-1

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?

Alessio K
  • 10,599
rentbuyer
  • 312

3 Answers3

2

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.

Rick
  • 1,896
  • 1
    Thank you very much, man! I appreciate your help! – rentbuyer Sep 03 '20 at 14:23
  • 1
    I hope you don't mind, but I expanded upon your answer to generate a more natural sounding English expression. I still think this should stay the accepted answer however. – user400188 Sep 05 '20 at 04:24
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 $p \to q \equiv \lnot p \lor q$.)

amWhy
  • 209,954
  • I think you made a mistake in equation $(1)$ where you compare $(A\land B\land C)\rightarrow D$ to $\lnot(A\land B\land C)\rightarrow D$. This carries on into the subsequent formulae of $(1)$ and $(2)$. – user400188 Sep 05 '20 at 11:14
  • 2
    @user400188: The mistake in the original post has since been corrected. – user21820 Sep 13 '20 at 16:02
0

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.

user400188
  • 1,936
  • 1
    I really don't understand that downvote... – Rick Sep 06 '20 at 09:18
  • 1
    @Rick I received 4 consecutive downvotes on my posts within a short timeframe and a vote to delete this answer. I am not sure why and asked about it in the meta chat room. Apparently there is a chance of everything been reversed within 24 hours by an automated system. If not, then I am to flag the post(s) and ask if something can be done about it manually. – user400188 Sep 06 '20 at 10:20