The Inverse

This post is part of a four-part series on logical reversal. The truth may lie on the other side or in the other direction, but there is more than one way to reverse a sentence: obverting, converting, inverting, and contraposing are four ways.

Opposite-George

George: […] It became very clear to me sitting out there today, that every decision I’ve ever made, in my entire life, has been wrong. My life is the opposite of everything I want it to be. Every instinct I have, in every aspect of life, be it something to wear, something to eat … It’s all been wrong.

( A waitress comes up to G )

Waitress : Tuna on toast, coleslaw, cup of coffee.

George : Yeah. No, no, no, wait a minute, I always have tuna on toast. Nothing’s ever worked out for me with tuna on toast. I want the complete opposite of tuna on toast. Chicken salad, on rye, untoasted, with a side of potato salad … and a cup of tea.

Elaine : Well, there’s no telling what can happen from this.

The Inverse

Inversion is a traditional logical operation on a sentence: it reverses both the subject and the predicate into their complements.

to-be-precise

Simple inversion can be applied to a categorical sentence such as “All gold is metal,” but the inverse, “All non-gold is non-metal,” isn’t logically equivalent. And here it isn’t true: copper is non-gold but isn’t non-metal.

Aristotelian logic does propose a “partial inverse” as a valid immediate inference. After inversion of a universal statement, the partial inverse is obtained by subalternation of the inverse (reducing its quantity from universal to particular) followed by obversion of the subaltern. This triple manipulation transforms a universal affirmative (A-type) “All gold is metal” into the particular negative (O-type) “Some non-gold isn’t metal.” And it turns a universal negative (E-type) “No cats are invertebrates” into (I-type) “Some non-cats are invertebrates.”

Contemporary logicians have debated whether such conclusions are indeed valid formal inferences. For example, could Aristotle infer from “All even numbers are numbers” that “some odd numbers aren’t numbers”? Or from “No cat is a flying monkey who beats grandmasters at chess” that “some non-cats are flying monkeys who beat grandmasters at chess”?

Even in a less contrived case, E.L. Hicks (1912) argues that “All ruminants are herbivores” doesn’t provide sufficient ground to infer the partial inverse, “Some non-ruminants are not herbivorous.” It may be true, but can it be immediately inferred from the premise?

In the premise we are talking about cows; in the conclusion about lions. How can we infer anything about the food of lions, or any other non-ruminant, from the fact that cows eat grass?

lion and cow

At issue in the contrived cases is the existential import of the partial inverse (its apparent claim that a category has actual instances). Traditional logic has a well-known weakness when it comes to empty sets (or universal sets, whose complements are empty), and Karl Schmidt (1912) argues that the partial inverse of “All A are B” is absurd only where B′ is empty “in the particular Universe of Discourse”; i.e., “Some A′ aren’t B” is a valid inference, provided B′ ≠ Ø.

Thus from “all ruminants are herbivores” it follows that “some non-ruminants are not herbivorous” – or there aren’t any non-herbivores.

Inverted conditionals

In contemporary logic, the simple inverse is applied to conditional sentences, and is not considered a valid inference. The inverse of “If P, then Q” (P → Q) is “If not-P, then not-Q” (¬P → ¬Q). For example the inverse of “If it’s raining, I’ll bring my umbrella” is “If it’s not raining, I won’t bring my umbrella.” While it might seem to be a logical inference, it doesn’t follow by form alone. “If you swallow poison, you’ll get sick” doesn’t imply that “if you don’t swallow poison, you won’t get sick” (as if you could avoid illness simply by refraining from ingesting poison).

Wrongly inferring the inverse of a conditional statement is a fallacy called denying the antecedent, sometimes simply called the inverse fallacy.

In fact, the inverse (¬P → ¬Q) is logically equivalent, not to the original conditional (P → Q), but to its converse (Q → P). For example “if no representation, then no taxation” is equivalent, not to “if representation, then taxation,” but to “if taxation, then representation.” (Americans are insisting on political representation in any body that deigns to tax them, not asking to be taxed by any body that represents them.) The inverse of the converse (or the converse of the inverse) is thus a valid inference – its name is the contrapositive.

Other meanings of the inverse

The inverse, in academic writing, does not always refer to the logical inverse.

An “inverse problem,” for example, is one that seeks to determine causes from their effects: what Smullyan calls retrograde analysis.

E.g., Ian Hacking, in a 1987 article on the “inverse gambler’s fallacy,” describes its relation to the gambler’s fallacy as follows:

The gambler’s fallacy … is the fallacy of someone who reasons that, relative to the evidence of a string of 35 non-double sixes, it is rather likely that a double six will occur at the next roll. But on the assumption of fairness, which I take to include independence of trials, it is not likely. The probability of double six, relative to the evidence, is still 1/36.

