Thanks for this clarification. When I first read the sentence shminux quoted, I imagined that any assignment from sentences to truth values constituted a model, but then realized I was confused when I got to the sentence in the next paragraph saying that, “because there is no model of the theory of order which is also a model of [(∃x)¬(x≤x)].”
I thought, “What about the ‘model’ where you directly assign (∃x)¬(x≤x) to true as well as the sentences in the theory?” But if I understand correctly now, what I just described does not constitute a model. All models are maps from all sentences in a language to a truth value, but not all such maps are models.
Is there a straightforward way to describe which maps are allowed as models (of a given theory)? Is it something like—a model is any truth assignment that does not mark as false any sentences in the theory or any sentences that can be derived from the theory by constructing new sentences according to the rules of the logic?
Yes! I cover this in the first part of Very Basic Model Theory, the next post. “Model” is the name for the subset of mappings from sentences to truth values that interpret the logical symbols in a very specific way. See the linked post for details.
Thanks for this clarification. When I first read the sentence shminux quoted, I imagined that any assignment from sentences to truth values constituted a model, but then realized I was confused when I got to the sentence in the next paragraph saying that, “because there is no model of the theory of order which is also a model of [(∃x)¬(x≤x)].”
I thought, “What about the ‘model’ where you directly assign (∃x)¬(x≤x) to true as well as the sentences in the theory?” But if I understand correctly now, what I just described does not constitute a model. All models are maps from all sentences in a language to a truth value, but not all such maps are models.
Is there a straightforward way to describe which maps are allowed as models (of a given theory)? Is it something like—a model is any truth assignment that does not mark as false any sentences in the theory or any sentences that can be derived from the theory by constructing new sentences according to the rules of the logic?
Yes! I cover this in the first part of Very Basic Model Theory, the next post. “Model” is the name for the subset of mappings from sentences to truth values that interpret the logical symbols in a very specific way. See the linked post for details.
Thanks!