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Thread: Modal Logic

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    Modal Logic

    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]

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    https://en.wikipedia.org/wiki/Dynami..._(modal_logic)

    Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later applied to more general complex behaviors arising in linguistics, philosophy, AI, and other fields.

    Language[edit]

    Modal logic is characterized by the modal operators ◻ p {\displaystyle \Box p} \Box p (box p) asserting that p {\displaystyle p\,\!} p\,\! is necessarily the case, and ◊ p {\displaystyle \Diamond p} \Diamond p (diamond p) asserting that p {\displaystyle p\,\!} p\,\! is possibly the case. Dynamic logic extends this by associating to every action a {\displaystyle a\,\!} a\,\! the modal operators [ a ] {\displaystyle [a]\,\!} [a]\,\! and ⟨ a ⟩ {\displaystyle \langle a\rangle \,\!} \langle a\rangle \,\!, thereby making it a multimodal logic. The meaning of [ a ] p {\displaystyle [a]p\,\!} [a]p\,\! is that after performing action a {\displaystyle a\,\!} a\,\! it is necessarily the case that p {\displaystyle p\,\!} p\,\! holds, that is, a {\displaystyle a\,\!} a\,\! must bring about p {\displaystyle p\,\!} p\,\!. The meaning of ⟨ a ⟩ p {\displaystyle \langle a\rangle p\,\!} \langle a\rangle p\,\! is that after performing a {\displaystyle a\,\!} a\,\! it is possible that p {\displaystyle p\,\!} p\,\! holds, that is, a {\displaystyle a\,\!} a\,\! might bring about p {\displaystyle p\,\!} p\,\!. These operators are related by [ a ] p ≡ ¬ ⟨ a ⟩ ¬ p {\displaystyle [a]p\equiv \neg \langle a\rangle \neg p\,\!} [a]p\equiv \neg \langle a\rangle \neg p\,\! and ⟨ a ⟩ p ≡ ¬ [ a ] ¬ p {\displaystyle \langle a\rangle p\equiv \neg [a]\neg p\,\!} \langle a\rangle p\equiv \neg [a]\neg p\,\!, analogously to the relationship between the universal (∀ {\displaystyle \forall \,\!} \forall \,\!) and existential (∃ {\displaystyle \exists \,\!} \exists \,\!) quantifiers.

    Dynamic logic permits compound actions built up from smaller actions. While the basic control operators of any programming language could be used for this purpose, Kleene's regular expression operators are a good match to modal logic. Given actions a {\displaystyle a\,\!} a\,\! and b {\displaystyle b\,\!} b\,\!, the compound action a ∪ b {\displaystyle a\cup b\,\!} a\cup b\,\!, choice, also written a + b {\displaystyle a+b\,\!} a+b\,\! or a | b {\displaystyle a|b\,\!} a|b\,\!, is performed by performing one of a {\displaystyle a\,\!} a\,\! or b {\displaystyle b\,\!} b\,\!. The compound action a ; b {\displaystyle a{\mathbin {;}}b\,\!} {\displaystyle a{\mathbin {;}}b\,\!}, sequence, is performed by performing first a {\displaystyle a\,\!} a\,\! and then b {\displaystyle b\,\!} b\,\!. The compound action a ∗ {\displaystyle a{*}\,\!} {\displaystyle a{*}\,\!}, iteration, is performed by performing a {\displaystyle a\,\!} a\,\! zero or more times, sequentially. The constant action 0 {\displaystyle 0\,\!} 0\,\! or BLOCK does nothing and does not terminate, whereas the constant action 1 {\displaystyle 1\,\!} 1\,\! or SKIP or NOP, definable as 0 ∗ {\displaystyle 0{*}\,\!} {\displaystyle 0{*}\,\!}, does nothing but does terminate.

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    Veteran Member Wiploc's Avatar
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    Everything I know about modal logic came from reading one of Plantinga's books.

    Using Plantinga's modal ontological argument, I can equally prove that god does exist and doesn't exist.

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    Veteran Member George S's Avatar
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    Quote Originally Posted by Wiploc View Post
    Everything I know about modal logic came from reading one of Plantinga's books.

