Modal Logic

Philosopher, Saul Kripke.Modal logic is a branch of formal logic that deals with possibility (◊, L), necessity (□, M), and existence. The most familiar logics in the modal family are constructed from a weak logic named K, for Saul Kripke.

This narrow reading confines modal logic to concerns with necessity and possibility. Modal logic was first developed to deal with these special, alethic modality concepts (possibily true and necessarily true), and was later extended to other concepts. Using K as a foundation, a variety of different logical systems have been developed.

Alethic modality indicates the speaker’s estimation of the logical necessity or possibility of the content of a proposition. Alethic logic deals with factual, or "truth", issues.

A proposition is said to be:
possible if it is not necessarily false (regardless of whether it actually is true or false);
necessary if it is not possibly false;
contingent if it is not necessarily false yet not necessarily true either (a limited case of possibility).

◊ P ↔ ¬ □ ¬ P
P is possible if and only if it is not necessary that not P.

□ P ↔ ¬ ◊ ¬ P
P is necessary if and only if it is not possible that not P.

Because anything is possible that is not necessarily false, almost anything is possible and almost nothing is logically impossible. Thus, possibility is a very weak condition in that much that is logically possible is not necessarily true. In a real sense, logical possibility signifies very little because it inheres so much while explaining so litttle.

It is not clear that there are any necessary propositions and, if there are, they are restricted to analytic propositions or other propositions that are true by virtue of their logical form. If there are necessary events, then natural rather than logical necessity is involved.

Metaphysical possibility is generally considered to be stronger than bare logical possibility in the sense that fewer things are metaphysically possible than are logically possible. A thing is physically possible if it is permitted by the laws of nature, which themselves have been established by the empirical observation of physical entities and phenomena.

Apologetic arguments rely on the fact that almost anything is logically possible, in order to make claims that God 'necessarily exists' according to definitions specially contrived to inhere supernatural existence. Such arguments have been repeatedly refuted, though it is the nature of religious belief to deny inconvenient refutation.

Whereas alethic logic deals with "truth" issues, epistemic logic deals with logical issues stemming from such epistemological concepts as knowledge, belief, assertion, and doubt. Epistemology is a branch of philosophy that deals with the nature, origin and scope of knowledge.

Deontic logic, by contrast, is concerned with obligation, permission, and related concepts (licitum, illicitum, debitum, indifferens). Deontic logic is a formal system that attempts to capture the essential logical features of concepts such as are expressed in "ought", "could", "permissible", "requires", "forbidden", and "obligatory".

Modality and Possible Worlds

In much more detail, videos and text: The Modal Ontological Argument.

McNamara, Paul, "Deontic Logic", The Stanford Encyclopedia of Philosophy (Spring 2006 Edition), Edward N. Zalta (ed.), url.
Lokhorst, Gert-Jan, "Mally's Deontic Logic", The Stanford Encyclopedia of Philosophy (Winter 2004 Edition), Edward N. Zalta (ed.), url.
Hendricks, Vincent and John Symons, "Epistemic Logic", The Stanford Encyclopedia of Philosophy (Spring 2006 Edition), Edward N. Zalta (ed.), url.
Garson, James, "Modal Logic", The Stanford Encyclopedia of Philosophy (Summer 2007 Edition), Edward N. Zalta (ed.), url.
Carnielli, Walter and Marcelo Esteban Coniglio, "Combining Logics", The Stanford Encyclopedia of Philosophy (Fall 2007 Edition), Edward N. Zalta (ed.), url.

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