The nature of possibility, along with purpose, is one of the most difficult questions in metaphysics. There are really only three significant theories of possibility in the history of philosophy, (1) one by Aristotle, which was apparently regarded as adequate for many centuries, (2) another that we find in quantum mechanics, which grew up quite by accident and unintentionally, fostered by various physicists, and is not to be found formulated in an entirely coherent form, and finally (3) another by the recent philosopher David Lewis (1941-2001). This problem has been touched in various ways at this site already, but a more focused and dedicated treatment of it is in order.
Possibility most importantly is to be found in two forms, de re, concerning things, which is the possibility of existence in objects, and de dicto, concernings things that are said, which is the possibility of truth in propositions. In the former case, we are concerned with the difference between existence and non-existence, while in the latter between truth and falsehood. In the former case again, we are faced with the metaphysical problem that possible objects may not exist, which means we can be tossed into the paradoxical and unwelcome theory of non-existent objects. In the latter case, we must consider the problem of truth and falsehood, with some sense of how the possibility of falsehood troubled Descartes. These two forms of possibility can be related in the sense that the truth of propositions can depend on the existence of facts in the world, with the dictum thus grounded on the res. What sort of "fact" makes something possible is a feature of the question.
The concept of possibility is part of the system of modal terms, which we may take to mean the "mode" in which things exist, apart from the simple indicatives of existence or non-existence -- this is separate from other modifications, such as quantification (see below), and it is different from the mode or mood of verbs, which is of interest in its own right. The most curious feature of modals is that, although we might think of all the concepts as equally primitive, 
it turns out that they can all be defined in terms of one another. We see this illustrated in the table at right, based on the logical square of opposition.
Modal logic introduced two symbols evident in the diagram for what might be thought to be the most primitive concepts, namely the diamond, 
, for "possible," and the square, 
, for "necessary." It turns out, however, that even the possible and the necessary can be defined in terms of each other. Thus if a proposition is necessary, P, this means that it is impossible that it not be true, ∼∼P ["not possible not P"].
Since impossibiliy is a kind of necessity, ∼P ≡ ["is equivalent to"] ∼P, the possible and the necessary are separated by no more than the placement of a negation. This example also enables us to imagine a rule that a negation can "pass through" a modal and change it into its opposite. This is like the similar rule in symbolic logic for quantifiers (called "Quantifier Exchange" by Benson Mates), that a negation can "pass through" and turn one into the other, e.g. ∼(x)Fx ≡ (∃x)∼Fx, "'it is not the case that everything is F' is equivalent to 'there is something that is not F'." We could examine an issue in metaphysics in these terms. If God is a "necessary being," (∃x)Gx, "it is necessary that there is a God," then the denial of this, ∼(∃x)Gx, is equivalent to ∼(∃x)Gx, "It is possible there is not a God," or (x)∼Gx, "it is possible that everything is not God." The denial of God as a necessary being, consequently, does not mean there is no God, only that God, if he existed, would be a contingent being.
Despite our ability to define all modals in terms of each other, the meaning of each is of some independent interest. The denial of the possible is, of course, the impossible. The denial of the necessary is the "contingent," which, as the possible means that something may exist or be true, the contingent means that something may not exist or not be true. The expression ∼(∃x)Gx would be that God, if he existed, would be a "contingent being," like us, whose non-existence is possible. Of course, something that is possible may nevertheless never exist; and a contingent being may never not exist, i.e. always exist. The possible and the contingent are thus both about things that may or may not exist, or may or may not be true. When we are not dealing with modals, propositions that can be either truth or false, because they are not tautologies (always true) or contradictions (always false), tend to simply be called "contingencies," with the "possibilities" category redundant.
The relationship between modals and indicatives is then of some interest. An important feature of the argument of Kant's Critique of Pure Reason is the principle that, if something is actual, then it is necessary that it be possible. The reasoning begins with the observation that P ⊃ ["implies"] P, "If P, then P is possible." A companion principle with necessity would be P ⊃ P, "If P is necessary, then P." Since the modals can be defined in terms of each other, we might suspect that these principles would turn out to be equivalent. Logically transposing the first formula, we get ∼P ⊃ ∼P, "If P is impossible, then not-P." Passing the negation through the possible operator, we get ∼P ⊃ ∼P. Substituting ∼P for P, the result is, ∼∼P ⊃ ∼∼P. And, eliminating the double negations, we get P ⊃ P, which is what was to be proven. It turns out that P ⊃ P ≡ P ⊃ P.
