Part 2. (1 point each) Well-Formed Formulae. For each of the following strings of symbols, determine whether it is a well-formed formula by selecting “Yes” if it is and “No” if it is not. For this section, P is a one place predicate, and R is a two-place predicate; the names are a,b, and c; and x, y, and z are variables. Parentheses and quantifier conventions may be used.

Question

BlackTom AI · Accepted Answer

Quantifier scope and parentheses matter: a quantifier must immediately bind a well-formed formula, and complex formulas built with → and & require parentheses (or an agreed convention) to make scope unambiguous. The string shown, ∀xyzPx → Py & Pz, is ambiguous and therefore not a proper wff as written.

1) Relevant concepts
- Atomic formulas: Px, Py, Pz are atomic (P is one-place).
- Formation rules: if φ is a formula then ∀x φ is a formula; if φ and ψ are formulas then (φ → ψ) and (φ & ψ) are formulas. Quantifiers bind the formula that follows them; parentheses or explicit scope conventions are needed to avoid ambiguity.
- A well-formed formula (wff) must be built by these rules with unambiguous scope.

2) Step-by-step parsing attempts and why they fail
- One plausible reading: ∀xyzPx is shorthand for ∀x∀y∀z Px, so the string becomes (∀x∀y∀z Px) → (Py & Pz). Under this parse the quantifiers bind only the atomic Px; Py and Pz on the right remain free. Although open formulas with free variables can be well-formed, this reading depends on the shorthand and still leaves the overall intended logical structure unclear.
- Another intended reading might be that the quantifiers were supposed to take wide scope over the whole implication: ∀x∀y∀z (Px → (Py & Pz)). That is a different formula with a different meaning, and to get that you must include parentheses (or an explicit convention) showing the quantifiers apply to the entire implication.

3) Why the string as written is not a wff
- Because the formation rules require unambiguous application of quantifiers to a formula. Without parentheses or a stated convention making the quantifier scope clear, the expression is syntactically ambiguous. A well-formed formula must have a unique, well-defined parse; this string does not.

4) Why the other choice (Yes) is incorrect
- Picking “Yes” would assume a particular scope convention without parentheses (either that the quantifiers scope only Px or that they scope the whole implication). Because the problem explicitly allows parentheses and quantifier conventions, the absence of parentheses makes the intended structure unclear; therefore we cannot treat it as a proper wff.

Conclusion: The expression is not a well-formed formula as written (due to ambiguous quantifier scope and missing parentheses). This matches the provided correct answer: No.

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