How do I set up a truth a truth table for:
All, programmers are fast typist.
Some, fast typist are clever.
Therefore, some programmers are clever.
I know this is an invalid argument but I"m not sure what the truth table should look like. thanks.
Since this is not my area of expertise, I searched Google under the key words "truth tables" to get these possible sources:
http://www.google.com/search?client=safari&rls=en&q=truth+tables&ie=UTF-8&oe=UTF-8
In the future, you can find the information you desire more quickly, if you use appropriate key words to do your own search. Also see http://hanlib.sou.edu/searchtools/.
To set up a truth table for this argument, you first need to identify the propositions involved. In this case, there are three propositions:
P: All programmers are fast typists.
Q: Some fast typists are clever.
R: Some programmers are clever.
Since P, Q, and R represent complex statements, you can further break them down using the following atomic propositions:
P1: Programmers
P2: Fast typists
P3: Clever individuals
Now, let's analyze the given argument:
1. All programmers are fast typists. (P)
2. Some fast typists are clever. (Q)
3. Therefore, some programmers are clever. (R)
To construct the truth table, you will need to consider all possible combinations of truth values for the three atomic propositions (P1, P2, and P3). Since P, Q, and R are connected by implication statements, you should also consider the truth values for the intermediate implications.
Here's how you can set up the truth table:
| P1 | P2 | P3 | P | Q | R |
|----|----|----|---|---|---|
| T | T | T | | | |
| T | T | F | | | |
| T | F | T | | | |
| T | F | F | | | |
| F | T | T | | | |
| F | T | F | | | |
| F | F | T | | | |
| F | F | F | | | |
To assign truth values for P, Q, and R, you need to evaluate the intermediate statements (P and Q) and the conclusion (R) using the truth values of P1, P2, and P3.
1. For proposition P: "All programmers are fast typists."
- The statement P will be false only if there exists at least one programmer who is not a fast typist.
2. For proposition Q: "Some fast typists are clever."
- The statement Q will be true if there exists at least one fast typist who is clever.
3. For proposition R: "Some programmers are clever."
- The statement R will be true if there exists at least one programmer who is clever.
By evaluating the intermediate statements and the conclusion based on the given conditions and your understanding of how the propositions are related, you can complete the truth table.