In Kant, the argument of the "Transcendental Deduction" is the "necessity of the possibility of experience," which means that the conditions for the possibility of experience are necessary -- conditiones sine qua non, the "conditions without which not," i.e. necessary conditions. These conditions in Kant's system involve the synthesis of experience according to the rules of the categories of the understanding. Synthesis binds together perception as part of the unity of consciousness. That must be argued for on its own terms, but it always leaves the requirement that the conditions for the possibility of experience are necessary.
Exactly where necessity would go for this in one of our formulae is a good question. It is tempting to say right way that P ⊃ P, "If P, then it is necessary that P is possible." However, although this may look good, it will not do. If we transpose the formula, to ∼P ⊃ ∼P, we end up with ∼P ⊃ ∼P, which says that if it is possible that P is impossible, then not-P. That does not look like a sound inference. The mere possibility that something is impossible cannot be enough to make it false. It is certainly possible that economic prosperity under socialism is impossible, but rather more is required for the conclusion that there is no economic prosperity under socialism, much less that it is impossible (as, in fact, it is). That requires a further step [note].
The alternative is (P ⊃ P), which puts necessity on the whole assertion. We can test this with a denial. What happens to ∼(P ⊃ P), "It is not necessary that P implies that P is possible"? Logically we can swap the conditional for the negation of a conjunction, ∼∼(P & ∼P). This reduces to (P & ∼P), which is the extraordinary assertion that it is possible for P to both true and impossible! Since ∼P ⊃ ∼P, we will have a contradiction and a good result for a reductio ad absurdum argument. So it is true that (P ⊃ P), "It is necessary that if P, then P is possible." How this works in Kant's argument, of course, depends on the details and credibility of the conditions he posits for the possibility of experience.
Prefixing necessity to any tautology, not surprisingly, gives us results similar to what we have just seen. Thus, (P ⊃ P), if denied, ends up giving us (P & ∼P), "It is possible that P and not-P." Since such a contradiction is not possible we have another indirect proof. Also, since P ⊃ P, we can always drop our prefixed necessity for convenience: (P ⊃ P) ⊃ (P ⊃ P) [note].
Adding quantifers to our modal symbols allows a formal definition of the difference between de re and de dicto. Thus the de re possibile is (∃x)Fx or (x)Fx; and the de re necessary is (∃x)Fx or (x)Fx. The de dicto possible is (∃x)Fx or (x)Fx; and the de dicto necessary is (∃x)Fx or (x)Fx. Since symbolic logic interprets the quantifiers as affirmations or denials of existence, the meaning of de re is clear enough, while the modal after the quantifier only applies to the predication.
Manipulating these symbols can be fun and instructive, but it actually supplies little that we need for a real understanding of their meaning or their metaphysics. Philosophers (especially Analytic philosophers) sometimes seem to think that because they have a symbol for something and can manipulate it, this means that they understand what it is. That is not necessarily the case, and this can give rise to serious confusions in philosophy -- for instance when it is applied to mathematics, resulting in the Sin of Galileo, that just because the math works, this means we know what is going on. Newton's math worked beautifully, but this did not mean that he or anyone else understood what gravity was or how it really worked. Newton thought it was the Will of God -- not a view to recommend itself to our contemporaries.
Possibility and necessity thus require substantive theory, which means metaphysics -- something else shunned by our contemporary (especially Analytic) philosophers. Historically, the first theory of possibility is in Aristotle. It is a function of his theory that all objects in the world consist of form and matter. What this means not what we would expect now. While we might think of "matter" as substance and "form" as an epiphenomenon, Aristotle's ideas about them are very nearly the opposite. Form is actuality, ἐνέργεια, from which we derive the word "energy." Matter is potential or power, δύναμις, from which we derive words like "dynamic." "Form" in Greek is εἶδος, one of the terms that Plato had used for his Forms. This would be translated into Latin as species, which passed into biology as the word for the basic taxonomic classification of organisms. Form represents οὐσία, which is usually translated "substance" (substantia in Latin) -- hence the term "substantial form" for Aristotle's entities -- but it has a closer linguistic affinity to "essence" (essentia in Latin), what something is. Both οὐσία and essentia are from the words for "to be" in Greek and Latin, εἶναι and esse, respectively. Matter, ὕλη, stripped of all form, is not substance or essence, although in Greek it can be called the ὑποκείμενον, the "underlying [thing]," even though this word looks like it would be parallel in origin and form to Latin substantia. This should all be sufficient to warn us that the meaning of these things, as they have shifted around historically, is a matter of some uncertainty.
