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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/false 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.

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.

1. Yes: typically in meta and informal logic, “$P$ implies $Q$” or “for all $x, P(x)$” tacitly does mean “$P$ being true implies that $Q$ is true” or “for all $x, P(x)$ is true”, just like the first four words in “It is true/false that I went to the cinema” are uttered only for emphasis.

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

   To be clear, I'd say that the proposition is understood—rather than assumed— to be true.

3. every proposition is a statement that is either TRUE or FALSE

   Some philosophy texts would disagree and instead say that every statement is a proposition, because they distinguish statements from propositions in that the former expresses/interprets the latter.

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/false 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.

1. Yes: typically in meta and informal logic, “$P$ implies $Q$” or “for all $x, P(x)$” tacitly does mean “$P$ being true implies that $Q$ is true” or “for all $x, P(x)$ is true”, just like the first four words in “It is true/false that I went to the cinema” are uttered only for emphasis.

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

   To be clear, I'd say that the proposition is understood—rather than assumed— to be true.

3. every proposition is a statement that is either TRUE or FALSE

   Some philosophy texts would disagree and instead say that every statement is a proposition, because they distinguish statements from propositions in that the former expresses the latter.

1. Yes: typically in meta and informal logic, “$P$ implies $Q$” or “for all $x, P(x)$” tacitly does mean “$P$ being true implies that $Q$ is true” or “for all $x, P(x)$ is true”, just like the first four words in “It is true/false that I went to the cinema” are uttered only for emphasis.

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

   To be clear, I'd say that the proposition is understood—rather than assumed— to be true.

3. every proposition is a statement that is either TRUE or FALSE

   Some philosophy texts would disagree and instead say that every statement is a proposition, because they distinguish statements from propositions in that the former expresses/interprets the latter.