Equivocation

The clean plate

Parent: You can have dessert when your plate is clean.

(Ten minutes later, the child brings back a plate rinsed under the tap, food scraped into the bin.)

Child: My plate is clean. So I get dessert.

“Clean your plate” meant eat what’s on it; the child’s “clean” means washed. The first premise uses one sense and the second premise the other, so the conclusion only follows if you let the word switch meanings halfway. This is the clear-cut case: the child knows exactly what was meant, and the shift is deliberate.

“Significant” results

An office manager reads a vendor’s report: “In our trial, typists using the new keyboard were significantly faster.” She writes to her boss: “The difference is significant, so switching all twelve workstations will make a real difference to our output.”

In the report, “significantly” is almost certainly the statistical sense: the difference is unlikely to be chance alone. In the email it means large or important. A difference can be statistically significant and still tiny. The manager isn’t trying to mislead anyone. The two senses are related and the word is the same, which is what makes this kind of shift easy to miss and makes it sound like an accurate summary.

Unlimited, then limited

A phone plan is advertised as “unlimited data.” After 50 GB in a month, the customer’s connection slows to a crawl. The company replies to the complaint: “Your plan is unlimited. We never cut off your data; we only manage speeds after 50 GB.”

In the advertisement, “unlimited” invites the reading use as much as you like. In the reply it means only never cut off completely. Each sense does a job: the strong one sells the plan, the weak one defends it. The equivocation runs across the whole exchange rather than inside one syllogism, which is also the structure of a Motte and bailey.

• Completely different meanings (homonyms): “bank” as a riverside or a lender. These are easy to spot, which is why they turn up mostly in textbook examples and jokes.
• Related meanings (polysemy): “significant”, “light”, “free”, “natural”. The Stanford Encyclopedia’s entry on ambiguity notes that lexical ambiguity is easy to detect when the meanings are wildly distinct, and harder when they are closely related. This is where real equivocations live.
• Technical and everyday senses: a word with a precise meaning in a field (“significant”, “normal”, “error”) slides into its ordinary sense, as in the keyboard example.
• Descriptive and evaluative senses: a word keeps its praise or blame while its content changes. When that happens through an explicit definition, it’s a Persuasive definition.
• An abstract noun taken as a concrete thing: the Reification entry notes that some sources file reification under ambiguity, as a word taken to refer to a different kind of thing than it does.

Equivocation and amphiboly. Both are fallacies of ambiguity. Hansen’s entry sums up the classic distinction: equivocation exploits the ambiguity of a term, amphiboly the indefinite structure of a sentence. “Semantical” ambiguity in one case, “syntactical” in the other. As the Amphiboly entry explains, an equivocation is also usually the arguer’s own doing, while an amphiboly usually involves misreading a sentence someone else wrote.

Equivocation and four terms. When the shifting word is the middle term of a syllogism (the term that links the two premises), the argument also has four terms instead of three, and the two labels meet. They aren’t the same thing: an argument can have four terms with no ambiguous word at all, and a word can shift meaning in an argument that isn’t a syllogism, as in the phone plan example. See Fallacy of four terms for how the sources disagree about filing the two.

History. Aristotle’s Sophistical Refutations puts ambiguity of words first among six ways that language can make a bad argument look good, with examples that turn on a shift in a single word, such as an argument that evils are good because what is needful is good and evils are necessary. A familiar modern example, used in Hansen’s entry, is “The end of life is death. Happiness is the end of life. So, death is happiness”, where “end” first means ceasing and then purpose.

• The word keeps one sense throughout. A word with many dictionary meanings only causes trouble if the argument actually uses two of them.
• No conclusion rests on the double meaning. Puns, jokes, slogans and poetry use two senses at once on purpose. That’s wordplay, not an argument.
• The shift is announced. “Significant in the statistical sense, though not in the practical one” uses both senses and says so.
• Both senses support the conclusion. If the argument works on either reading, the ambiguity does no harm.
• The meaning is general, not ambiguous. Some words have one broad meaning that applies to different cases (“in” a car, “in” trouble). Sennet’s entry reports that telling ambiguity from a general meaning is difficult, and quotes Saul Kripke’s warning against claiming an ambiguity just to escape a counterexample. Crying equivocation can itself be a dodge.

The test: restate the key word in plain words each time it appears. Are the restatements the same, and if not, does the argument still work?

Positive news

A doctor calls with test results: “Good news. Your strep test came back negative, which is about as positive as a result gets. You can go back to work tomorrow.”

“Negative” and “positive” are used in two senses in one breath, the clinical and the everyday. But the conclusion (go back to work) rests only on the test result, not on the pun. This is wordplay with no conclusion resting on the double meaning. Swap the joke for a plain sentence and the reasoning is unchanged.

Statistically significant, and saying so

The vendor’s analyst writes: “The speed difference was statistically significant, so it probably isn’t just noise. It’s also only about two words per minute, so whether it’s worth replacing keyboards is a separate question.”

This uses the same word as the office manager’s email, in a technical sense that is easy to confuse with the everyday one. But the analyst keeps the sense fixed and announces the difference: the first conclusion (probably not noise) is exactly what statistical significance supports, and the practical question is kept apart. Compare the office manager, whose conclusion needed the other sense.

The eye and the ear match words, not meanings. When the same word appears in two premises, the connection seems to be made before anyone checks what the word means each time. The more closely the two senses are related, the more natural the slide feels, which is why technical terms with everyday cousins are such a common source.

Ambiguity is also ordinary. Most words have several senses, and listeners usually pick the right one from context without noticing a choice. That same fluency hides the moment when context changes and the sense changes with it. And a shift can be useful to the arguer, as in the phone plan example: the strong sense persuades and the weak sense defends, so there’s often a quiet incentive not to notice.

• Restate the word both times. “When you say the plan is unlimited, do you mean I can use as much data as I like at full speed, or that you won’t cut me off?”
• Try the conjunction test. Sennet describes a standard check for ambiguity: put both uses in one sentence. “The colors and the feathers are light” sounds wrong, which shows “light” has two meanings. “The trial result and the effect on our output are both significant” should sound just as odd.
• Ask which sense the conclusion needs, then check whether the premise is true in that sense.
• Remember that a broken argument isn’t a false conclusion. The new keyboards might speed up the office after all; the report just didn’t show it (see Valid vs. true).