Exercises
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Exercises
Use a truth table to determine if the argument forms are valid.
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Use either modus tollens or modus ponens to fill in the blank in the following arguments.
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Identify the rule of inference that guarantees the validity of each of the following arguments or the fallacy that shows it isn't.
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Determine if the argument
p → q
q → p
∴ q ∨ p
is valid or not using a truth table.
The statement version of the argument that you need to create a truth table for looks like:
(p → q ∧ q → p) ∧ (q ∨ p)
Determine if the argument
p → q
q → p
∴ q ∨ p
is valid or not using a truth table.
Determine if the argument
~p → c
∴ q
is valid or not using a truth table.
Use either modus tollens or modus ponens to fill in the blank in the argument:
If a function is differentiable then it's continuous.
g isn't continuous.
∴ ____________________________________________________
Translate the argument into its logical form then put that side by side with the modus tollens and modus ponens forms and see if they match.
Use either modus tollens or modus ponens to fill in the blank in the argument:
If a function is differentiable then it's continuous.
g isn't continuous.
∴ ____________________________________________________
Start be letting p = "g is differentiable" and q = "g is continuous". Now I'll use those to translate the original argument into its logical form and put that side by side with the modus tollens form.
| Argument | Modus Tollens |
| p → q ~q ∴ ____________ |
p → q ~q ∴ p |
Notice how the first two rows match exactly. To complete the last row, we need to put ~p which is "g is not differentiable."
Use either modus tollens or modus ponens to fill in the blank in the argument:
If a number is divisible by 6 then it's divisible by 3.
186 is divisible by 6.
∴ ____________________________________________________
Use either modus tollens or modus ponens to fill in the blank in the argument:
If a number is divisible by 6 then it's divisible by 3.
186 is divisible by 6.
∴ ____________________________________________________
Start be letting p = "186 is divisible by 6" and q = "186 is divisible by 3". Now I'll use those to translate the original argument into its logical form and put that side by side with the modus tollens form.
| Argument | Modus Ponens |
| p → q p ∴ ____________ |
p → q p ∴ q |
Notice how the first two rows match exactly. To complete the last row, we need to put q which is "182 is divisible by 3".
Use either modus tollens or modus ponens to fill in the blank in the argument:
If p isn't a perfect square then $\sqrt{p}$ isn't an integer.
7 isn't a perfect square
∴ ____________________________________________________
Use either modus tollens or modus ponens to fill in the blank in the argument:
If p isn't a perfect square then $\sqrt{p}$ isn't an integer.
7 isn't a perfect square
∴ ____________________________________________________
Start be letting p = "7 isn't a perfect square" and q = "$\sqrt{7}$ isn't an integer". Now I'll use those to translate the original argument into its logical form and put that side by side with the modus tollens form.
| Argument | Modus Ponens |
| p → q p ∴ ____________ |
p → q p ∴ q |
Notice how the first two rows match exactly. To complete the last row, we need to put q which is "$\sqrt{7}$ isn't an integer".
Identify the rule of inference that guarantees the validity of each of the following arguments or the fallacy that shows it isn't.
If your team wins then you'll go out to dinner afterward.
If you go out to dinner afterward, you'll get home late.
∴ If your team wins then you'll get home late.
Start by translating the argument into its logical form. You'll need three variables for thsi one.
Identify the rule of inference that guarantees the validity of each of the following arguments or the fallacy that shows it isn't.
If your team wins then you'll go out to dinner afterward.
If you go out to dinner afterward, you'll get home late.
∴ If your team wins then you'll get home late.
First, we need to conver the statement to its logical form using
p = "your team wins"
q = "you go out to dinner afterward"
r = "you get home late"
That makes the logical form
p → q
q → r
∴ p → r
This is a valid argument since it matches the "Transitivity" form.
Identify the rule of inference that guarantees the validity of each of the following arguments or the fallacy that shows it isn't.
If you don't order dessert then you'll be hungry later.
You aren't hungery later.
∴ You did order dessert.
Notice how all the standard forms start with p, not ~p. To match one of those forms start with p = "you don't order dessert" even though that's a "negative" statement.
Identify the rule of inference that guarantees the validity of each of the following arguments or the fallacy that shows it isn't.
If you don't order dessert then you'll be hungry later.
You aren't hungery later.
∴ You did order dessert.
First, we need to conver the statement to its logical form. Notice how all of the forms start with p . . . and none of them start with ~p . . .. With that in mind, we'll start with
p = "you don't order dessert"
q = "you'll be hungry later"
even though the p statement is negative.
That makes the logical form
p → q
~q
∴ ~p
This is the valid Modus Tollens form.
Identify the rule of inference that guarantees the validity of each of the following arguments or the fallacy that shows it isn't.
If you graduate on time then you'll get a good job.
You didn't graduate on time.∴ You won't get a good job.
First, we need to conver the statement to its logical form using
p = "you graduate on time"
q = "you'll get a good job"
That makes the logical form
p → q
~p
∴ ~q
This is the "Fallacy of the Inverse" fallacy.
