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I understand how in propositional logic, every proposition is a statement that is either TRUE or FALSE, and we can make use of logical connectors to turn separate propositions into fromulas which intern are more complex propositions.

My question is, just to see if i get this right: Is any proposition (be it a compound of predicates with quantors or just the basic well-formed formula), that's written down without any indication of its truth value, assumed to be true?

This would clarify why in higher level math topics (depending on formal logic and axiomatic set theory, like real analysis, linear algebra, etc..), when we state a theorem, we only write down a proposition without specifying truth values.

Mauro ALLEGRANZA
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cmatteo
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  • Can you give a specific example? – Mark Saving Jan 31 '22 at 22:56
  • @MarkSaving For instance if i see $\forall x, (x \in \emptyset \lor x \notin \emptyset)$ in a textbook i simply read it as being true but from a formal logic standpoint, I've read that any proposition is either True or False. So i was wondering if there's a convention for the case where people don't explicitly state the truth value – cmatteo Jan 31 '22 at 23:10
  • I didn’t mean, “Give me an example of a proposition.” I meant, “Give me an example of a proposition in the context that you are seeing them in the textbook.” – Mark Saving Jan 31 '22 at 23:37
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    When in a textbook we state a Theorem, we are asserting it as TRUE, because the proof follows the statement: maybe it is left as exercise, or it is "left to the reader" because it is quite simply. – Mauro ALLEGRANZA Feb 01 '22 at 09:43

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Is any proposition that's written down without any indication of its truth value, assumed to be true?

Yes: in non-formal logic, issued statements are tacitly understood—rather than assumed— as true, just like the first four words in “It is true that I went to the cinema” are typically omitted except for emphasis.

every proposition is a statement that is either TRUE or FALSE

Side point: some philosophy texts seem to distinguish statements from propositions in that the former express/interpret the latter over a domain of discourse.

ryang
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