Modal logics

Modal logics are a form of non-classical logics that serve as an extension of classical propositional logic. Modal logics have all the same truth-functional connectives as classical logic (¬, ∧, ∨, and ⊃) as well as the modal operators ◇ and □. Modal operators are designed to handle propositions related to modality; namely, propositions that make reference to necessity and possibility. The diamond operator ◇ is often used to symbolize “it is possible that” or “possibly”, while the box operator □ is used to symbolize “it is necessary that” or “necessarily”. It is common to interpret possibility to mean “in at least one possible world” and necessity to mean “in all possible worlds”.

In this case, if “w” denotes a possible world then ◇ can be interpreted as “∃ w” and □ as “∀ w”.

Types of modal logics
Below is a list of commonly used modal logics. Please note that this list is not exhaustive.

Basic modal logic
Basic modal logic treats modal operators as basic extensions of propositional logic with their own valid inferences. For example, if it's necessary that a proposition is true then you can validly infer that proposition. □P implies P. Given the necessitation of a proposition it would also mean that the negation or the falsity of said proposition would be impossible; so, □P is logically equivalent to ¬ ◇ ¬P. The converse would also hold in basic modal logic with ¬ □ ¬P  being logically equivalent to ◇P. Any true proposition would have to be possible so P implies ◇P. Tautologies could be argued to be always necessary, with logical contradictions like the form (P ∧ ¬P) being necessarily negated and such not possible.

With these basic inferences, one can build a system of natural deduction that incorporates the basic implied logic of these modal operators to check the validity of modal arguments.

Normal modal logics
Normal modal logics are more complicated, utilizing Kripkean possible world semantics to build formalized models to determine validity. Validity in this sense is more or less the same as in classical logic, being that an argument is valid if, under the circumstances that the premises are true, it is impossible for the conclusion to be false. In the context of normal modal logics, a model is an ordered triple  of a universe W, composed of all possible worlds, an accessibility relation, R, indicating which worlds are possible with respect to each other, and an interpretation function, I, which assigns truth values to the atomic formulas at each world w in the universe. The truth of non-atomic formulas is evaluated according to a valuation function, and a formula is valid in a model just in case it is true at every world in the model. For example, an argument may be invalid in a model with a reflexive accessibility relation, but valid in a model with a non-reflexive accessibility relation, or vice versa.

This added complexity introduced with interpretation makes it so this logic requires the use of Kripkean possible world semantics and models to determine validity, though one can also make use of a modal tableaux with specialized rules for normal modal logics. Within the framework of the possible worlds semantics, □P is said to be true at a world w if and only if P is true at every world accessible from w.

The basic normal modal logic is system K. Other systems are built on system K by the addition of further axioms. All K-models satisfy axiom K: □(φ ⊃ ψ) ⊃ (□φ ⊃ □ψ). Other important systems and their associated axioms include:
 * System D: (□φ ⊃ ◇φ)
 * System T: (□φ ⊃ φ). In T-models, the accessibility relation is reflexive (i.e. a world is always accessible to itself).
 * System B: (φ ⊃ □◇φ). (System B also includes axiom T.) In B-models, the accessibility relation is reflexive and symmetric (if world w1 is accessible from w2, then w2 is accessible from w1).
 * System S4: (□φ ⊃ □□φ). (System S4 also includes axiom T.) In S4-models, the accessibility relation is reflexive and transitive.
 * System S5: System S5 uses axioms T, B, and 4. In S5-models, the accessibility relation is reflexive, symmetric, and transitive. For this reason, it is an equivalence relation.

Non-normal modal logics
Non-Normal modal logics are much like normal modal logics in that they utilize Kripkean possible world semantics and work within a framework of various interpretations. What makes non-normal modal logics differ is that the models and proofs utilize the use of "non-normal" worlds which are characterized by having the feature to which everything is possible and nothing is necessary. Validity is defined in terms of truth preservation across "normal" worlds.

Conditional Logic
Classical logic features various inferences that are seen as problematic by certain philosophers and logicians. Take, for example, antecedent strengthening, which is valid in classical logic. Such an argument goes as follows...

(P ⊃ Q) ⊨ ((P ∧ R) ⊃ Q).

This would be like arguing from "If Jane takes a trip to Africa then she will take a safari" to the conclusion that "If Jane takes a trip to Africa, and dies at the airport, then she will take a safari". Needless to say, some would find such an argument a bit odd. In ordinary language, one could say, "If Jane takes a trip to Africa, then she will take a safari, but if she dies at the airport, then she won't take a Safari". Since, after antecedent strengthening, the same conclusion can no longer be inferred, the English "if... then" does not seem to be described adequately by the material conditional.

Conditional Logic is a type of modal logic that attempts to avoid such problematic inferences by introducing a new symbol ">" to function on the basis that the valuation of (A > B) is true if and only if for all worlds that express a relation via "A" the valuation of "B" on such world is also true.

