutually exclusive (because this is an inclusive disjunctive).
Gabriel Blanchard is CLT’s editor at large; he has worked for the company since February of 2019. He lives in Baltimore, MD.
If you enjoyed this piece, you might also enjoy our “great ideas” series. Indexed here, it introduces the many sub-topics of philosophy, science, law, art, history, and religion, from time immemorial to the present day. And speaking of time immemorial, be sure to check out our Texts in Context series too—a crash course in history that puts the men and women and books of the CLT Author Bank in their original setting.
Published on 23rd January, 2025.