Thread: question about how to determine Tautologies, Contradictions & Contingent propositions

Hi,

I am looking at a question here and I don't entirely know how to answer this ...

We know that:
1) Tautologies are true for all possible values of the logical variables
2) Contradictions are false for all possible values of the logical variables
3) Contingent propositions are neither

Make the following propositions to one of the above.

a ∧ ¬a

a ∧ ¬(b ⇒ a)

a ⇒ b ⇔ ¬a ∨ b

(a ∨ b) ∧ ¬b

a ⇔ ( ¬a ∧ b)

So lets say that:
- a = we are all happy
- b = we bought a new iPad

The first proposition (a ∧ ¬a) would be:
"We are all Happy AND we are all not happy"
These 2 are not true together because ... we can not be happy and not happy at the same time ... so this one is a contradiction

Can someone please explain how I find out the others?

Thanks,

Defining a to be "we are all happy" does not help determine if a ∧ ¬a is a contradiction. As "we are all happy" can be either true or false, so can a.

You need to build truth tables for each of these formulas. Tautologies are those formulas that only have True in the final column, contradictions only have False, and contingent propositions have True in some rows and False in others.

So if I create a truth table I start with this:

a | b | ¬a|
---------------
T | T | F |
T | F | F |
F | T | F |
F | F | F |

Is this correct?

But I don't know how to continue ...

a | b | ¬a|
---------------
T | T | F |
T | F | F |
F | T | F |
F | F | F |

Is this correct?

No, the negation of F should be T in the last two rows. Also, the last column should correspond to a given formula, e.g., a ∧ ¬a. Since this formula has only one variable, it is not necessary to include b, and the table will have only two rows.

Something like this:

a | ¬a|
---------------
T | F |
T | F |
F | F |
F | F |

right?

Originally Posted by iwan1981

We know that:
1) Tautologies are true for all possible values of the logical variables
2) Contradictions are false for all possible values of the logical variables
3) Contingent propositions are neither

Make the following propositions to one of the above.
a ∧ ¬a
a ∧ ¬(b ⇒ a)
a ⇒ b ⇔ ¬a ∨ b
(a ∨ b) ∧ ¬b
a ⇔ ( ¬a ∧ b)

Here is an online resource

Note I asked it to do a truth table for the second one.
That tells us it is a contradiction.
You can type in each of the others one at a time.

Something like this:

a | ¬a|
---------------
T | F |
T | F |
F | F |
F | F |

As I said, you can remove two of the four rows. Also, this is the truth table for the formula ¬a (the name of the last column), which is not one of the formulas in your first post. The truth table for a ∧ ¬a is

Code:

a | ¬a | a ∧ ¬a
--+----+-------
T |  F |   F
F |  T |   F

or just

Code:

a | a ∧ ¬a
--+-------
T |   F
F |   F

Originally Posted by Plato

Here is an online resource

Note I asked it to do a truth table for the second one.
That tells us it is a contradiction.
You can type in each of the others one at a time.

Thanks Plato! great link!

So this means with this information emakarov gave me:
- tautologies are those formulas that only have True in the final column
- contradictions only have False,
- contingent propositions have True in some rows and False in others.

(p ∨ q) ∧ ¬q = contingent

@ Plato ... how do we use ⇔ with this tool?
⇒ = IMPLIES ... but can't figure out the ⇔

Edit:
Found it ... "<=>" or "EQUIVALENT"

a ∧ ¬a = Contradiction
a ∧ ¬(b ⇒ a) = Contradiction
a ⇒ b ⇔ ¬a ∨ b = Tautology
(a ∨ b) ∧ ¬b = Contingent
a ⇔ ( ¬a ∧ b) = Contingent

Thanks!