Counterfactuals
In classical logic, the formula (P ⊃ Q) is true if P is false or Q is true. However, in counterfactual claims, like 'If Reagan had lost in 1984, Mondale would have been president', the proposition expressed by the antecedent (in this case, that Reagan lost in 1984) is always false, making counterfactual conditionals trivially true on this analysis. Such problems have motivated the development of new theories about conditionals in counterfactual contexts. For convenience, counterfactuals will here be represented by the notation (P > Q), which can be read 'if it had been the case that P, then it would have been the case that Q'.

Strict conditional analysis
The strict conditional analysis posits that (P > Q) is logically equivalent to □(P ⊃ Q). In words, to say 'if P had been the case, Q would have been the case', is to say that in every possible world where P is true, Q is true.

The strict conditional analysis carries over problems from the material conditional. It retains, for instance, the validity of antecedent strengthening. That this remains problematic in counterfactuals is easy to demonstrate: 'If Jane had taken a trip to Africa, she would have gone on a Safari' does not entail that 'If Jane had taken a trip to Africa and died at the airport, she would have taken a Safari'.

Variably strict conditional analysis
While there are multiple forms of the variably strict conditional, a common one interprets (P > Q) as: in the closest possible world where P is true, Q is true. This analysis renders antecedent strengthening (and some other schemas) invalid, at the cost of imposing a new requirement that it be possible to measure distances between possible worlds, to determine how close they are to the actual world.

Quantified Modal Logics
Quantified Modal logics are an extension of normal modal logics that introduces domains into worlds that can be quantified over using the quantifiers of first-order logic "∀" and "∃". This is basically the combination of standard predicate logic with modal logic so that you can express statements like "necessarily all men are mortal" in terms such as □∀x(Men(x) ⊃ Mortal(x)). As the modal operators quantify over worlds, the standard quantifiers quantify over objects within said worlds also referred to as "domains".

Quantified Modal Logics also tends to introduce the semantics and syntax for identity from first-order logic, i.e. " x = x ". The notion of identity within metaphysics related to modality, and modal logic itself is controversial with specifics logics being constructed that treat identity relations as necessary and others which treat it as contingent/variable. was a very influential figure within quantified modal logics introducing what is referred to as Barcan's Formula as an axiom for quantified modal logic: formula being that ∀x□Fx → □∀xFx. This translates to if all x is necessarily F then necessarily all x is F.

Relevance to historical and scientific inferences
Counterfactual statements are used in the context of scientific or historical inference when one tries to predict what the result of a given phenomenon will be, even if said phenomenon never happened or is yet to happen. An example would be like "If the plum pudding model was true, then during Rutherford's Gold Foil Experiment all the alpha particles would have been deflected backwards". Obviously, the plum pudding model isn't true, and the results of Ernest Rutherford's experiment did not turn out this way. Regardless we can still talk meaningfully about how things alternatively could have been — and that is in essence what a counterfactual statement is. Despite not being the case, however, a counterfactual statement can still possess the quality of being true or false. One could argue for example that the claim that the alpha particles would have deflected backwards is not a true consequence of JJ Thomson's plum pudding model; and so, such a counterfactual would be false.

Other examples come into play in historical debates such as "if the social democrats and communists adequately banded together, then they could have prevented the Nazis from coming to power". The truth or falsity of such a claim still becomes a matter of debate even though it's a matter of fact that the Nazis did come to power and that the social democrats and communists didn't adequately band together.

Counterfactual statements imply a degree of modality, and so there is an interest to find ways to ground the meaning of such statements in formalized semantics -- hence some of the motivations for modal logics; especially in regards to conditional logics. There is also the matter of dealing with statements that claim whether a given natural phenomenon is even possible, and grounding the meaning of such terms when they are not actualized requires the use of modality, and so, a logic that handles such statements needs the means to represent modality.

Philosophy and modality
It is one thing to have the means to represent modal sentences and counterfactuals in formal logic, however, there is an issue of explaining exactly how they are meaningful and what is it exactly that grounds them as true or false. This naturally leads to questions about what a "possible world" even is, and what it even means for something to be necessary.

Some philosophers such as David Lewis take the extreme view called modal realism arguing that all possible worlds are essentially their own separate universes all equally existing, with nothing special about the world we occupy. In a sense, all possible worlds are actual worlds, and what grounds a counterfactual as true is that it corresponds to the events in another relevant possible world. Other philosophers argue that possible worlds are merely sets of logical propositions, some argue that possible worlds are only hypothetical recombinations of our existing world. No one metaphysical account of "possible worlds" is without its problems, for example, Lewis's view may have to commit us to the idea that unicorns exist solely because they can exist in other possible worlds. .

Modality comes up in questions of metaphysics and epistemology; for example, in the question "why is there is something rather than nothing?" one can take the Lewis-esque response and argue that both nothing worlds and something worlds equally exist and so there is no real "rather than" in matters to why things exist. Modality also becomes relevant in "Tracking theories of Knowledge" such as that of Robert Nozick's which utilize counterfactuals as relevant conditions to knowledge.

