[R1] it is contingent that °p° [R2] it is possible but not necessary that °p° [R3] it is n/possible that °p° [F] >

<>p \and <>¬p [I] The contingency of something is the possibility of both it and the opposite. [C1] This may be a better interpretaion of the word "possible" in everyday langauge than possibility as defined in §7: When one says that something is possible one sometimes seems to exclude it necessarily being so. [C2] Actually, the symbol should be two horizontally arranged, at each other pointing unfilled triangles with a common top. [§28:D:N] *Non-contingency* [S] >/

[R] it is not contingent that °p° [F] >/

¬>

><¬p [I] If something is contingent, so is the opposite. [§30:T:F] >

[]p ¬\or []¬p [I] The contingency of something is equivalent to neither it nor its opposite being necessary. [§31:T:F] >

[]p ¬\or p [I] The contingency of something is equivalent to it being neither necessary nor impossible. [§32:T:F] >/

[]p \or! []¬p [I] The non-contingency of something is equivalent to either it or its opposite being necessary. [§33:D:N] *Accidence* [S] |>p <*> [R1] it is accidental that °p° [R2] it is a coincidence that °p° [F] |>p <==> p \and >

p \or! |>¬p \or! []¬p [I] There are four mutually exclusive alternatives: something is necessarily true, it is accidently true, the opposite is accidently true, or the opposite is necessarily true. [§35:T:F] |>p <==> p \and <>¬p [I] The accidental nature of something is equivalent to it being true, still the opposite being possible. 5. Strict implication and compatibility --------------------------------------- [§36:C] There are a few connections between the modality of the constituent sentences of a disjunction or conjunction and the modality of the compound sentence in system T. [§37:T:F] [] (p \and q) <==> []p \and []q [§38:T:F] <> (p \and q) ==> <>p \and <>q [§39:T:F] <> (p \or q) <==> <>p \or <>q [§40:T:F] [] (p \or q) ==> []p \or <>q [§41:T:F] [] (p \or q) <== []p \or []q [§42:C] The conditionals of sentential logic are sometimes called "_material_ implications", which is somewhat improper, since the original concept of implication, _logical_ implication, is a relation between two sentences (belonging to metalogic), rather than a type of sentence. There are, in modal logic, _necessary_ conditionals, and they use to be called "strict implications" in this missleading usage. [§43:D:N] *Strict implication* [S] p --3 q [R] that °p° strictly implies °q°'ing [F] (p --3 q) <==> [](p ==> q) [I] Strict implication is necessary material implication. [C1] A better term would be "strict (straight) conditional". [C2] Actually, the symbol should be a right-pointing arrow whose head is like a symmetric digit 3, a so-called fish-hook arrow. [§44:D:N] *Strict consequence* [S] p E-- q <*> [R] that °p° is a strict consequence of °q°'ing [F] (p E-- q) <==> [](p <== q) [C1] A better term would be "strict reverse conditional". [C2] Egentligen ska symbolen vara en spegelbild av symbolen för strikt implikation. [C2] Actually, the symbol should be the mirror image of the symbol for strict implication. [§45:D:N] *Strict equivalence* [S] p E--3 q <*> [R] that °p° is strictly equivalent to °q°'ing [F] (p E--3 q) <==> [](p <==> q) [C1] A better term would be "strict biconditional". [C2] Actually, the symbol should be the fusion of the symbols for strict implication and strict consequence. [§46:T:F] (p E-- q) <==> (q --3 p) [§47:T:F] (p E--3 q) <==> (p --3 q) \and (p E-- q) [§48:A:N] *Law of strict implication* <*> [F] (p --3 q) ==> ([]p ==> []q) [I] If °p° strictly implies °q°, then the necessity of °p° implies that of °q°. [§49:T:F] (p --3 q) ==> (<>p ==> <>q) [I] If °p° strictly implies °q°, then the possibility of °p° implies that of °q°. [§50:T:F] (p --3 q) ==> (q ==> p) [I] If °p° strictly implies °q°, then the impossibility of °q° implies that of °p°. [§51:T:F] (p --3 ¬p) <==> p [I] A sentence strictly implying its opposite is impossible. [§52:C] The paradoxes of material implication is the fact that a true sentence is implied by any sentence and the fact that a false sentence implies any sentence. Strict implication avoids them but has analogous paradoxes of its own. [§53:T:F] ¬(q --3 (p --3 q)) [§54:T:F] ¬(¬p --3 (p --3 q)) [§55:T:F] []q --3 (p --3 q) [§56:T:F] p --3 (p --3 q) [§57:C] Compatibility and strict incompatibility can be defined similarly. [§58:D:N] *Compatibility* [S] p () q [R1] that °p° is compatible with °q°'ing [R2] that °p° is consistent with °q°'ing [F] (p () q) <==> <>(p \and q) [I] Compatibility is possible