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Math Help - question about how to determine Tautologies, Contradictions & Contingent propositions

  1. #1
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    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 ∧ (ba)

    abab

    (ab) ∧ b

    a ⇔ ( ab)

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

    The first proposition (aa) 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,
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  2. #2
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

    Defining a to be "we are all happy" does not help determine if aa 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.
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  3. #3
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

    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 ...
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  4. #4
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

    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.
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  5. #5
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

    Something like this:

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

    right?
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

    Quote Originally Posted by iwan1981 View Post
    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 ∧ (ba)
    abab
    (ab) ∧ b
    a ⇔ ( ab)
    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.
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  7. #7
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

    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
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  8. #8
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

    Quote Originally Posted by Plato View Post
    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"
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  9. #9
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    Re: question about how to determine Tautologies, Contradictions & Contingent proposit

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

    Thanks!
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