Quantifier Translations

Remember the Aristotelian forms, how to recognize them in English, and what the
standard translation for each of them is. **Most** sentences involving
quantification (not all!) will have one of the
four Aristotelian forms as their overall structure. Sometimes the subject
property and/or predicate property of the overall sentence will also have one of
the Aristotelian forms as its structure.

**Aristotelian forms:**

Type |
English Pattern |
Translation |

Universal Affirmative | All A are B Every A is a B |
∀x (A(x) → B(x)) |

Particular Affirmative | Some A is B There is an A that is B |
∃x (A(x) ∧ B(x)) |

Universal Negative | No A is B Every A is not B |
∀x (A(x) → ¬B(x)) |

Particular Negative | Some A is not B There is an A that is not B |
∃x (A(x) ∧ ¬B(x)) |

**General strategy:**

1. Identify the overall form of the sentence. Usually this will be one of the four Aristotelian forms.

Example: "Every cube is larger than something else." This is a universal affirmative. So the translation will have the form ∀x (A(x) → B(x)).

2. Identify the subject property A(x) and the predicate property B(x). If they don't contain additional quantifiers, go ahead and translate them.

In our example: A(x) is "x is a cube" and B(x) is "x is larger than something else." A(x) doesn't contain any quantifier expressions, so we can just go ahead and translate it as Cube(x).

3. If A(x) or B(x) contains one or more additional quantifiers, leave it in quasi-English but express it as a property of an unknown object x (or y or another variable).

B(x): x is larger than something else. The predicate, B(x), contains a quantifier expression ("something"), so we leave it in quasi-English for the time being.

4. If you left A(x) or B(x) in English, rephrase to get the quantifier at the beginning.

Continuing with the previous example: B(x) was "x is larger than something else." Rephrasing to get the quantifier "something" at the beginning, we get: "something else is such that x is larger than it."

5. Now translate the rephrased English A(x) or B(x) into symbolic notation. (Often it will help to do this by identifying its Aristotelian form.)

"Something else is such that x is larger than it": particular affirmative. Subject property, A(y): y is something else, that is, y is something other than x. Translation: y # x. Predicate property, B(y): x is larger than it, that is, x is larger than y. Translation: Larger(x, y). Complete translation: ∃y (y # x ∧ Larger(x, y)).

6. Once you have A(x) and B(x) completely translated, plug them back into the standard translation for the general form you identified in step 1. (By "plug them back in" I mean: starting from the general form, replace A(x) by your translation for the subject property, and replace B(x) by your translation of the predicate property.)

In this example: We have a universal affirmative, ∀x (A(x) → B(x)).

Plugging in our translations for A(x) and B(x), we get:

∀x (Cube(x) → ∃y (y # x ∧ Larger(x, y))).

Curtis Brown | Symbolic Logic I | Philosophy Department | Trinity University

cbrown@trinity.edu