I now turn to a second and less familiar fallacy. Its diagnosis seems not to occur in any literature. I call it the inverse gambler’s fallacy.

… Think of a gambler coming into a room, walking to the fair device, and seeing it roll double six. A kibitzer asks, ‘Do you think this is the first roll of the evening? Or have there been many rolls?’ The gambler reasons that since double six occurs seldom, there have probably been many rolls.

In this case, “inverse” indicates (fallacious) reasoning about the past, rather than about the future: an inverse problem rather than a forward problem.

“Inverse” is also used to indicate the converse. Robert Sheldon, e g., in his outline for a 2001 course at the University of Alabama at Huntsville, asks his students for a first paper to consider the impact of science, especially Darwinism, on religion. He then asks for a second paper “to consider the inverse of the first assignment, namely, the impact of religion on science.”

Clearly, by inverse he means converse. Of course, for many statements, the two are logically equivalent, but even considering the sense of the statement (I take inverse and converse to have a different sense even if their truth value is equivalent) there is some justification for this other meaning of “inverse.”

In mathematics, for example, the inverse of a function, f, is another function, f¹, that undoes the first, like the square root undoes the square of a positive number. The square of 5 is 25 and the square root of 25 is 5. Thus the inverse in math is quite close to what might otherwise be called a converse relationship like grandparent/grandchild: If the grandparent of X is Y, then the grandchild of Y is X.

inverse function

But even mathematicians are not entirely consistent in their use of the term. To invert a number, for example, means to turn it over, forming its reciprocal, or multiplicative inverse: thus 710 is the reciprocal of 107.

In this sense of inverse as reciprocal, one can say that two quantities have an “inverse relationship” when their product is constant. For example, if one quantity is tripled, the other will be cut to one third.

Confusion between the mathematical meanings of reciprocal and inverse is also evident in their French translations. The inverse (la réciproque) of the sine function is arcsin, whereas its reciprocal (son inverse) is cosecant.

Two further examples

Ernest Gellner, in a chapter called “The Devil in Modern Philosophy” (2003), outlines Locke’s attack on innate ideas.

In reality, it is only as an effect of our senses that we have ideas: it is only through modification of our brains that we think, will, act. (Sensationalism and physical determinism are fused by him.) From this, a verification principle, in effect, of meaning is extracted and declared to be but the inverse of Aristotle’s dictum about what is in the mind being found first in the senses. If a word or idea refers to nothing sensible, then it is meaningless.

Here, a strict logical inverse may be intended: Locke takes a view of sensation where the sensible modifies our brains and becomes meaningful ideas in the mind (S → M) and adds the logical inverse: what is not sensible cannot become any meaningful idea in the mind (¬S → ¬M).

However, as expressed, it comes closer to a contrapositive: i.e., following Aristotle, what is in the mind is found first in the senses (M → S); therefore, what was not found first in the senses (what “refers to nothing sensible”) is not meaningful, or not in the mind, properly constituted (¬S → ¬M).

Änne Söll’s, in her 2014 article, “Fashion, Media, and Gender in Christian Schad’s Portraiture of the 1920s,” describes the relationship between portraiture and fashion:

Schad’s portraits of fashionable ladies and starlets were also used as covers for magazines such as Die Dame, Jugend, and Sport im Bild, as the portrait of actress Mulino von Gluck from 1930 shows (Plate 23). The employment of portrait painting as covers is indicative of painting’s status as the superior artistic medium. Yet, portrait painting was never considered a medium for fashion but rather the inverse. The portrayal of fashion was thought to be a by-product of portrait painting, since its main aim was to convey the personality and soul of the sitter, not the extravagance or banality of his or her attire.

CHRISTIAN SCHAD FRÄULEIN MULINO VON KLUCK

Perhaps a formal logical inverse is intended:

A → F
If conveying attire was one of portrait painting’s main aims, then fashion could be considered to push the production of portrait painting (as its chosen medium).

¬A → ¬F
If conveying attire was not one of portrait painting’s main aims (but it aimed to convey personality and soul), then fashion could not be considered to push portrait painting; instead, the converse: portrait painting could be considered to push fashion (albeit merely as its byproduct).

Here again, the logic might better be described as the contrapositive, or modus tollens. If fashion drove portraiture as its medium, we’d expect attire to be prominent in portraits (F → A). But portraiture in fact had other chief aims (¬A). Therefore, fashion did not drive portraiture (∴¬F).

But there is also a heavy accompanying sense of conversion as fashion and portrait painting switch sides around which serves or produces the other.

It seems in academic argument there is some breadth of meaning of inverse, and one shouldn’t always understand this polyvalent term to mean the logical inverse. Sometimes a more general turning on its head is intended, sometimes a retrograde analysis, sometimes the contrapositive, and sometimes the converse, especially when it has this sense of a converse relation – as in the mathematical inverse.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s