Construct a complete truth table for each of the following formulas. Note the main connective at the top, and beneath the table say whether it is truth-functionally true, false, or indeterminate, citing the appropriate definition as justification.

## Construct a complete truth table for each of the following formulas

PLEASE FOLLOW DIRECTIONS OF EVERYTHING THAT IS HIGHLIGHTED YELLOW

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For all truth tables, list sentence letters in alphabetical order from left to right and make sure to list the possible combinations of truth-values beneath them according to the convention. Tables that do not conform to those conventions WILL NOT BE GRADED.

**Part one:**

Construct a complete truth table for each of the following formulas. Note the main connective at the top, and beneath the table say whether it is truth-functionally true, false, or indeterminate, citing the appropriate definition as justification.

1. ( C V ~ ~ E ) -> ( ~ E -> C )

2. [ ( ( C -> D ) & ( D -> E ) ) & C ] & ~E

3. ( A <-> B ) V [ ( C & ~B ) -> ~ ~ A ]

**part two:**

4. A disjunction with one truth-functionally true disjunct must itself be true. True or false? Explain. Make sure to cite all appropriate definitions.

5. A material conditional with a truth- functionally false consequent must itself be truth-functionally false. True or false? Explain. Make sure to cite all appropriate definitions.

**Part three:**

6. Is this set truth-functionally consistent or inconsistent? Construct a complete truth table to evaluate. Cite the definition in your justification.

{ (A & B) & C, C V (B V A), A <-> ( B -> C ) }

**Part four:**

7. Translate the following argument into SL. No need to construct a truth table, but you will need to provide your own legend assigning sentence letters to the simple sentences in the argument. Remember that your simple sentences must be connective-free. That includes flushing out all negations.

**Argument:**

Sophie doesn’t believe in trolls, but she does believe in Bigfoot. Jason believes in both trolls and Bigfoot. If Sophie and Jason both believe in trolls, then neither is a critical thinker. Therefore, Sophie is a critical thinker but Jason isn’t.

Part five:

Construct a complete truth table for the following arguments. Label it with the main conditional at the top. Beneath the table, specify whether the argument is valid or not, and explain why, citing the appropriate rows of the table (e.g., “Row 4 has all true premises and a false conclusion, therefore the argument is invalid”. Or “no rows show all true premises and a false conclusion, so the argument is valid.”)

8.

A V B

A -> C

B -> D

————

C V D

9.

A -> C

B -> D

~ C V ~ D

—————-

~ A V ~ B

10.

A <-> ( B V C)

~ B & ~ C

—————

A

(Try this last one with ~A as the conclusion for comparison. No need to turn that second table in. Just something to know.)