Q&A: What Is a Logical Proposition?
What Is a Logical Proposition?
Question
Hello Rabbi,
- Is it correct to define a "proposition" in logic as "a sentence (statement) that has a truth value," meaning that it is either true or false? If so, then sentences like "There is a table in the room" are propositions, and so is "Socrates is a human being."
Is "I promise you…" a proposition? Is "You love watermelon" a proposition?
2. In mathematics, can there be circular definitions, or is that still a problem? For example: a vertex of a polygon is the point of intersection of two sides, while a side is the segment connecting two adjacent vertices of a polygon.
According to the first definition, vertex relies on side, and according to the second definition it is exactly the other way around.
If that is possible, then definitions in mathematics (aside from those that rely on primitive notions—point, line, and plane) are like dictionary definitions: the definition of word A may rely on the definition of word B, and at the very same time the reverse is also true.
I would be happy to hear your clarification on this matter.
Best regards,
Mordechai
Answer
- Yes. Those are definitely propositions. Why not?
- No. You cannot give a circular definition.
Discussion on Answer
I already answered everything. I don't understand what you're asking. Obviously this is a circular definition. What's the question?
My question is how the circularity can be resolved.
What would you change in the definition of side or vertex so that they would not depend on each other?
It's not a matter of changing it. You have to decide what the primitive notions are that are used without definition, and define everything else on their basis. There must be, at the foundation, a set of undefined terms.
1. If so, do you agree with the definition that a proposition is "a sentence with a truth value"?
2. How can one get out of this circularity:
Vertex = the meeting point of two sides.
Side = the segment connecting two adjacent vertices.
The two concepts—vertex and side—depend on each other.
Thanks