Translate Each Of The Following Sentences Into First-Order L

Translate Each Of The Following Sentences Into First Order Logic

Translate each of the following sentences into FIRST-ORDER LOGIC:

(a) “Malone does not read every book that is recommended to him.”

(b) “Every painter admires at least one painter.”

(c) “Every solid is soluble in some liquid.”

(d) “Not every river in America is longer than every river in Europe.”

(e) “No number is greater than itself.”

In addition to specifying a non-empty universe of discourse, what is required to give an interpretation of a schema?

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The task at hand involves translating natural language sentences into formal first-order logic expressions. This process requires identifying the relevant predicates, quantifiers, and logical connectives that accurately represent the structure and meaning of each sentence. Additionally, understanding the considerations for interpreting a schema involves recognizing the necessary components beyond the universe of discourse, which include the assignment of meanings to predicates and constants within the domain.

To begin, translating sentence (a): “Malone does not read every book that is recommended to him,” requires defining predicates such as Reads(Malone, x) for “Malone reads a book x,” and perhaps RecommendedTo(x, Malone) for “x is recommended to Malone.” The logical form thus involves a universal quantifier over books recommended to Malone, combined with a negation of the reading predicate:

¬∀x (RecommendedTo(x, Malone) → Reads(Malone, x))

This formula states that it is not the case that Malone reads every book recommended to him, aligning with the original meaning.

For sentence (b): “Every painter admires at least one painter,” predicates such as Painter(x) and Admires(x, y) are used. The logical statement is:

∀x (Painter(x) → ∃y (Painter(y) ∧ Admires(x, y)))

This expresses that for every x who is a painter, there exists some y who is also a painter such that x admires y.

Sentence (c): “Every solid is soluble in some liquid,” involves predicates such as Solid(x) and SolubleIn(x, y). The first-order logic translation is:

∀x (Solid(x) → ∃y (Liquid(y) ∧ SolubleIn(x, y)))

This states that for every solid x, there exists a liquid y in which x is soluble.

Sentence (d): “Not every river in America is longer than every river in Europe,” requires predicates RiverInAmerica(x), RiverInEurope(x), and LongerThan(x, y). Its logical form is:

¬∀x (RiverInAmerica(x) → ∀y (RiverInEurope(y) → LongerThan(x, y)))

This indicates that it is not true that every river in America is longer than every river in Europe.

Sentence (e): “No number is greater than itself,” involves a predicate Number(x) and a comparison GreaterThan(x, y). The formalization is:

¬∃x (Number(x) ∧ GreaterThan(x, x))

which states that there does not exist any number x that is greater than itself.

In the context of interpreting a schema, beyond specifying a non-empty domain, it is essential to assign meaning to the non-logical symbols—predicates, functions, and constants—by defining their interpretations as sets, functions, or elements in the domain. This interpretation provides the semantics needed to evaluate the truth of formulas within a given model, making the schema meaningful and applicable within a particular context or domain of discourse.

References

  • Enderton, H. B. (2001). A First Course in Logic. Harcourt/Academic Press.
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  • Miller, D., & others. (2018). First-order Logic. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/logic-firstorder/
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