Conditional Statements in Logic

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Questions and Answers

What is the purpose of using truth tables in propositional logic?

To prove the validity of an argument

To diagram the structure of an argument

To determine the falsity of an argument

To check for logical validity of an argument (correct)

What is the truth value of ¬P when P is false?

True (correct)

Cannot be determined

Both A and C

False

What is the purpose of adding the truth table for ¬P ∨ Q?

To make it easier to figure out the truth table for the whole claim ¬P ∨ Q (correct)

To prove the validity of the argument

To simplify the truth table for P

To determine the falsity of the argument

What type of claim is ¬P ∨ Q?

Disjunction Signup and view all the answers

What is an argument logically valid if?

It is impossible for all the premises to be true and the conclusion false Signup and view all the answers

What do we do when checking for logical validity using truth tables?

We write the truth tables for each of the premises and the conclusion Signup and view all the answers

What do we look for when checking for logical validity using truth tables?

A line on the truth table where all the premises are true and the conclusion is false Signup and view all the answers

What is the purpose of Step 1 in using truth tables to check for validity?

To write out a table that includes the propositional variables, each premise, and the conclusion Signup and view all the answers

What is the truth value of the conditional 'x > 3 → x > 1' when x = 2?

False Signup and view all the answers

What is the equivalent of P → Q in terms of negation and disjunction?

¬P ∨ Q Signup and view all the answers

What is the correct interpretation of the symbol '∧' in propositional logic?

And Signup and view all the answers

What is the purpose of Step 2 in the provided truth table exercise?

To fill in the truth table for the propositional variables Signup and view all the answers

What is the purpose of constructing a truth table for a complex claim?

To determine the truth value of the claim Signup and view all the answers

What is the logical rule applied in the given argument?

Modus Tollens Signup and view all the answers

What is the equivalent of X ∧ Y in terms of negation and disjunction?

¬(¬X ∨ ¬Y) Signup and view all the answers

What is the purpose of the ¬Q column in the truth table?

To represent the negation of Q Signup and view all the answers

What is the truth value of the statement 'P → Q' when P is true and Q is false?

False Signup and view all the answers

What is the correct interpretation of the symbol '¬' in propositional logic?

Not Signup and view all the answers

What is the conclusion of the given argument?

¬P Signup and view all the answers

What is the purpose of Step 6 in the provided truth table exercise?

To check every line in which all the premises are true Signup and view all the answers

What is the equivalent of P → Q in terms of negation and conjunction?

¬(P ∧ ¬Q) Signup and view all the answers

What is the role of the P → Q column in the truth table?

To represent the conditional statement P → Q Signup and view all the answers

What is the purpose of using the conditional symbol '→' in propositional logic?

To represent implication Signup and view all the answers

What is the purpose of constructing a truth table?

To determine the validity of an argument Signup and view all the answers

What is the correct interpretation of the symbol '∨' in propositional logic?

Or Signup and view all the answers

What is the logical operator represented by the symbol ¬?

Negation Signup and view all the answers

Study Notes

Conditional Claims

A conditional claim 'If P then P' is a tautology, always true
The truth table for a conditional claim P → Q has four possible scenarios:
- P: True, Q: True → P → Q: True
- P: True, Q: False → P → Q: False
- P: False, Q: True → P → Q: True
- P: False, Q: False → P → Q: True

Tautologies and Contradictions

A tautology is a claim that is always true
A contradiction is a claim that is always false

Truth Tables and Validity

Truth tables can be used to check the validity of an argument
An argument is logically valid if it is impossible for all premises to be true and the conclusion false
To use truth tables to check for logical validity:
1. Write out a table including the propositional variables, premises, and conclusion
2. Fill in the truth table for the propositional variables
3. Add in the truth table for each premise
4. Add in the truth table for the conclusion
5. Check every line in which all premises are true, to see if the conclusion is also true

Steps to Check Validity

Step 1: Write out the truth table for the argument
Step 2: Fill in the truth table for the propositional variables
Step 3: Add in the truth table for the first premise
Step 4: Add in the truth table for the second premise
Step 5: Add in the truth table for the conclusion
Step 6: Check every line in which all premises are true, to see if the conclusion is also true