steve_bank
Diabetic retinopathy and poor eyesight. Typos ...
Picking up from another thread. It sounds useful. A formal way to deal with informal probabilities and uncertainties.
https://en.wikipedia.org/wiki/Modal_logic
Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. A modal—a word that expresses a modality—qualifies a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p").[1] Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"),[2][3] deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p")[4] and doxastic modalities, or modalities of belief ("It is believed that p").[5]
A formal modal logic represents modalities using modal operators. For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, "Possibly", attached to the sentence "It will rain today".
It is fallacious to confuse necessity and possibility. In particular, this is known as the modal fallacy.
The basic unary (1-place) modal operators are usually written "□" for "Necessarily" and "◇" for "Possibly". Following the example above, if P {\displaystyle P} P is to represent the statement of "it will rain today", the possibility of rain would be represented by ◊ P {\displaystyle \Diamond P} \Diamond P. This reads: It is possible that it will rain today. Similarly ◻ P {\displaystyle \Box P} \Box P reads: It is necessary that it will rain today, expressing certainty regarding the statement.
In a classical modal logic, each can be expressed by the other with negation.
◊ P ↔ ¬ ◻ ¬ P ; {\displaystyle \Diamond P\leftrightarrow \lnot \Box \lnot P;} \Diamond P\leftrightarrow \lnot \Box \lnot P;
In natural language, this reads: it is possible that it will rain today if and only if it is not necessary that it will not rain today. Similarly, necessity can be expressed in terms of possibility in the following negation:
◻ P ↔ ¬ ◊ ¬ P {\displaystyle \Box P\leftrightarrow \lnot \Diamond \lnot P} {\displaystyle \Box P\leftrightarrow \lnot \Diamond \lnot P}
which states it is necessary that it will rain today if and only if it is not possible that it will not rain today. Alternative symbols used for the modal operators are "L" for "Necessarily" and "M" for "Possibly".[6]
https://en.wikipedia.org/wiki/Modal_logic
Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. A modal—a word that expresses a modality—qualifies a statement. For example, the statement "John is happy" might be qualified by saying that John is usually happy, in which case the term "usually" is functioning as a modal. The traditional alethic modalities, or modalities of truth, include possibility ("Possibly, p", "It is possible that p"), necessity ("Necessarily, p", "It is necessary that p"), and impossibility ("Impossibly, p", "It is impossible that p").[1] Other modalities that have been formalized in modal logic include temporal modalities, or modalities of time (notably, "It was the case that p", "It has always been that p", "It will be that p", "It will always be that p"),[2][3] deontic modalities (notably, "It is obligatory that p", and "It is permissible that p"), epistemic modalities, or modalities of knowledge ("It is known that p")[4] and doxastic modalities, or modalities of belief ("It is believed that p").[5]
A formal modal logic represents modalities using modal operators. For example, "It might rain today" and "It is possible that rain will fall today" both contain the notion of possibility. In a modal logic this is represented as an operator, "Possibly", attached to the sentence "It will rain today".
It is fallacious to confuse necessity and possibility. In particular, this is known as the modal fallacy.
The basic unary (1-place) modal operators are usually written "□" for "Necessarily" and "◇" for "Possibly". Following the example above, if P {\displaystyle P} P is to represent the statement of "it will rain today", the possibility of rain would be represented by ◊ P {\displaystyle \Diamond P} \Diamond P. This reads: It is possible that it will rain today. Similarly ◻ P {\displaystyle \Box P} \Box P reads: It is necessary that it will rain today, expressing certainty regarding the statement.
In a classical modal logic, each can be expressed by the other with negation.
◊ P ↔ ¬ ◻ ¬ P ; {\displaystyle \Diamond P\leftrightarrow \lnot \Box \lnot P;} \Diamond P\leftrightarrow \lnot \Box \lnot P;
In natural language, this reads: it is possible that it will rain today if and only if it is not necessary that it will not rain today. Similarly, necessity can be expressed in terms of possibility in the following negation:
◻ P ↔ ¬ ◊ ¬ P {\displaystyle \Box P\leftrightarrow \lnot \Diamond \lnot P} {\displaystyle \Box P\leftrightarrow \lnot \Diamond \lnot P}
which states it is necessary that it will rain today if and only if it is not possible that it will not rain today. Alternative symbols used for the modal operators are "L" for "Necessarily" and "M" for "Possibly".[6]
