Last Update: 29 September 2010
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Possible answer:
This is called an "argument"
(in the sense of a legal argument,
not a yelling match)
Let p = "You get a passing grade in all assignments."
Let q = "You (will) pass the course".
Then the form of this argument can be shown by the following sequence of propositions:
This "argument form" is also called a "rule of inference"
prem 2 | conc | prem 1 |
---|---|---|
p | q | (p→q) |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
Note that row 1 is the only row where both premises are T.
In that row, the conclusion is also T.
i.e., if all of its premises are T,
then its conclusion must be T
or: it is impossible for all of its premises to be simultaneously T, yet its conclusion is F.
(A → B)
(B → C)
∴ (A → C)
You should construct a truth table to convince yourself that this is truth-preserving, hence valid.
Is this valid?
We can prove validity by syntactically deriving all "intermediate" conclusions:
Let p = You pass all assignments.
Let q = You pass the course.
Let r = You'll be successful in life.
Then:
line num |
proposition | comment mark |
justification |
---|---|---|---|
1. | (p → q) | : | Prem P1 |
2. | (q → r) | : | Prem P2 |
3. | (p → r) | : | From lines 1,2; by HS |
4. | p | : | Prem P3 |
5. | r | : | From lines 3,4; by MP |
∴ the argument is not "sound":
Def: