02x07 - Translating Propositions Into Atomic Formulas And Negated Propositions Into Negated Formulas

Welcome to Fast Philosophy. This video is part of our Translating Into Propositional Logic series and explains how to translate natural language propositions and negated propositions into formal atomic formulas and negated formulas.
The simplest translation is translating a single natural language proposition into a formal atomic formula. For example, ‘Socrates likes pineapple’ is a single proposition in natural language. So, using a key to show what the sentence letter represents, the single proposition is translated into an atomic formula.
‘Socrates likes pineapple and quince’ is not a single proposition. This natural language sentence expresses two propositions: that ‘Socrates likes pineapple’ and that ‘Socrates likes quince’. So, using a key, we can translate the natural language sentence into two atomic formulas then connect them using the caret, which stands for ‘and’. More on this in later videos. The point is that any sentence containing a connective cannot be an atomic formula.
Atomic formulas must not contain a negation either. For example, the proposition ‘Socrates does not like pineapple’ must be translated as ‘~P’ where P = ‘Socrates likes pineapple’. This is because we must make it clear whenever an operator is used. ‘~P’ can be read: ‘It is not the case that Socrates likes pineapple’.
Negation can also be found in some prefixes. For example, ‘Socrates dislikes pineapple’ is the negation of ‘Socrates likes pineapple’. This is because the prefix ‘dis’ negates ‘likes’. However, a prefix which negates in one sentence may not operate to negate in another sentence. For example, whereas ‘dis-’ negates ‘likes’, ‘dis-’ does not negate ‘cover’ in ‘discover’. ‘Socrates discovered the pineapple’ cannot be understood as meaning: ‘It is not the case that Socrates covered pineapple’. You often have to employ your intuition when translating negating prefixes.
Intuition is also required when translating in order to retain as much information as possible. For example, take the natural language sentence ‘If Socrates is at home, then Plato is at work’. It is tempting to translate this sentence, using a key in which S = ‘Socrates is at home’ and P = ‘Plato is at home’, as ‘(S⊃~P)’. But this is an uncharitable translation and does not retain the maximal amount of information possible. Although it may be true that when Plato is at work he is not at home, it may also be true that he works from home and nothing in the sentence tells us otherwise. Furthermore, we have been given more information than just that Plato is not at home. We have been told that Plato is at work. So, our translation should use the key P = ‘Plato is at work’. The correct, charitable translation which retains the maximal amount of information is ‘(S⊃P)’.
Recall that we may have any number of negation operators. For example, ‘There are no disinterested students of logic’ may be translated, using a key in which ‘L = There are interested students of logic’, as ‘~~L’. This can be read: ‘It is not the case that it is not the case that there are interested students of logic’.

[RajSahaRks]

"Dave is Playing with a red Ball" what will be the atomic formulae for this sentence?