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For one of my homework challenge questions, I have a true or false type question. The question is as follows:

True or False:

The Question

I'm not particularly sure how to solve this. I have no information about the domain or predicates to go off of.

For the sentence to be a falsehood, it would mean that no interpretation of it can be true. However as I have no information about the interpretation, I am a bit stuck on where to begin to figure it out.

Any guides or pointers to solve this question would be very much appreciated!

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  • You err: an interpretation is false once there is a single instance of falsehood. You are trying to say the definition of false is the reverse of true which is misguided. True means that there is no interpretation of a false value. You tried to reverse that definition. It does not work in reality. If I say All Swans are white is false you are thinking that the statement is false if there is no interpretation of white swans. So does happens if there are some white swans? Is the statement all swans are white NOW true because there claim no interpretation of white swans is unsatisfiable?
    – Logikal
    Feb 18 '20 at 8:48
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    @Logikal I'm not sure what you are trying to say. A sentence that is false in every interpretation is a logical falsehood. However my issue was that I wasn't sure if I could just apply any interpretation to the sentence above in an attempt to make the sentence true.
    – Bongo
    Feb 18 '20 at 8:59
  • @Logikal - What does it mean "an interpretation is false once there is a single instance of falsehood" ? Sentences are True or False, not interpretations. A formula must be interpreted to have meaning, i.e. truth value. Thus, what makes sense is to ask "if a formula is true or false in a specific interpretation". Feb 18 '20 at 9:02
  • "All Swans are white" is a sentence of natural language; it is not a formula that we have to "interpret". It has a truth value (presumably : False). Feb 18 '20 at 9:03
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With "logical falsehood" you mean a formula that is always false, i.e. false in every interpretation.

We can assume as domain of the interpretation the set N of natural numbers and interpret the sentence G with a formula that is always true in N: e.g. "for all x (x >= 0)".

This means that, in the above interpretation, the formula is True.

Alternatively, we can consider the domain of human beings and interpret G with: "for all x, x is Mortal".


Logically valid means "true in every possible interpretation"; similarly we can read "logically falsehood" as unsatisfiable that means "no interpretation make the formula true".

Thus, having found some interpretations that make the formula true, we have to conclude that the formula is satisfiable, i.e. it is not a "logical falsehood".

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  • So because it's left blank, we can basically assume that the domain can be an infinite amount of options that make it true and therefore not a falsehood?
    – Bongo
    Feb 18 '20 at 8:33
  • sorry, i mean because there is no interpretation given, we can come up with our own examples to test the sentence
    – Bongo
    Feb 18 '20 at 8:40
  • @Bongo Indeed. When asked to evaluate the truth of "No sentence of this form is a logical falsehood," you should either (1) build some sentence of that form that is a logical falsehood, xor (2) argue why it is impossible to find such a counterexample. Feb 18 '20 at 23:09

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