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First, sorry if this is a truly silly or question.

Basically, this is the result of a debate I was having. I have only my Comp sci education of a decade ago to fall back on. I was curious if the community could help. This is really a 2 part question and I'm not sure really how to tackle it.

So ...

If we define a statement as a collection of grammatically correct words.

And we acknowledge from that definition that some statements may have no truth value whatsoever. And for the purpose of argument we either accept or disregard something that may be subjective.

Part 1

Is the set of all true statements the same size as the set of all false statements?

This is somewhat different than a propositional logic question.

  • If I go to the mall, all fish can't swim.

The if implies causality. In natural language it is equivalent to

  • If I go to the mall, all fish can't swim, because I went to the mall.

Simply adding a not doesn't cover it. My going to the mall has no affect on fish.

But I could say well

  • The Statement "If I go to the mall, all fish can't swim. " is false.

Is it's negation. But then what is the negation of

  • My going to the mall does not affect fish.

or

  • My going to the mall does not affect fish's ability to swim.

So. Is there a 1 to 1 mapping of true and false?

Part 2

What if we disregard statements that are logically duplicates of other statements.

So....

  • My going to the mall does not affect fish's ability to swim.

is logically equivalent to

  • he Statement "If I go to the mall, all fish can't swim. " is false.

but is not logically equivalent to

  • My going to the mall does not affect fish.

Does that change the answer?

EDIT:

I don't even think those two are equivalent.

  • My going to the mall does not affect fish's ability to swim.

  • My going to the mall does not affect fish's ability to swim past 1/m/s.

  • My going to the mall does not affect fish's ability to swim deeper than 2000 meters

I'm having a real problem with part 2.

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  • $\begingroup$ Hint: if the set of words is finite, and each sentence's length is finite, then...? $\endgroup$ – Fimpellizieri Oct 9 '16 at 3:23
  • $\begingroup$ @Fimpellizieri this was my initial line of thinking as well. But a sentence's possible length is not finite. To part one, You could always add the (This Statement..... Is false/true) at the end. At least with part 1, the set of possible statements are infinite. $\endgroup$ – 8bitwide Oct 9 '16 at 3:30
  • $\begingroup$ Sorry, do you mean to consider a sentence with infinite words? My hint didn't mean to imply the set of possible statements to be finite (it clearly isn't), only that it is countable. $\endgroup$ – Fimpellizieri Oct 9 '16 at 5:03
  • $\begingroup$ @Fimpellizieri. I'm not sure that is even true. If you take the view that logically equivalent statements are different. I can negate a statement infinitely; and add an addendum from a infinite set of other statements( including their negations). I don't see how that is countable. I also think the point is silly. I'm leaning toward Part 1 being equal sets. But part 2 not. But this is mostly intuition. I'm lacking a formal framework to work it out. $\endgroup$ – 8bitwide Oct 9 '16 at 5:19
  • $\begingroup$ Smith and Jones are the only two people who could possibly have committed the murder. Is "If Smith didn't do it, then Jones did it" a true or false statement, according to you? Where is the causality? Did Smith's inaction cause Jones to commit murder? What about "If Smith passes the lie detector test, then he's innocent"? Can the result of Smith's lie detector test today cause him to refrain from commiting a crime the day before yesterday? Where do you get the idea that "if" implies causality? $\endgroup$ – bof Oct 9 '16 at 5:48

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