More complex symbolizations
Real sentences stack connectives inside each other. This section gives you a method for taking them apart, patterns for “neither”, “not both”, and “exactly two”, and practice with whole arguments.
Five guidelines for paraphrasing
- Decide whether it’s a truth-functional compound. If it isn’t, it’s its own paraphrase and gets a single sentence letter. If it is, ask the same question about each of its parts, and their parts.
- Use only the five connectives: “it is not the case that”, “and”, “or”, “if … then”, and “if and only if”.
- Show the grouping with parentheses and square brackets, so it’s clear which sentences each connective joins.
- Put arguments in standard form: premises listed first, a line, then the conclusion.
- Make every part a complete sentence on its own. Replace pronouns with what they refer to, and when two wordings make the same claim, pick one wording and use it every time.
Guideline 1 is where many mistakes start. A sentence can contain “and” without being a conjunction: “Rock and roll” is a name, and “Ben and Amy met” isn’t two separate claims. And some connectives, like “thinks that” or “because,” aren’t truth-functional at all.
Working from the outside in
For a long sentence, find the main connective first, the one that governs the whole thing. Then treat each part as a new, smaller sentence and repeat until every piece is atomic.
- The whole sentence is “if … then”. Antecedent: it rains and the field floods. Consequent: the game is off.
- The antecedent is itself a conjunction: it rains and the field floods.
- Every remaining piece is atomic. Paraphrase: If (it rains and the field floods) then the game is off.
- Symbolize with R, F, G: (R & F) ⊃ G.
The paraphrase builder below works the same way. Choose the main connective in the top box; new boxes appear for its parts. Fill those in, and so on down. Your paraphrase and its symbolization update as you go.
Grouping and ambiguity
English sentences with more than one connective can be ambiguous. “Paul plays bass and Ella sings or Kofi drums” could mean Paul plays bass, and (Ella sings or Kofi drums) or (Paul plays bass and Ella sings), or Kofi drums. Those are different claims: they come apart when Kofi drums and Paul doesn’t play.
SL has no ambiguity, so your paraphrase must pick a reading and show it with parentheses. Context, tone of voice, or stress usually tells you which one is meant.
Extended conjunctions and disjunctions aren’t ambiguous. “Ada, Bea, and Cy all passed” is true only if all three passed, however you group it. SL still requires parentheses, but any grouping is correct: (A & B) & C or A & (B & C).
For each sentence below, write both readings.
Counting patterns
Many sentences say how many of a group of claims are true. These patterns come up again and again:
| English | SL |
|---|---|
| both A and B | A & B |
| at least one of A, B | A ∨ B |
| neither A nor B | ∼A & ∼B or ∼(A ∨ B) |
| not both A and B | ∼(A & B) or ∼A ∨ ∼B |
| exactly one of A, B | (A ∨ B) & ∼(A & B) |
| both or neither | (A & B) ∨ (∼A & ∼B) |
| all of A, B, C | (A & B) & C |
| at least two of A, B, C | [(A & B) ∨ (A & C)] ∨ (B & C) |
| exactly two of A, B, C | list each pair with the third denied, joined by ∨ |
The trick for “exactly”: when a sentence says how many but not which ones, list every way it could be true and join them with ∨. Or combine “at least” with “not more than.”
All the exercises in this set share one key, about three films Maya may have seen.
Longer sentences
Here everything combines: “only if”, “unless”, “neither”, and nested conditionals. Work from the outside in. If you’re stuck, the first hint tells you the main connective.
One claim, one letter
Guideline 5 matters most for arguments. If “Rita is late” and “Rita will be late” get different letters, SL treats them as unrelated claims, and a valid English argument can turn into an invalid symbolic one.
- Replace pronouns. “If Jen enters the race, she’ll win it” becomes “If Jen enters the race then Jen will win the race.”
- Unify wordings. When two phrasings make the same claim in context, pick one and use it everywhere.
- Watch for pronouns with no fixed referent. In “Kim will only read a novel if it’s set in Japan,” “it” doesn’t name any one novel. SL can’t split this sentence up, so it gets one letter. Predicate logic (Chapter 7) handles it properly.
Whole arguments
Symbolize each premise and the conclusion with the key given. Once every line is right, you’ll see whether the symbolic argument is valid in SL. That check uses truth tables, which Chapter 3 teaches; here it’s just a preview.
Words like “so”, “therefore”, and “hence” mark the conclusion. They aren’t connectives and don’t get symbolized.
Review
Every 2.3 symbolization, mixed together. The quick reference has the full table of English phrases and their SL forms.