P→(Q∨(C&S))

Question

P→(Q∨(C&S))

BlackTom AI · Accepted Answer

Main connective is →, so the formula parses as P → (Q ∨ (C & S)).

1) Relevant knowledge points and concepts
- Atomic formulas: P, Q, C, S are propositional atoms.
- Binary connectives: → (implication), ∨ (disjunction), & (conjunction).
- Well-formed formula (WFF) formation rule: if A and B are WFFs, then (A ∧ B), (A ∨ B), (A → B) etc. are WFFs. Parentheses are used to indicate the scope of binary connectives; common parenthesis-dropping or precedence conventions (e.g. & and ∨ bind tighter than →, and → is right-associative) can be applied when parentheses are omitted.

2) Step-by-step parsing / reasoning
- Identify the outermost symbol structure: the string begins with P followed by →, so the top-level connective is →, with left operand P and right operand the rest: (Q ∨ (C & S)).
- Inspect the right operand: it is parenthesized as (Q ∨ (C & S)). The outer connective there is ∨, so that subformula has left operand Q and right operand (C & S).
- Inspect (C & S): the connective is &, with left operand C and right operand S.
- Each atomic symbol (P, Q, C, S) is a WFF; combining them with the binary connectives as parenthesized yields valid WFFs at every step.

3) Intermediate “calculation” steps (syntactic composition)
- C and S are WFFs; form (C & S).
- Q and (C & S) are WFFs; form (Q ∨ (C & S)).
- P and (Q ∨ (C & S)) are WFFs; form P → (Q ∨ (C & S)).

4) Why this is correct
Because at each composition step we apply the formation rule for binary connectives using explicitly provided parentheses, the entire string is a well-formed formula. The parentheses remove any ambiguity about grouping: the implication applies to P and the complete disjunction Q ∨ (C & S).

5) Why other readings would be incorrect
If parentheses were omitted one might misread P → Q ∨ C & S as (P → Q) ∨ (C & S) or P → (Q ∨ (C & S)) depending on precedence. Those alternative parses are not applicable here because explicit parentheses specify the grouping unambiguously.

6) Conclusion
The string is a WFF and is unambiguous (given parentheses-dropping conventions). This matches the provided answer.

P→(Q∨(C&S))Single choice

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