The virtue of Aristotle's theory is that it gives us an ontological ground for possibility. This seems to have been enough to settle the question in the minds of most, for many centuries. However, there is a fundamental paradox in Aristotle's treatment. Not just anything is possible. There are limitations on what can come-to-be. Possibility thus must have a structure, yet the only structure in the system is what emerges in the form of actuality. This posits the requirement exactly backwards. Aristotle's ultimate matter, "prime matter," is what is devoid of all actuality, all substrance, yet it must already contain within itself the structure of all that is possible. In modern terms, the laws of nature must be inherent within it. But prime matter cannot actually exist separately. The only provision that Aristotle made for such a structure is the existence of God, who is pure actuality, and who might be said to be responsible for the laws of nature. But we are then struck by another paradox. Since Aristotle's God is pure actuality, he cannot do anything that he is not already doing. Unlike the God of Abraham and Isaac, he is without δύναμις and so cannot be the "Almighty." More like the "No-mighty." All power and possibility is ontologically vested in matter, which untimately looks like nothing and was in fact interpreted by the Neoplatonists, only only as Not Being, but as Evil. It might be argued that what God is already doing is to effect the actuality of the laws of nature. This means that in each case of a material objects moving in obedience to the laws of nature, such as falling weights, they are directly affected by the control of God. This begins to sound like Occasionalism, which some may accept with some complacency, but which otherwise begins to look like a violation of Ockham's Razor.
Looking from Aristotle back towards Plato, the earlier philosopher's entire theory of Forms can be interpreted as a theory of possibility. The Forms are all the things that are possible in the world. The paradox there is that Plato's Forms seem to be too specific. If there are television sets or dental floss, then there must be the Form of the Television ("the Television Itself," as Plato would say) and the Form of the Dental Floss ("the Dental Floss Itself"). This begins to sound a little silly, as Plato himself realized (although he didn't know about television sets or dental floss). However, if the Forms only consist of those abstract laws of nature that make television sets, etc. possible, then we get something more like it. 
Plato, however, liked the concreteness of the Forms because he thought of them as beautiful. Mathematical laws of nature can themselves be beautiful, but it's not quite the same thing. You need to understand the math.
In recent thought, discussions of possibility inevitably draw in problems from quantum mechanics. The wave function, described by Schrödinger's Equation, is something whose nature in reality is matter of uncertainty, intepretation, and dispute, if not fear. Originally, light was posited as a wave, with some certainty because of the experiments of Thomas Young demonstrating interference effects. Then Einstein argued that the Photoelectric Effect showed that light was a particle, as Newton has thought -- even Richard Feynman incautiously asserted that this meant that light was a particle because, "the wave theory collapsed." Louis de Broglie, however, showed that particles could act like waves; and finally Niels Bohr posited the principle of Complementarity, that subatomic entities could appear as waves or particles, but not both at the same time. This is where we stand now.
Now we get to the good part. 
According to Bohr, the wave function collapses into particles when it is observed. According to Werner Heisenberg, the square of the wave function is then the probability of finding a particle at a particular place and momentum. Probability is a quantified possibility, so there we have our theory of possibility.
