I'm reading Logic by Wifred Hodges (very good IMHO)

There is an exercise express the following as a truth functor -

She needs all the help she can get being a single parent.

My answer would be

She is a single parent [1] => she needs all the help she can get [2]

Truth Table

[1] [2] [1] => [2]
T   T   T
T   F   F
F   T   T
F   F   T

BUT the answer in the book is

She needs all the help she can get ^ She is a single parent

Truth Table

[1] [2] [1] ^ [2]
T   T   T
T   F   F
F   T   F
F   F   F

I don't understand where I am wrong. I think that [1] implies [2] so that my answer seems right but I'm a bit lost. Can anyone give me some pointers please.

Many Thanks for your thoughts

  • 2
    I think that the "translation" suggested by Hodegs is that tha statement "She needs all the help she can get being a single parent" presuppose that she is "single": thus the choice of "and". With "imply", the statement will be true also when she is not "single". May 17 '14 at 14:21
  • 1
    I'm with the OP. It's an implication. It says that "She is a single parent, therefore she needs all the help she can get."
    – user4894
    May 17 '14 at 17:21
  • 1
    The issue is with the term "being," which doesn't translate straightforwardly into any one logical connective as used here. Interpreted as ordinary language, it is suggestive of both implication and assertion. The logical consequence of this double suggestion is the same as the AND truth table, so that's arguably the more correct answer. But this is something of a trick question as I see it!
    – senderle
    May 20 '14 at 13:13

When we make statements in natural language there are ambiguities and layers of meaning in even simple statements, but translating them into formal logic removes those.

In this case the original statement tells us that she is a single parent and that she needs all the help she can get (the AND statement from the book). It also seemingly implies that if one is a single parent, then one needs all the help you can get (your IMPLIES statement). However, in terms of logic, the AND statement is much stronger --knowing that both the AND version and the IMPLIES version are true tells us exactly the same information about this person as just the AND version alone.

To elaborate: If you have IF A THEN B you don't necessarily have A or B, you just have a guarantee that in the case you have A you also have B. So knowing that you do have A AND B is a lot more info than knowing IF A THEN B, because if you have A AND B, you always also have IF A THEN B. In fact if you even just have B, then you still will always have IF A THEN B.

The reason this is counter-intuitive is that in natural language, when we say "If A then B" we're usually not talking about one A and B, but a whole set of A's and B's --for instance, all single parents --which does in fact convey additional information. You can't express that in basic propositional logic --you would need predicates and quantifiers in order to do so.

  • +1 thank you. But i don't understand why AND and IMPLIES has the same informational value as just AND alone. I would have though that IMPLIES gives additional information - they two are related in some way May 17 '14 at 21:39
  • @CrabBucket I edited the answer to add more info. May 18 '14 at 3:49
  • Thank you again. So the difference is that this is a particular rather than a universal. So 'single parents need all the help they can get' would be implied rather than and. Is that correct? May 18 '14 at 9:19
  • Please note, I edited my answer again to remove an inaccuracy: you can't do universals or predicates in propositional logic, but you can do them in first order logic. In the case of your new statement "single parents need all the help they can get", you would really need something like FOR ALL X ( (X is a parent AND X is single) IMPLIES X needs help) which is beyond the reach of propositional logic. May 19 '14 at 13:26

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