Question about LSP (Liskov Substitution Principle) and subtypes?

Q(x) is true for any x of T and q(y) is true for any y of S => S is a subtype of T The answer is No What the expression means is that a common supertype R of S and T could be defined, and that then the LSP (shame on how that name became mainstream) would hold for T->R and S->R In typing theory, there are types, that include semantics, and there are implementations of the types that abide to the semantics, perhaps by inheriting implementations In practice, the only reasonable way to specify the semantics of a type (the q(x) part) is through an implementation, so we are left with semantic-less signatures in the form of interfaces and classes that inherit for implementation purposes, and implement the interfaces they like, with no way to check if they are doing it correctly Researches have tried to define formal languages to specify types, so tools can check if an implementation abides to type definitions, but the effort is so large that it would do as good to compile the formal language into executable code. It's a Catch-22 situation that I think will never be solved Back to your original question, in languages that allow what today is called "Duck Typing", the answer is undecidable, because an object of any type can be passed to any function, and the typing is right if the correct signatures are implemented and the result is right. Let me explain In a language like Eiffel you could place a postcondition on List.append() that List.length() must increase after the operation.

That is not the way languages like Perl, JavaScript, Python, or even Java work. That lack of type-strictness allows for much more succinct code than stricter type definitions would.

Q(x) is true for any x of T and q(y) is true for any y of S => S is a subtype of T The answer is No. What the expression means is that a common supertype R of S and T could be defined, and that then the LSP (shame on how that name became mainstream) would hold for T->R and S->R. In typing theory, there are types, that include semantics, and there are implementations of the types that abide to the semantics, perhaps by inheriting implementations.In practice, the only reasonable way to specify the semantics of a type (the q(x) part) is through an implementation, so we are left with semantic-less signatures in the form of interfaces, and classes that inherit for implementation purposes, and implement the interfaces they like, with no way to check if they are doing it correctly.

Researches have tried to define formal languages to specify types, so tools can check if an implementation abides to type definitions, but the effort is so large that it would do as good to compile the formal language into executable code. It's a Catch-22 situation that I think will never be solved. Back to your original question, in languages that allow what today is called "Duck Typing", the answer is undecidable, because an object of any type can be passed to any function, and the typing is right if the correct signatures are implemented and the result is right.

Let me explain... In a language like Eiffel you could place a postcondition on List.append() that List.length() must increase after the operation. That is not the way languages like Perl, JavaScript, Python, or even Java work. That lack of type-strictness allows for much more succinct code than stricter type definitions would.

It does not make sense; your statement using and is symmetric in S and T. But I think you meant to say the following If it is the case that for any proposition q such that q(x) is provable for all x of type T, then q(y) is also provable for all y:of type S, than we may consider S a subtype of T. I would prefer to use mathematical logic rather than informal English, but if I have got the definition right, this is behavioral subtyping, which these days is often called "duck typing."

It's a perfectly good subtyping principle and again leads to the idea that in any context that expects a value of type T, you may instead supply a value of type S, and it's OK because the value of type S is guaranteed to satisfy all properties that are expected by the context.

I think no, you can't use it as a definition. Besides if q(x) is true for any x of T and q(y) is true for any y of S it could also mean that T is a subtype of S. To be sure of which is a subtype of which (assuming you know that there is an inheritance relationship between them) you also have to know something about which is more "generic" or which is more "specialized" than the other.

I cant really gove you an answer,but what I can give you is a way to a solution, that is you have to find the anglde that you relate to or peaks your interest. A good paper is one that people get drawn into because it reaches them ln some way.As for me WW11 to me, I think of the holocaust and the effect it had on the survivors, their families and those who stood by and did nothing until it was too late.