Truth-Tree Rules for Non-PL Operators
We've emphasized that you do not need to memorize the truth-tree rules. If you know how truth-trees work, and you know the basic truth-tables, then you can figure out the rules. Whenever a wff appears on a tree, we are asserting the truth of that wff. So if (A v B) appears on a tree, then either A is true or B is true. We express that by creating branches for A and B.
To really test your understanding of this, you can figure out what the truth tree rules would be for other possible connectives. For example, suppose we had a connective "*" which we call "asterisk". Suppose the truth-table for the asterisk is:
| p | q | (p*q) |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | F |
Note that this is not the basic truth-table for any of the connectives of PL, so would be a new connective, were we to allow this into PL. What would be the truth-tree rule for (p*q) and what would be the rule for ~(p*q)?
For (p*q), note that this wff is true when both p and q are true, or when p is false (and thus ~p is true) and q is true. So the rule would look like this:
| (p * q) | ||||
| / | \ | |||
| p | ~p | |||
| q | q |
What about ~(p*q)? The truth table is:
| p | q | ~(p*q) |
| T | T | F |
| T | F | T |
| F | T | F |
| F | F | T |
So ~(p*q) is true when either p is true and q is false (so ~q is true) or when both p and q are false.
| ~(p * q) | ||||
| / | \ | |||
| p | ~p | |||
| ~q | ~q |
In the following lovely exercise, you'll be given several non-PL operators, and you'll be asked to describe the truth-tree rules for them. If you've followed the above, it will be a piece of cake. If you haven't, just reread!
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