Logic
Necessity and Possibility
'Will surely come' and 'might come' are not simply different degrees of confidence. They are different modalities, different ways a statement relates to reality. Modal logic formalizes the difference between what is, what must be, and what could be.
- **Law:** 'should have foreseen' versus 'could have foreseen' are different liability standards with different modalities
- **Counterfactuals:** 'If I had not been late...' is reasoning about nonexistent possibilities, formalized via possible worlds
- **Program verification:** modal logics CTL and LTL check that a system 'always' or 'eventually' reaches a state
Necessity
**Modal logic** extends classical logic with operators for **necessity** (□) and **possibility** (◇). Classical logic asks whether a statement is true. Modal logic asks whether it could have been otherwise, whether it must be so, whether another course of events is possible.
**Necessity (□p):** 'Necessarily p' means p is true in all possible worlds, under all circumstances. 2+2=4 is necessarily true. There is no world where it is false. 'Snow is white' is contingent. It could have been otherwise.
**Analytic vs synthetic:** necessary truths are often analytic, true by the meaning of the terms. 'A bachelor is unmarried' follows from the definition 'bachelor = unmarried man'. But necessary synthetic truths also exist (a contested point in philosophy).
Which statement is a necessary truth (□p)?
Possibility
**Possibility (◇p):** 'Possibly p' means p is true in at least one possible world. 'Possibly I am in Paris right now' means a world exists where this is the case, even though in our world I am not in Paris. Possibility is the modal dual of necessity.
**Duality of operators:** - ◇p ≡ ¬□¬p (possibly p ≡ not necessarily not-p) - □p ≡ ¬◇¬p (necessarily p ≡ not possibly not-p) This is the De Morgan analog for quantifiers: □ as ∀ (all worlds), ◇ as ∃ (some world).
**Kinds of possibility:** logical (consistency), metaphysical (how the world could have been), physical (compatible with the laws of nature), epistemic (compatible with what we know). 'Time travel' is logically possible and physically contested.
If □p (necessarily p), what follows?
Modal operators
**The modal operators** □ and ◇ are added to propositional logic, creating formulas like □(P → Q), ◇(P ∧ Q), □P → □□P. The interpretation depends on the kind of modality: alethic (truth), deontic (duty), epistemic (knowledge), temporal (time).
**Modal logic systems:** different axioms produce different systems: - **K:** the basic one: □(P→Q) → (□P→□Q) - **T:** K plus □P → P (necessary implies true) - **S4:** T plus □P → □□P (necessary is necessarily necessary) - **S5:** S4 plus ◇P → □◇P (possible is necessarily possible)
**Why different systems?** For alethic modality (logical necessity) S5 fits. If something is possible, that is a fact independent of the world from which you view it. For epistemic modality, S4 is more appropriate. Knowledge does not necessarily know that it is possible. For deontic modality, different rules apply.
The axiom □P → P (necessary implies true) holds for which modality?
Possible worlds
**Possible-worlds semantics** (Kripke) is a way to interpret the modal operators. A 'world' is a complete description of how reality could be. □p is true in world w if p is true in every world reachable from w. ◇p is true if p is true in at least one reachable world.
**The accessibility relation R:** not all worlds 'see' each other. World w₁ is reachable from w₀ if w₁ is an alternative that w₀ 'regards as possible'. Properties of R (reflexivity, transitivity, symmetry) determine the system's axioms.
**Properties of R and axioms:** reflexivity (wRw) yields axiom T. Transitivity (wRv, vRu → wRu) yields 4. Symmetry (wRv → vRw) yields B. S5 is when R is an equivalence relation (all worlds reach each other).
'Possible worlds' are parallel universes that really exist
Possible worlds are a formal tool for analyzing modal concepts; their ontological status is a separate philosophical question
Most logicians use possible worlds instrumentally, as a way to define truth conditions for □ and ◇. David Lewis's modal realism (worlds are real) is one position among many. You can be a modal logician without believing in 'real' parallel worlds.
In a model with three worlds: w₀ → w₁, w₀ → w₂, where p is true only in w₁. Is □p true at w₀?
Key Ideas
- **Necessity (□)** is truth in every possible world; **possibility (◇)** is truth in at least one
- **Duality:** ◇p ≡ ¬□¬p; □p ≡ ¬◇¬p, the De Morgan analog for quantification over worlds
- **Different systems** (K, T, S4, S5) correspond to different properties of the accessibility relation
- **Kripke semantics:** worlds plus accessibility plus valuation, a formal interpretation of modal operators
Related Topics
Modal logic connects to:
- Deontic logic — Modality of obligation: □ = must, ◇ = permitted
- Quantifiers — □ as ∀ over worlds, ◇ as ∃ over worlds
- Temporal logic — □ = always, ◇ = eventually (future or past)
Вопросы для размышления
- Which of your beliefs do you treat as necessary truths and which as contingent? By what criterion?
- When you say 'that is impossible', do you mean logical, physical, or practical impossibility?
- How would your thinking change if you systematically distinguished 'true' from 'necessarily true'?