problem has to do with three valued logic, ie the values 1, 1/2, 0 interpreted as true, unknown, and false.
i need to establish the connectives for the 3 valued logic, construct truth tables for the negation, conjunction, disjunction, the conditional, and biconditional.
not good at logic stuff, let me know if you can help!
Of course, I can help you with that! In three-valued logic, the values 1, 1/2, and 0 are interpreted as true, unknown, and false, respectively. To establish the connectives for three-valued logic, we can start by defining the truth tables for the basic logical connectives: negation, conjunction, disjunction, conditional, and biconditional.
1. Negation ("not"):
The negation of a statement p can be denoted as ¬p. The truth table for negation in three-valued logic would be as follows:
p | ¬p
----|----
1 | 0
1/2 | 1/2
0 | 1
2. Conjunction ("and"):
The conjunction of two statements p and q can be denoted as p ∧ q. The truth table for conjunction in three-valued logic would be as follows:
p | q | p ∧ q
----|-----|------
1 | 1 | 1
1 | 1/2 | 1/2
1 | 0 | 0
1/2 | 1 | 1/2
1/2 | 1/2 | 1/2
1/2 | 0 | 0
0 | 1 | 0
0 | 1/2 | 0
0 | 0 | 0
3. Disjunction ("or"):
The disjunction of two statements p and q can be denoted as p ∨ q. The truth table for disjunction in three-valued logic would be as follows:
p | q | p ∨ q
----|-----|------
1 | 1 | 1
1 | 1/2 | 1
1 | 0 | 1
1/2 | 1 | 1
1/2 | 1/2 | 1/2
1/2 | 0 | 1/2
0 | 1 | 1
0 | 1/2 | 1/2
0 | 0 | 0
4. Conditional ("if-then"):
The conditional of two statements p and q can be denoted as p → q. The truth table for the conditional in three-valued logic would be as follows:
p | q | p → q
----|-----|------
1 | 1 | 1
1 | 1/2 | 1/2
1 | 0 | 0
1/2 | 1 | 1
1/2 | 1/2 | 1/2
1/2 | 0 | 1/2
0 | 1 | 1
0 | 1/2 | 1
0 | 0 | 1
5. Biconditional ("if and only if"):
The biconditional of two statements p and q can be denoted as p ↔ q. The truth table for the biconditional in three-valued logic would be as follows:
p | q | p ↔ q
----|-----|------
1 | 1 | 1
1 | 1/2 | 1/2
1 | 0 | 0
1/2 | 1 | 1/2
1/2 | 1/2 | 1
1/2 | 0 | 0
0 | 1 | 0
0 | 1/2 | 0
0 | 0 | 1
These truth tables provide a systematic way to determine the truth values of logical statements in three-valued logic. I hope this explanation helps you better understand the connectives and truth tables for three-valued logic! Let me know if you have any further questions.