I would appreciate anyone who would be willing to critique my translation of the following statements into English.
If D stands for “Doug is tall” and E stands for “Edie is short,” what English sentences are represented by the following expressions? a. (D ∧ E) ∨ ¬D
The symbolic expression “(D ∧ E)” stands for “Doug is tall and Eddie is short.” The symbolic expression ”¬D” meas that “it is not the case that Doug is tall.” The statement as a whole, with the main logical operator being “or,” or symbolically, “∨,” would read as “Either it is the case that Doug is tall and Eddie is short, or it is not the case that Doug is tall.” Hence the sentence as a whole is disjunctive.
Note: Doug’s tallness or shortness is contigent upon Eddie’s shortness; “tallness” and “shortness” are necessarily terms that are measured in a relative, and not absolute, fashion.
b. (D ∨ ¬E) ∧ ¬(D ∧ E)
The symbolic expression “(D ∨ ¬E)” reads in English as “Either Doug is tall or Eddie not short.” The symbolic expression “¬(D ∧ E)” reads in English as “it is false that Doug is tall and Eddie is short.” In essence, this clause stipulates that only one of the atomic statements D or E will receive a negation, but not both. As a whole, the original statement reads “Either Doug is tall or Eddie is not short, but it is not the case that Doug is tall and Eddie is short.” Thus it follows that if Doug is tall, then Eddie is not short (Perhaps Doug, a tall guy, is Eddie’s father – Eddie’s tallness seems contingent upon Doug’s height).
c. ¬D ∧ [ (E ∧ D) ∨ ¬E ]
Our main logical operator here is the symbol “∧,” which means “and.”
On the left side of our main logical operator, we have a negated atomic statement, which reads in English as “it is false that Doug is tall.”
On the right side of our original statement’s main logical operator, we have nested well formed formula that is comprised of several atomic statements and a heirarchy of logical operators, with the main one being the logical symbol for disjunction, which reads in English as “or,” and is symbolized as “∨.”
The entire right hand clause is enclosed with brackets.
Treating the right hand well formed formula as an existential entity within the universe of the orignial statement, we will interpret it from left to right. The paranthetical statement on the left hand side of the well formed formula enclosed by the brackets on the right side of the original statement is a conjunction of two affirmative atomic statements. In English, they read “Eddie is short and Doug is tall.” Moving to the right of our main logical operator, “or,” we have a negated atomic statement which reads “it is false that Eddie is short.”
So, as a whole, the existential WFF on the right hand side of the original statement reads “Either Eddie is short and Doug is tall, or it is false that Eddie is short.”
The orginal statement already stipulates that it is false that Doug is tall. It is a universal affirmative statement. By the law of non contradiction, it cannot be both true and false that Doug is tall. Yet the left hand side of the original statement tells us that it is false that he is tall, so this must be assumed to mean that anything on the right side of the original statement that says otherwise cannot be the case. The right side of the equation existentially proposes as an option that perhaps Doug is tall and Eddie is short. Yet this cannot be true due to the assumed universal negation that Doug is tall. Therefore, the statement found on the right hand side of the original statement, (E ∧ D), does not actually exist as an option within its respective disjuntive territory. So, within the brackets, we are left only with the negation of the atomic statement that Eddie is short, or symbolically, “¬E,” which reads “it is not the case that Eddie is short.”
The original statement, therefore, reads “It is not the case that Doug is tall and it is not the case that Eddie is short.”
$(b) \;\; (D \lor \lnot E) \land \lnot (D \land E) \equiv (D\lor \lnot E) \land (\lnot D \lor \lnot E)\tag{Demorgan's}$
$$\equiv \lnot E \lor \underbrace{(D \land \lnot D)}_{\text{false}}$$ $$ \equiv \lnot E$$ Which means that "It is not the case that Eddie is short."
$(c) \;\; \lnot D \land [ (E \land D) \lor ¬E ]$
$$\equiv \lnot D \land [\underbrace{(E\lor \lnot E)}_{\text{true}} \land (D \lor \lnot E]$$
$$\equiv (\underbrace{\lnot D \land D}_{\text{false}})\lor (\lnot D \land \lnot E) $$
which means (c) "D is not tall, and E is not short."