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I do not think you can find a brief answer to this debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

 

Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of the possibility of the negation, which is quite "standard" in modal logic, but only moves the problem one step back.

 

Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

And also a contradiction is "defined" as a logical impossibility.

So again, we have a sort of "circularity".

 

Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic [see SEP's entry] :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered. [emphasis added].

I do not think you can find a brief answer to this debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of the possibility of the negation, which is quite "standard" in modal logic, but only moves the problem one step back.

Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered.

I do not think you can find a brief answer to this debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

 

Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of the possibility of the negation, which is quite "standard" in modal logic, but only moves the problem one step back.

 

Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

And also a contradiction is "defined" as a logical impossibility.

So again, we have a sort of "circularity".

 

Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic [see SEP's entry] :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered [emphasis added].

2 added 1 character in body
source | link

I do not think you can find a brief answer to thithis debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of the possibility of the negation, which is quite "standard" in modal logic, but only moves the problem one step back.

Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered.

I do not think you can find a brief answer to thi debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of possibility, which is quite "standard" in modal logic, but only moves the problem one step back.

Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered.

I do not think you can find a brief answer to this debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of the possibility of the negation, which is quite "standard" in modal logic, but only moves the problem one step back.

Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered.

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source | link

I do not think you can find a brief answer to thi debated issue.

See at least Modal Logic, Varieties of Modality, The Epistemology of Modality and Possible Worlds.

Def (A) of necessity

A statement that cannot be untrue

is quite useless; "cannot be" means "impossible". Thus, necessity is simply the negation of possibility, which is quite "standard" in modal logic, but only moves the problem one step back.

Def (C)

A statement whose negation implies a contradiction

reduces necessity to (logical) entailment. But the relation of entailment is basically the formalization of "logical necessity" : B necessarily follows from A iff A entails B (i.e. iff B is logical consequence of A).

Def (B)

A statement which is true in all possible worlds

is the base of the modern semantics for the languages of modal logic :

The power and appeal of basic possible world semantics is undeniable. In addition to providing a clear, extensional formal semantics for a formerly somewhat opaque, intensional notion, cashing possibility as truth in some possible world and necessity as truth in every such world seems to tap into very deep intuitions about the nature of modality and the meaning of our modal discourse.

Unfortunately, the semantics leaves the most interesting — and difficult — philosophical questions largely unanswered.