Thank you for the great content! A quick question regarding the example at 9:53, is not in the I(G), though 3 is greater than 1. Do I understand correctly that it is still considered to be false because it is not defined in the model M? Many thanks!
Yes, is not in the model even though 3 is greater than 1. We might say that this model doesn't do a good job of describing reality (or the relations between numbers) because a model that actually models numbers would have it such that is included in the interpretation.
What if the domain is empty? I read somewhere, when searching around, that domains in predicate logic aren't allowed to be empty. Also, when looking at natural deduction and proof systems, Ex(x = x) follows from the axioms alone. Authors of some books show Ax[x = x] as a rule of inference, or x = x, but the same seems to follow, i.e. Ex[x = x] 1. Ax[x = x] 2. x = x 3. Ex[x = x] So something always exists in the domain that is itself. Can you explain or confirm?
Yes, usually there is the assumption that (1) names are referring and (2) the domain isn't empty. When you drop the assumption that names are referring (1), then you get Free Logic. When you drop (1) and (2), you get Inclusive Logic (or Universally Free Logic). I've never really looked at (2) in much detail, I just know that it exists.
Thank you for the great content! A quick question regarding the example at 9:53, is not in the I(G), though 3 is greater than 1. Do I understand correctly that it is still considered to be false because it is not defined in the model M? Many thanks!
Yes, is not in the model even though 3 is greater than 1. We might say that this model doesn't do a good job of describing reality (or the relations between numbers) because a model that actually models numbers would have it such that is included in the interpretation.
Thank you so much! Excellent video!
Glad it was helpful! Thanks for the positive comment.
What if the domain is empty? I read somewhere, when searching around, that domains in predicate logic aren't allowed to be empty.
Also, when looking at natural deduction and proof systems, Ex(x = x) follows from the axioms alone.
Authors of some books show Ax[x = x] as a rule of inference, or x = x, but the same seems to follow, i.e. Ex[x = x]
1. Ax[x = x]
2. x = x
3. Ex[x = x]
So something always exists in the domain that is itself.
Can you explain or confirm?
Yes, usually there is the assumption that (1) names are referring and (2) the domain isn't empty. When you drop the assumption that names are referring (1), then you get Free Logic. When you drop (1) and (2), you get Inclusive Logic (or Universally Free Logic). I've never really looked at (2) in much detail, I just know that it exists.
@@LogicPhilosophy
I thought I was doing something wrong, but I'm glad that I've found that out, thank you.
Great work, this helped me quite a lot!
+MrTobbeman22 you're welcome! Good luck!
really cool stuff, my lifesaver!
Thanks. Glad it helped!
Domain: All types of thanks.
G(x):___________x you.
∀x[G(x)]. is always true.
Best comment ever. Many thanks!
Great comment it helped me reviewing too xD