Not so fast. The metaphysics depends on what the wave function is, and that is a matter that sometimes seems to be of little interest to physicists. Bohr himself said that "nothing exists" until it is observed, and it is not unusual to get the sense that only particles really exist and that the wave function is a mathematical construct that has nothing to do with reality. But then this leaves a rather large hole in reality, with a rich harvest of paradoxes. Electrons, as waves, can be in two places at once, simply because a wave is a spatially extended entity. As particles, however, electrons are discrete entities that can only be in one place at a time -- and the place is no more than a point of location, since electrons are Dirac Point Particles. So what are we to say about an experiment where an electron is apparently in two places at once and, according to Bohr, can only be understood as a wave and not, because of Complementarity, as a particle? Some acknowledgement must be made of the physical reality of the wave, as de Broglie and Schrödinger would have affirmed themselves. But I suspect that most physicists would rather than worry about it, much less answer the question. Bohr's anti-realism is an easy way out.
So what is the problem? Well, physicists don't like metaphysics; and in fact, to do their science, they don't need to decide, know, or care about the ontological status of the wave function. And philosophers are so intimidated by physics that they also let the question drop. 
Who are they to tell physicists what to think? Well, somebody needs to state the obvious. The only classic Western metaphysical doctrine that posits a dualism like the wave/particle duality is that of Immanuel Kant. A Kantian interpretation of quantum mechanics would say that wave function represents a level of reality, which encompasses possibility and probability, among things in themselves. This is outside what we can experience in phenomenal objects, which is where particles as particles appear. This still does not tell us exactly what is going on there, but then Kant did not believe that we could have complete theoretical knowledge of things in themselves. The wave function is buried behind the limitations of reason.
The possible contempt of physicists for entertaining a theory that involves things in themselves is inconsistent with their (usual) complacency about investigations that aim to find effects in the observative universe either from other universes or from areas in this universe that are outside of observable space -- a condition made possible by "inflationary" theories that posit an expansion of the universe, for a while, that exceeded the velocity of light. Also, quantum mechanics posits "non-locality," which means that quantum effects can ignore space. Since that is exactly would Kant would allow about things in themselves, physicists have already, insensibly, slipped into Kantian thinking. The wave function itself is non-local, since it can collapse instantaneously over cosmological distances, violating Special Relativity. This has been proven through Bell's Theorem. Thus, the Kantian theory of space, already allowed by non-locality, already accepts the wave function at an ontological level below the observable, or (in Kantian terms) phenomenal, universe.
The real heat over possibility in quantum mechanics, however, does not come over the ontological meaning of the wave function. When the wave function collapses, with observation, and we get discrete particles, physicists wonder why we get certain state and not others. Actuality weeds out an infinite number of other possibilities. For some reason, people find this disturbing, or annoying. So we get the "many worlds" interpretation of quantum mechanics. When the wave function collapses, we get all the possibilities becoming actual, each in its own parallel universe. I am not sure what kind of advantage this theory is supposed to confer on the physics, especially when it is a particularly eggregious violation of Ockham's Razor, as was noted by Martin Gardner.
The appeal of "many worlds" quantum mechanics may be of a piece with the appeal of the general metaphysics of David Lewis. This may have begun with Lewis's colleague at Princeton, Saul Kripke, who decided, with a hint from Leibniz, to approach modal logic as the quantification of "possible worlds". Necessity is when something is true in "all possible worlds"; impossibility in "no possible worlds"; and possibility/contingency in "some possible worlds." Like Leibniz, Kripke did not think himself that there was more than one actual world; but Lewis look the leap to saying that all possible worlds are in fact actual worlds, as different, parallel universes. This is our third classic theory of possibility, and of course it overlaps the loose ends of the ontology in quantum mechanics. Kirpke himself found this theory disturbing; and it does strike most people in philosophy has overblown, even while providing wonderful grist for science fiction writers. It is, however, challenging in its simplicity. It eliminates all the peculiarities required for a theory of possibility, i.e. that possible things may not ever actually exist and yet must be grounded in a physical reality that would make them possible. Now they do exist, and we can forget about possibility as a separate ontological business. Not only does the collapse of the wave function generate an infinite number of new universes, but an infinite number of new universes have always existed and are always being newly generated.
