Rewrite the following sentences as propositional logic Use H

Rewrite the following sentences as propositional logic. Use H = the horse is fresh, K = the knight will win, and A = the armor is strong. If the horse is fresh, then the knight will win. The knight will win only if the horse is fresh and the armor is strong. A fresh horse is a necessary condition for the knight to win. The knight will win if and only if the armor is strong. A sufficient condition for the knight to win is that the armor is strong or the horse is fresh.

Solution

Propositional Logic is concerned with statements to which the truth values, “true” and “false”, can be assigned. The various connectives used in Propositional Logic are: AND (), OR (v), NOT/Negation (~), If-Then/Implies () and If and Only if (<=>).
Let\'s see the results of these connectives for two statements A and B.
AND - A AND B (AB) is true only when both the statements A and B are true. In rest of the cases, it\'s false.
OR - A OR B (AvB) is true if at least one of the statements among A and B is true. It\'s false when both are false.
NOT - NOT A (~A) is true when A is false and vice versa.
If-Then - If A then B (AB) is false if A is true and B is false. In rest of the cases, it\'s true.
If and Only if - A<=>B is true only when both A and B are true or both are false. In rest of the cases, it\'s false.

There are also what are called necessary conditions and sufficient conditions. A necessary condition is expressed as \"A only if B\" i.e. A would occur only when B would occur i.e. B is a necessary condition for A. This means, if A has occurred then B has definitely occurred or if B has not occurred then A has definitely not occurred. It is written as AB (Implication connective).
A sufficient condition is expressed as \"A if B\" i.e. A would occur if B has occurred i.e. B is a sufficient condition for A. It is written as BA.
Now let\'s see the solution-

There are three statements given: H (Horse is fresh), K (Knight will win) and A (armour is strong)

a. The question is a straightforward If-Then condition, i.e. If H is true then K is true. Propositional logic is: HK
b. The question is \"only if\" condition, i.e. it is a case of a necessary condition \"A only if B\" (Please refer to the explanation above) Here, A is K and B is a composite statement of (H AND A). Hence the question becomes \"K only if H AND A\". Propositional logic: K (HA).
c. This is also a case of necessary condition. It is given that H is a necessary condition for K i.e. \"K only if H\". Propositional logic: KH
d. This is a straighforward \"If and only if\" question. K is true if and only if A is true. Propositional logic is: K<=>A
e. This is a case of sufficient condition \"A if B\". (Please refer to the explanation above) Here, A is K and B is a composite condition of (H OR A) i.e. if either H or A is true, K would be true. Propositional Logic is: (BvA)K

Please let us know if you have any follow-up questions.