conjunction. [C1] See [§59:D:C1]. [C2] Actually, the symbol should be a big unfilled circle. [§59:D:N] *Strict incompatibility* [S] p (/) q <*> [R1] that °p° is strictly incompatible with °q°'ing [R2] that °p° is strictly inconsistent with °q°'ing [F] (p (/) q) <==> [](p ¬\and q) [C1] The 2-place truth function "°p ¬\and q°", i.e. "not both °p° and °q°", means that °p° excludes °q°, that °p° is incompatible with °q°. It may improperly be called a "_material_ incompatibility". If it is necessary, we have "_strict_ incompatibility". The opposite concept, compatibility, does not, however, have any correspondence in pure sentential logic. These concepts are not to be mixed up with _logical_ compatibility and incompatibility, which are relations between sentences. [C2] Actually, the symbol should be a big unfilled circle, crossed by an oblique stroke. [§60:T:F] (p (/) q) <==> ¬(p () q) 6. Logical and other necessity ------------------------------ [§61:M] Besides two new axioms, in this version of modal logic one rule of inference is added, *the rule of necessitation*|: If °p° is a theorem, then °[]p° is a theorem. [§62:C] The rule of necessitation implies that provability is a sufficient condition for necessity. One does indeed speak about _logical necessity_. Which other kinds of necessity there may be is unclear and a very controversial philosophical question. Some philosophers hold that _analytically_ true sentences (being true wholly because of the meanings of their words), a wider concept than logically true sentences, are those that should be regarded as necessary. According to other philosophers, it is sentences that are true _a priori_ (true independently of experience), a yet somewhat wider concept, that coincide with necessary truths. But perhaps also the _causes_ in nature are a source of necessity for those sentences they make true? Or perhaps also laws of nature that do not involve causes are necessarily true? One may certainly speak about _physical necessity_. A less ambitious way of interpreting naïve modal logic is as a qualitative preliminary stage before probability theory, where one does not calculate numerical probabilities but only discern three mutually exclusive alternatives: a) °p° is a necessity: °P(p) = 1° b) °p° is an accident: °0 < P(p) < 1° c) °p° is an impossibility: °P(p) = 0° [§63:C] Different axiom systems of varying strength have been put forward for modal logic, among others C. I. +Lewis+'s systems S1 - S5. The differences are chiefly noticeable in what they say about double modalities. The system S4 results, when the axiom []p ==> [][]p is added to the axiom system T, which means that for everything necessarily so, it is necessary that it so is. Then for example these new theorems can be derived: <>p <==> <><>p that means, only that which actually is possible _may_ be possible, and <>[]<>p ==> <>p (p --3 q) ==> ([]p --3 []q) The system S5 results if one, instead, adds the axiom: <>p ==> []<>p that means, what is possible is possible by necessity. All theorems of S4 are valid in S5 and, in addition, also among others []p <==> <>[]p [] (p \or <>p) <==> ([]p \or <>p) <> (p \and []q) <==> (<>p \and []q) ([]p --3 []q) ==> ([]p E-- []q) In even more liberal systems of modal logic <><>p is valid, which means that that anyting could be possible, even if it is not actually possible. [§64:C] When modalities are added to predicate logic, a difference arises between modality _de re_ and modality _de dicto_, depending on whether the modal connective stands before or after a quantifier. It seems reasonable that \all x ([] P: x) ==> [] \all x (P: x) \all x (<> P: x) ==> <> \all x (P: x) but is it also the case that <> \exist x (P: x) ==> \exist x (<> P: x) which can be proved in S5 (*Barcan's formula*)? That would involve that everything that can exist actually exists. [§65:C] In predicate logic with identity yet other difficulties involving modalities arise. For example, identity is always necessary: x = y ==> [](x = y) Substitution may therefore alter the truth value of a modal sentence: [] (The Evening Star = The Evening Star) and The Evening Star = The Morning Star but [/] (The Evening Star = The Morning Star) 7. Alethic and other modalities ------------------------------- [§66:C] Other families of concepts similar to the alethic modalities are the existential, epistemic, doxastic and deontic modalities. [§67:C] *Alethic* modalities: 1. °p°: actuality 2. °[]p°: necessity 3. °<>p°: possibility 4. °[/]p°: non-necessity 5. °p°: impossibility 6. °>

