Use a truth table to test whether the following argument is valid. (Present the truth table, and say whether the argument is valid or invalid. When presenting your truth table you must fill in the truth values in the matrix in the way presented in lecture and in §3.3 of the textbook where it says “Here is a trick for filling in the truth values in the matrix”.) If the argument is invalid, give a counterexample.(¬A → ¬C)((C ∨ ¬A) ↔ E) ∴ (E∧¬C)
Question
Use a truth table to test whether the following argument is valid. (Present the truth table, and say whether the argument is valid or invalid. When presenting your truth table you must fill in the truth values in the matrix in the way presented in lecture and in §3.3 of the textbook where it says “Here is a trick for filling in the truth values in the matrix”.) If the argument is invalid, give a counterexample.(¬A → ¬C)((C ∨ ¬A) ↔ E) ∴ (E∧¬C)
Solution
To test the validity of the argument, we will construct a truth table for the premises and the conclusion.
Let's assign truth values to the propositions A, C, and E and fill in the truth table:
| A | C | E | ¬A | ¬C | ¬A → ¬C | C ∨ ¬A | (C ∨ ¬A) ↔ E | ¬C ∧ E |
|---|---|---|---|---|---|---|---|---|
| T | T | T | F | F | T | T | T | F |
| T | T | F | F | F | T | T | F | F |
| T | F | T | F | T | T | F | F | T |
| T | F | F | F | T | T | F | T | F |
| F | T | T | T | F | F | T | T | F |
| F | T | F | T | F | F | T | T | F |
| F | F | T | T | T | T | T | T | T |
| F | F | F | T | T | T | T | T | F |
In the truth table, we can see that there is at least one row where all the premises are true (row 3), but the conclusion is false. Therefore, the argument is invalid.
Counterexample: A = T, C = F, E = T
In this case, the premises (¬A → ¬C) and ((C ∨ ¬A) ↔ E) are both true, but the conclusion (E ∧ ¬C) is false.
Similar Questions
Use symbols to write the logical form of each argument given below, and thenuse a truth table to test the argument for validity.If the positive number is less than 2, then its square is not larger than 4.If square of number is larger than 4, then the number is not less than 2.∴ The positive number is not less than 2 and its square is not larger than 4
A truth table is used to:a.Determine the validity of a logical argumentb.Determine the truth value of a compound propositionc.Determine the logical connectives in a propositiond.Determine the inverse of a logical statement
Construct a truth table for the formula .
Determine whether the following symbolized arguments are valid or invalid byconstructing a truth table for each:1. K ⊃ ~K ./ ~K2. R⊃ R ./ R
True or false? In a two-column proof, the right column states your reasons.A.TrueB.False
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