Translation from English to PL
Translating from a natural language to a formal language is a key skill for those interested in the applicability of the formal language. After all, most of us are interested in logic because it helps us understand the structure of arguments originally expressed in our mother tongue, a natural language. Once an argument is translated into a formal or symbolic language, we can then examine its structure in a precise manner, and determine whether the argument in question is valid or invalid.
The first step in translation is to recall that we introduced the operators of PL as abbreviations for certain terms in English. We also drew the distinction between simple propositions and complex propositions. With that background, we can state the basic principles of translation succinctly:
| 1. | Translate simple propositions with individual sentence letters. |
| 2. | Translate "and" with "&" |
| 3. | Translate "not" with "~" |
| 4. | Translate "or" with "v" |
| 5. | Translate "If p then q" with "(p ⊃ q)", where "p" and "q" stand for arbitrary propositions. |
| 6. | Translate "p if and only if q" with "(p ≡ q)", where "p" and "q" stand for arbitrary propositions. |
Let's look at some examples. It's best to carry out translations in a set of steps. First we identify the simple propositions and translate them; then we identify the operators and translate them.
| Translation: | Commentary |
| It is raining and Fred is hungry. | English sentence to be translated |
| A: It is raining | First simple proposition and sentence letter abbreviation. |
| B: Fred is hungry | Second simple proposition and sentence letter abbreviation |
| A and B | Replace simple propositions with sentence letters. |
| (A & B) | Replace "and" with the symbol "&". Add parentheses. |
| Translation: | Commentary |
| If Thisbe passes this course, she'll graduate on time. | English sentence to be translated |
| A: Thisbe passes this course | First simple proposition and sentence letter abbreviation. |
| B: She'll graduate on time | Second simple proposition and sentence letter abbreviation |
| If A then B | Replace simple propositions with sentence letters. |
| (A ⊃ B) | Replace "if...then__" with the symbol "⊃". Add parentheses. |
We'll make things a little more difficult soon, but for now, try your hand at translating from English to PL in the following first translation exercise.
More translation 
| table of contents | List of Exercises |