The Certainty of Logic
Basic logic uses quantifiers like 'for all' and 'there exists' to make precise statements about scope and truth, overcoming the ambiguity of everyday language.
THE IDEA
THE CERTAINTY OF LOGIC
In basic logic, a statement has a specified scope: the people or things it concerns. Two precise ways to qualify a statement within that scope are:
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The statement is true of everything in your world. Perhaps “all mathematicians are awkward”.
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The statement is true of at least one thing in your world. Perhaps “there is at least one mathematician who is friendly”. (I hope I count as this.)
Compare these two statements:
Everyone in the US is obese. Someone in the US is obese.
The basic statement concerns being obese. “In the US” narrows the scope from the entire world to just the US, and then we say whether we’re talking about everyone in that scope, or just someone, at least one person. If two people are obese it is still true that “someone is obese”.
As usual, the way we turn this into formal language is a bit tricky, because we need something that is more rigid than our fluid, flexible, spoken language. Formally these two types of statement would be rendered using “for all” and “there exists” like this:
For all people X in the US, X is obese. There exists a person X in the US such that X is obese.
This sounds terribly pedantic in normal language, but makes things easier to manipulate in formal mathematics. “For all” and “there exists” are called quantifiers in mathematics; they quantify the scope of our statement.
When to use it
In a scientific experiment, when stating a hypothesis, use quantifiers to specify whether a result applies to all subjects or only some.
In a debate, identify when someone is overgeneralizing by asking whether they mean 'all' or 'some'.
In policy making, precise language with quantifiers prevents misinterpretation of who is affected.
When writing mathematical proofs, correctly applying 'for all' and 'there exists' ensures logical rigor.
FROM THEORY TO EVERYDAY LIFE
In practice
Mathematician Quantifiers
The statement 'All mathematicians are awkward' uses the universal quantifier, meaning it applies to every mathematician. In contrast, 'There exists a mathematician who is friendly' uses the existential quantifier, meaning at least one mathematician satisfies the property.
Obesity in the US
Compare 'Everyone in the US is obese' (universal) and 'Someone in the US is obese' (existential). The first claims all people in the US are obese, while the second only requires at least one.
A MOMENT TO REFLECT
Put it to work.
How can you apply quantifiers to clarify ambiguous statements in everyday arguments?
Think of a time when someone made a universal claim that was actually only true for some. How would you rephrase it using quantifiers?
What are the implications of using 'for all' versus 'there exists' in scientific hypotheses?
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)