Propositional Formula and Translation

Question

On the basis of the following abbreviation: $A$ - Sanders is leader; $B$ - Florida is in the South; $C$ - Joe is Sanders' brother; $D$ - Sanders is from Florida, translate the following symbolic formula into natural English.

$A \Rightarrow (B \Rightarrow (C \Rightarrow D))$

I have very little background for alternative usage of "if-then" and therefore, would be strongly welcomed if anyone could suggest ideas on how to translate it into normal sentence?

Answer 1

The first thing I would do is eliminate some of the implications using the fact that for any propositional terms $p$ and $q$, the implication $p \to q$ is logically equivalent to $\neg p \vee q$, where $\neg$ reads as "not" an $\vee$ as "or" in natural English; for some explanation of this fact you can consult this MSE post. If I let $\equiv$ be the symbol denoting the relation "is logically equivalent to", we have the following equivalences:

$$\begin{align} A \to (B \to (C \to D)) &\equiv A \to( B \to (\neg C \vee D))\\ &\equiv A \to (\neg B \vee (\neg C \vee D)) \\ &\equiv A \to (\neg B \vee \neg C \vee D )\end{align} $$

Therefore one could translate your given logical formula in natural English as:

If Sanders is leader, then either Florida is not in the south, or Joe is not Sanders brother or Sanders is from Florida.

Answer 2

$$\begin{align} A \to (B\to (C \to D)) &\equiv A \to ((B \land C) \to D)\\ \\ &\equiv (A\land B \land C) \to D\tag 1\\ \\ &\equiv \lnot (A \land B \land C) \lor D\\ \\ &\equiv (\lnot A \lor \lnot B \lor \lnot C) \lor D\tag 2\\ \\ &\equiv \lnot A \lor \lnot B \lor \lnot C \lor D\end{align}$$

$(1)$ This is based on the equivalence of $$p \to (q \to r) \equiv (p\land q) \to r$$

$(2)$ this is based on $(1)$ and the fact that $\lnot p \to q \equiv p \lor q$. (Note this is because $p \to q \equiv \lnot p \lor q$.)

Answer 3

To expand upon Ricks answer, $A\rightarrow(B\rightarrow(C\rightarrow D))$, which is equivalent to $\lnot A\lor\lnot B\lor\lnot C\lor D$, is in turn the same as $\lnot (A\land B\land C)\lor D$ (using de Morgans law). This can be shortened to $(A\land B\land C)\rightarrow D$ using the definition of implication Rick mentioned.

$(A\land B\land C)\rightarrow D$ translates much more naturally to English:

If Sanders is leader, Florida is in the South, and Joe is Sanders' brother, then Sanders is from Florida.