    Using Plantinga's modal ontological argument, I can equally prove that god does exist and doesn't exist.
    I remembered from IIDB: https://frdbarchive.org/search.php?k...&submit=Search

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    Contributor Speakpigeon's Avatar
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    Quote Originally Posted by steve_bank View Post
    Picking up from another thread. It sounds useful. A formal way to deal with informal probabilities and uncertainties.
    As you wouldn't know, modal logic doesn't deal with probabilities, informal or not. Uncertainty yes, probabilities, no. In fact, with both uncertainty and certainty.

    informal probability
    An informal probability summarizes your state of knowledge, no matter how much or how little knowledge that is. You can make an informal probability in a second, based on your present level of confidence, or spend time making it more precise by looking for details, anchors, reference classes. It is perfectly valid to assign probabilities to things you don't have numbers for, to things you're completely ignorant about, to things that are too complex for you to model, and to things that are poorly defined or underspecified. Giving a probability estimate does not require *any* minimum amount of thought, evidence, or calculation. Giving an informal probability is not a claim that any relevant mathematical calculation has been done, nor that any calculation is even possible.
    https://www.lesswrong.com/posts/uMXY...-probabilities
    EB

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    Contributor Speakpigeon's Avatar
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    Quote Originally Posted by Wiploc View Post
    Everything I know about modal logic came from reading one of Plantinga's books.

    Using Plantinga's modal ontological argument, I can equally prove that god does exist and doesn't exist.
    I wouldn't want to use Plantinga as a reliable authority or even honest pundit to disparage my pet peeve personified.

    It's a fact of life that people use modality to reason and argue in the context of uncertainty and certainty. There is a branch of logic working on the formalisation of this sort of argument. I think myself most of what these people do is crap. But, it remains a fact of life that people use modality to reason and argue in the context of uncertainty and certainty. That should be pause for thought, not hasty dismissal.
    EB

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    Veteran Member George S's Avatar
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    Quote Originally Posted by Speakpigeon View Post
    Quote Originally Posted by Wiploc View Post
    Everything I know about modal logic came from reading one of Plantinga's books.

    Using Plantinga's modal ontological argument, I can equally prove that god does exist and doesn't exist.
    I wouldn't want to use Plantinga as a reliable authority or even honest pundit to disparage my pet peeve personified.

    It's a fact of life that people use modality to reason and argue in the context of uncertainty and certainty. There is a branch of logic working on the formalisation of this sort of argument. I think myself most of what these people do is crap. But, it remains a fact of life that people use modality to reason and argue in the context of uncertainty and certainty. That should be pause for thought, not hasty dismissal.
    EB
    Well we at least agree about Plantinga. My great-granddaughter (2 yrs old) can reason better than he.

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    We all use modal logic, we just don't formalize it. Part of informal logic.

    What Model Logic does is apply a symbolic system. I can see where it could be useful in decision making in technology in some cases.

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    Veteran Member Cheerful Charlie's Avatar
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    Modal logic in the matter of possible worlds is another aspect of modal logic. It can be abused by woo peddlers. For example, we might consider the concept of possible worlds where Donald Trump does not exist. But to some theologicans God is necessary in all possible worlds. And then you have people like David Lewis who claims all possible worlds are equally real as our present actualized world.

    Possible world semantics of Saul Kripke is where one starts to understand modern possible world modal logic.
    Cheerful Charlie

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    Contributor Speakpigeon's Avatar
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    Quote Originally Posted by Cheerful Charlie View Post
    Modal logic in the matter of possible worlds is another aspect of modal logic. It can be abused by woo peddlers. For example, we might consider the concept of possible worlds where Donald Trump does not exist. But to some theologicans God is necessary in all possible worlds. And then you have people like David Lewis who claims all possible worlds are equally real as our present actualized world.

    Possible world semantics of Saul Kripke is where one starts to understand modern possible world modal logic.
    Oh, good, so perhaps you could explain the wisdom of Kripke's notion of "rigid designator"?

    My brain sort of screams at this notion and I'd like to enjoy peace of mind...

    Rigid designator
    In modal logic and the philosophy of language, a term is said to be a rigid designator or absolute substantial term when it designates the same thing in all possible worlds in which that thing exists and does not designate anything else in those possible worlds in which that thing does not exist.
    https://en.wikipedia.org/wiki/Rigid_designator
    EB

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