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Truth table, implication and equivalence

  1. Jul 1, 2013 #1
    Hello, I have some questions about the truth tables for impliocation and equivalence.

    for implication we have:

    p | q | p=> q

    T | T | T
    T | F | F
    F | T | T
    F | F | T

    Here I do not understand the last two lines, how can we say that p implies q when p is false, and q is either true or false, if we only know that p is false and q is true, shouldn't p=> be unknown instead of T?
    The same for p is false and q is false?, shouldn't p=>q then be unknown.

    I have the same problem for equivalence:

    p | q | p<=> q

    T | T | T
    T | F | F
    F | T | F
    F | F | T

    Here I only have the problem with the last line when both p and q are false. How can we then say that p implies q?
  2. jcsd
  3. Jul 1, 2013 #2


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    You have to move away from the English language idea of "implies" and into a more general mathematical view of it. In fact, it would behoove you to realize that "P implies Q" is just a convenient way of saying P -> Q, which to many people tends to imply that you can deduce Q from P, which as you can see isn't always true.

    The idea that if given a false statement and a true or false conclusion, then the conclusion doesn't matter, because the statement is false. Think of it like a contract. If you run a mile, then I'll give you water. What if you ran half mile and I gave you water? What if you ran half a mile and I don't give you water? Well, I didn't lie, because my conclusion was only guaranteed when you fulfilled your obligation, so therefore no matter the case, I did a 'truthful' thing.
  4. Jul 1, 2013 #3
    Thank you, I think I understand it now.

    I have another question though. I think I used the wrong arrows, for what I was supposed to explain I should have use -> and <-> instead of => and <=>. Is there any easy way to explain the difference of these two kinds of implications?

    Is it correct to use the implication a>0 -> "a is positive" or a >0 => "a is positive"
  5. Jul 2, 2013 #4


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  6. Jul 3, 2013 #5


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    Implication and Equivalence

    Quine has called this an unfortunate choice of terminology dating back at least to Russell of calling the statement connective '[itex]\supset[/itex]' or '→' "implication". This invites confusion with the notion of "logical implication" which is the relationship between formulas A and B when it is not possible for A to be true and B false.

    Similarly, by calling the sentence connective '[itex]\leftrightarrow[/itex]' or '[itex]\equiv[/itex]' "equivalence" we invite confusion with the notion of "logical equivalence" which is the relationship between formulas A and B when A logically implies B and B logically implies A.

    Sometimes the symbol '[itex]\Rightarrow[/itex]' is used for logical implication and the symbol '[itex]\Leftrightarrow[/itex]' is used for logical equivalence. Notice, however, that in this case these symbols belong not to the object language (sentential calculus, predicate calculus, etc.) but to the meta-langauge. Logical implication and logical equivalence are relationships between formulas not sentence connectives.

    Using '[itex]\Rightarrow[/itex]' for logical implication and '[itex]\Leftrightarrow[/itex]' for logical equivalence, we can capture their relationship with '→' and '[itex]\leftrightarrow[/itex]' as follows:

    A [itex]\Rightarrow[/itex] B if and only if 'A → B' is logically true.
    A [itex]\Leftrightarrow[/itex] B if and only if 'A [itex]\leftrightarrow[/itex] B' is logically true.
    Last edited: Jul 3, 2013
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