Translate each of the following sentences into symbolic...

Translate each of the following sentences into symbolic notation, using the suggested symbols as abbreviations. 1. The Reds will win only if the Dodgers collapse. $(R, D)$ 2. The Steelers will win if their defense holds up. $(S, D)$ 3. If it rains or snows, the game will be called off. $(R, S, O)$ 4. If she came home with a trophy and a prize, she must have won the tournament. ( $T, P, W)$ 5. If you order the dinner special, you get dessert and coffee. ( $S, D, C)$ 6. If you order the dinner special, you get dessert; but you can have coffee whether or not you order the dinner special. ( $S, D, C)$ 7. If the house comes up for sale, and if I have the money in hand, I will bid on it. $(S, M, B)$ 8. If you come to dinner, I will cook you a lobster, if you want me to. (D, L, W) 9. You can be a success if only you try. $(S, T)$ 10. You can be a success only if you try. $(S, T)$ 11. Only if you try can you be a success. $(S, T)$ 12. You can be a success if you are the only one who tries. $(S, O)$ 13. Unless there is a panic, stock prices will continue to rise. $(P, R)$ 14. I won't scratch your back unless you scratch mine. ( $I, Y$ ) 15. You will get a good bargain provided you get there early. $(B, E)$ 16. You cannot lead a happy life without friends. (Let $H=$ You can lead a happy life, and let $F=$ You have friends.) 17. The only way that horse will win the race is if every other horse drops dead. (Let $W=$ That horse will win the race, and let $D=$ Every other horse drops dead.) 18. You should take prescription drugs if, but only if, they are prescribed for you. $(T, P)$ 19. The grass will die without rain. ( $D, R=$ It rains.) 20. Given rain, the grass won't die. ( $R, D=$ The grass will die.) 21. Unless it doesn't rain, the grass won't die. ( $R, D=$ The grass will die.)

Translate each of the following sentences into symbolic notation, using the suggested symbols as abbreviations.
1. The Reds will win only if the Dodgers collapse. $(R, D)$
2. The Steelers will win if their defense holds up. $(S, D)$
3. If it rains or snows, the game will be called off. $(R, S, O)$
4. If she came home with a trophy and a prize, she must have won the tournament. ( $T, P, W)$
5. If you order the dinner special, you get dessert and coffee. ( $S, D, C)$
6. If you order the dinner special, you get dessert; but you can have coffee whether or not you order the dinner special. ( $S, D, C)$
7. If the house comes up for sale, and if I have the money in hand, I will bid on it. $(S, M, B)$
8. If you come to dinner, I will cook you a lobster, if you want me to. (D, L, W)
9. You can be a success if only you try. $(S, T)$
10. You can be a success only if you try. $(S, T)$
11. Only if you try can you be a success. $(S, T)$
12. You can be a success if you are the only one who tries. $(S, O)$
13. Unless there is a panic, stock prices will continue to rise. $(P, R)$
14. I won't scratch your back unless you scratch mine. ( $I, Y$ )
15. You will get a good bargain provided you get there early. $(B, E)$
16. You cannot lead a happy life without friends. (Let $H=$ You can lead a happy life, and let $F=$ You have friends.)
17. The only way that horse will win the race is if every other horse drops dead. (Let $W=$ That horse will win the race, and let $D=$ Every other horse drops dead.)
18. You should take prescription drugs if, but only if, they are prescribed for you. $(T, P)$
19. The grass will die without rain. ( $D, R=$ It rains.)
20. Given rain, the grass won't die. ( $R, D=$ The grass will die.)
21. Unless it doesn't rain, the grass won't die. ( $R, D=$ The grass will die.)

Key Concepts

Propositional Variables

Propositional variables are symbols that represent basic statements or propositions in logic. They are treated as indivisible units with a truth value (true or false) and are fundamental to translating verbal statements into symbolic notation.

Implication (Conditional Statements)

Implication or conditional statements express a dependency or a condition, typically in the form 'if P then Q'. In symbolic logic, this is usually denoted as P ? Q. It is a key concept that underlies much of mathematical logic and reasoning, and proper translation of English conditional phrases (such as 'if', 'only if') requires careful attention to the direction of the implication.

Conjunction

Conjunction refers to the logical operation 'and', which combines two propositions to form a compound statement that is true only when both component propositions are true. In symbolic notation, the conjunction is usually represented by the symbol ?.

Disjunction

Disjunction is the logical operation 'or', used to combine two or more propositions such that the compound statement is true if at least one of the components is true. The disjunction is typically symbolized by ?, and it plays an important role in expressing options or alternative conditions in logical translations.

Necessary and Sufficient Conditions

Understanding necessary and sufficient conditions is crucial for correctly translating phrases like 'only if' and 'if'. A condition is necessary if it must be true for the outcome to occur, and sufficient if its truth guarantees the outcome. Properly distinguishing between these conditions ensures the correct formation of logical implications.

Biconditional Statements

Biconditional statements capture situations where a condition is both necessary and sufficient for another. They are expressed as 'if and only if' and symbolized by ?. This equivalence between two propositions reflects a two-way dependency and is essential in cases where mutual implication is intended.

The 'Unless' Construction

The 'unless' construction in everyday language is typically translated in formal logic as a conditional involving negation. It means 'if not' and is often rendered as the implication of the negation of one proposition leading to another. Properly handling 'unless' is critical to avoid misrepresenting the logical relationships set by the statement.

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