/

p°: accidence Power law: °[]p ==> p ==> <>p° [§68:C] *Existential* modalities: 1. °P: x°: example 2. °\all x (P: x)°: universality 3. °\exist x (P: x)°: existence 4. °¬ \all x (P: x)°: non-universality 5. °¬ \exist x (P: x)°: emptyness 6. °\exist x (P: x) \and \exist x (¬ P: x)°: 7. °\all x (P: x) \or \all x (¬ P: x)°: 8. °P: a \and \exist x (¬ P: x)°: Power law: °\all x (P: x) ==> P: x ==> \exist x (P: x)° [§69:C] *Epistemic* modalities: 1. - 2. °p° is verified 3. °p° is not falsified 4. °p° is not verified 5. °p° is falsified 6. °p° is decided 7. °p° is undecided 8. - Power law: If °p° is verified, then °p° is not falsified. Alternative interpretations of verified / falsified: a) °p° is proven / °¬p° is proven (logical/mathematical knowledge) b) °p° is confirmed / °¬p° is confirmed (empirical knowledge) c) °S° knows that °p° / °S° knows that °¬p° d) it is known that °p° / it is known that °¬p° [§70:C] *Doxastic* modalities: 1. - 2. °S° believes that °p° 3. °S° does not believe that °p° 4. °S° does not believe that °¬p° 5. °S° believes that °¬p° 6. °S° has no belief concerning whether °p° 7. °S° has a belief concerning whether °p° 8. - Power law: If °S° believes that °p°, then °S° does not believe that °¬p°. The power law shows that only consistent belief is covered here. [§71:C] *Deontic* modalities: 1. - 2. °a° is obligatory 3. °a° is permitted 4. °a° is not obligatory 5. °a° is forbidden 6. °a° is normatively indifferent 7. °a° is obligatory or forbidden 8. - Power law: If °a° is obligatory, then °a° is permitted. 8. Short history ---------------- [§72:C] Like the theory of assertoric syllogism , modal logic was founded by +Aristoteles+, who was clear on [§7:D:F] and [§27:D:F], though less accurate in the treatment of modal syllogism. Among later ancient philosophers, +Diodorus+ is notable for interpreting the possible as that which is true or will be true, and the necessary as that which is true and never will be false. In the islamic culture, +Avicenna+ refined the temporal interpretation of altehtic modality in the first decade of the eleventh century. (Philosophy in both India and China before the twentieth century was completely separate from the western philosophy and it is difficult to discern any direct counterpart to modal logic there.) During the twelfth, thirteenth and fourteenth centuries the theory of modal syllogism was further developed and applied to theological questions by scholastic philosophers. In modern symbolic logic modal concepts were first paid attention to by Clarence Irwing +Lewis+ (1918), who definied strict implication and the axiom systems S1--S5. Ruth Barcan +Marcus+ pioneered the study of modal predicate logic in the fourties, and Georg Henrik von +Wright+ research on deontic modality around 1950. New ways of interpreting modal concepts by means of possible worlds were introduced independently by Stig +Kanger+ and Saul +Kripke+ in the fifties. As the most eminent critic of the soundness and applicability of alethic modal logic Willard Van Orman +Quine+ has appeared during the second half of the twentieth century. 8. Summary ---------- +------------------------------------------------------------------------------+ | Summary of naïve alethic modal logic | +------------------------------------------------------------------------------+ | P = primitive concept, D = definition, A = axiom, T = theorem, M = meta-rule | | [§6:P] []p ["it is necessary that °p°"] | | [§7:D] <>p <==> ¬[]¬p ["it is w/possible that °p°"] | | [§17:A] []p ==> p [Necessity implies actuality.] | | [§18:T] p ==> <>p [Actuality implies possibility.] | | [§27:D] >

<>p \and <>¬p ["it is contingent that °p°"] | | [§33:D] |>p <==> p \and >

p \or! |>¬p \or! []¬p [The four modal alternatives] |
| [§43:D] (p --3 q) <==> [](p ==> q) ["that °p° strictly implies °q°'ing"] |
| [§48:A] (p --3 q) ==> ([]p ==> []q) [Law of strict implication] |
| [§58:D] (p () q) <==> <>(p \and q) ["that °p° is compatible with °q°'ing"] |
| [§61:M] If °p° is a theorem, then °[]p° is a theorem. [Logical necessity] |
| [§67-71] Parallel modal concepts: |
| Alethic: necessity possibility impossibility contingency |
| Existential: universality existence emptiness |
| Epistemic: verified not falsified falsified undecided |
| Doxastic: believe °p° don't b. °¬p° b. °¬p° no belief |
| Deontic: obligatory permitted forbidden indifferent |
+------------------------------------------------------------------------------+
10. References
--------------
(Missing numbers refer to sources only relevent to the original
Swedish version of this text.)
[1] Robert +Audi+ (editor): _The Cambridge Dictionary of Philosophy_