Ontological arguments using modal logic
A particularly motivated logician may come to represent Anslem's ontological argument in favour of the Christian God using modal logic. Here is the formalization provided by Paul Herrick in his 2013 Introduction to Logic.

G = God exists.

1. G ⊃ □G (this premise being if God exists it is therefore necessary that God exists).

2. ¬G ⊃ □¬G (If God does not exist, then it is necessary God does not exist)

3. ◇G (it is possible God exists)

C: G (therefore God exists)

The argument is valid because if we assume God does not exist it results in a contradiction, namely in the premises of □¬G and ◇G (because ◇G is equivalent to ¬□¬G). This argument invites its fair share of objections, especially in the form of "Guanilo's Parody" which takes the same structure as Anslem's argument but instead of arguing for the existence of God, it argues for the existence of a fictional "Lost Island" exposing an underlying informal fallacy with the argument as presented. Another objection is the form of rejecting premise 3, as some have argued as a consequence of paradoxes related to an omnipotent God as conceptualized would possess contradictory properties making God a modally-necessary falsehood, meaning it is not possible for God to exist or ¬◇G.

Varying Truth values on "objective statements"
… all these idioms [i.e. propositional attitudes and modal logic] reduce to a pretty hollow mockery if we never quantify into them. Efforts to save the situation prove to involve us either in considerations of essence and accident and kindred dim distinctions, or else in elaborate further grammatical apparatus which I forebear to enlarge upon here.

The scientific concept of possibility is a key ingredient in physics, particularly as it used in phase spaces—in the latter (classical) mechanics of Helmholtz, Kelvin, Mach, and Duhem; and in quantum mechanics wherein phase spaces have been explicated further.

However, the philosophical notion — as with most philosophical notions it is a priori—is woozy and vague; since inter alia it is based on viciously-circular intentional notions, escapes all means of bringing evidence to support or refute it (e.g. how, exactly, do we observe possibility, or necessity), creates unintelligible sentences which are neither true nor false, confuses use with mention, needlessly complicates scientific theory, and is referentially opaque, at the very least according to Quine.

Scientific sentences should be objective. Their truth values should not vary due to the affect(s) of: speaker, place, context, or time — they are presumably either true or false. The following are supposed examples of objective sentences (statements, to be precise), independent of all of the above criteria:
 * 1) Iron is a metal,
 * 2) Iron is a vegetable,

The former is true and the latter false. However, if we construct (1) and (2) by using the a priori modal notions of “necessity” and “possibility”, we thereby cause their truth values to vary. Consider: each of these modal constructions transforms a sentence that formerly expressed a fact, a fixed truth value, into sentences whose truth value either varies, or more perniciously, is altogether absent. For example, is Iron necessarily a metal? What if we say that it is a “contingent”? But this word ‘contingent’ is just another modal term, as problematic as the rest. Moreover, how do we even acquire evidence about ‘necessarily iron is a metal’? About the ‘necessity of being a metal’?
 * Possibly Iron is a metal,
 * Necessarily Iron is a metal,
 * Possibly Iron is a vegetable,
 * Necessarily Iron is a vegetable,

A possible response to such a critique could come from the historical-causal theories of meaning from the likes of Saul Kripke and Hilary Putnam.

Kripke argues that once an object/subject gets “christened” with a name by a linguistic community that name becomes a “rigid designator” referring to the same object or class of objects across all possible worlds. Putnam argues something similar is his essay arguing for semantic externalism utilizing his twin earth thought experiment.

If these two philosophers are correct then the truth values would not vary if iron constitutes a rigid designator. The truth value of "It is necessary that iron is a metal" would be the the same as "iron is a metal". Since necessity also implies possibility, the truth value of "it is possible that iron is a metal" would also remain the same.

You can extend this reasoning for the example “iron is a vegetable” adding “it is necessary” or “it is possible” in front of such a sentence wouldn’t change the truth value from being false.

Quine's criticism of C.I. Lewis's "Modal Implication"
The conditional is a logical connective of the object language of first-order logic. The conditional can only be used, not mentioned.

C.I. Lewis’ “paradox of material implication” arises due to C.I. Lewis’ confusing use with mention. For example, C.I. Lewis’ construal of implication: does not represent implication, for it is a syntactic formula of the object language, while implication is a semantical concept belonging to the metalanguage. Lewis has not resolved his paradox, he has simply proposed a new connective for the object language. Lewis fails to see the significant difference between the material conditional and material implication.
 * 1) ‘□ (p ⊃ q)’,

Implication may be defined recursively (entertaining only one primitive notion: logical truth) :
 * 1) A sentence is logically false just in case its negation is logically true.
 * 2) Two or more sentences are logically incompatible just in case their conjunction is logically false.
 * 3) Finally, one sentence logically implies another sentence just in case it is logically incompatible with the other’s negation.