Lewis's metaphysics, however, does not address things like non-locality in quantum mechanics, or the actual physical inderminacy that goes with it. Those features go with the actual uncollapsed wave function, which now does not have any physical reality apart from the existing parallel universes. Thus, if we have particles whose spins are correlated, i.e. one must be positive, the other negative, but this has not been determined yet, Lewis must say that they have already been determined but that, before the apparent collapse of the wave function to us, there are already an infinite number of universes in which each of the possible states of collapse are already in effect. This raises difficult questions about identity and causality. Kripke already had noted that identity across possibile worlds must be maintained, so that it is meaningful to say that the same Socrates might have won his case as the Socrates who instead was condemned to death. Such names are "rigid designators," and without them it would not even to true to say, "Socrates might have escaped death." The reference of the term "Socrates" would not be the same. It is not clear from Lewis's theory what semantic or ontological connection there could possibly be between a "Socrates" in one world and another in another.
There can be similar problems with causality, and with will. A cause can make one thing happen rather than another. But in Lewis's metaphysics, only one thing can happen in a particular universe because the "possibility" of anything else happening simply means that something else happens in another universe. Or do causes make an infinite number of new worlds happen every time they effect some change? It's enough to give anyone a headache. Nor do we have any real choices in our own universe. A different me (whatever that means) in a different universe will do something different, but this me in the here and now only does the single thing that is actual in this universe. Nothing is indeterminate and, in a strong sense, probability doesn't exist -- my throw of the dice can have only one outcome in this universe.
Metaphysical theories about possibility and other worlds are different from actual physical theories about other universes. 
The ideas floated in quantum mechanics that universes can come to be out of nothing, or that a new universe might pinch off from an old one -- for instance that universes are "branes," i.e. like membranes, on a higher dimension multiverse -- don't involve problems of identity and posit such causal entanglement that observational or experimental evidence is sometimes adduced for such theories -- although they are mostly speculative. Both the physical and the metaphysical theories, however, suffer from a fundamental physical flaw: conservation laws. If a universe comes to be from nothing -- or if infinite universes are created every time a wave function collapses -- there is a lot of energy involved there. Quantum mechanics came to allow "virtual particles," which borrow energy from nothing in order to come into some kind of (non-real) existence. Since the amount of energy is inversely proportional to the time it can be borrowed, only massless particles (like photons or gravitons) are indefinitely durable; and large masses exist, even virtually, only for tiny amounts of time. It is hard to build a universe on those terms; and even a universe that pinches off from an old one isn't going to have much real energy to work with. In the presentations I have have seen of these kinds of theories, I don't remember any explanation of how large amounts of real energy, enough for a universe, can be taken from nothing for cosmological lengths of time. This was not previously allowed by quantum mechanics and flies in the face of the conservation of mass, the conservation of energy, and the Einsteinian conservation of mass-energy. The idea that a small, pinched-off region of space might make for a miniature universe of microscopic beings also ignores the problems of physical differences of scale.
Thus, despite the simplicity and elegance of David Lewis's theory, I think it is at odds with common sense, with reasonable metaphysical requirements, and with some basic things in physical science. Lewis's theory suffers from the curious disability that it is at once reductionistic, eliminating a separate physical or metaphysical reality for possibility, and unnecessarily multiplies entities, violating Ockham's Razor, in the form of infinities upon infinities of other worlds. Despite the unsettled state of its interpretation, quantum mechanics gives us something to really work with for the nature of possibility, namely the wave function. And the only metaphysical theory in Western philosophy that really provides a match for the wave/particle duality, complete with non-locality, is Kant's.
This puts possibility among things in themselves, hidden from our direct observation. Of course, disappointing or overly convenient as this may seem, it is exactly the terms on which the wave function already exists. If we look directly at it in the physical world, in terms of the standard understanding and expectations of modern physics, we don't see the wave or the possibilities, we see the real particles that result from the collapse of the wave. No one can complain that this is Kant's fault.
What this means is that there is only one actual world, but that there is a mathematical expression, and not just a vague potential, for all possible worlds a parte post, i.e. going forward from the present. The wave function is deterministic; and Laplace, who said that he could calculate the whole future if given the conditions of the present, would have no complaint with Erwin Schrödinger. However, the wave function collapses randomly, and Werner Heisenberg has the last word. But not quite. Both determinism and randomness leave out freedom, although anyone might become confused about this. While I had a professor once, who wrote a book about freedom, who said that freedom was simply randonmness, the random is not rational; and it is not an edifying theory of freedom that it is irrationally random and trival -- the mule chosing one stack of hay over another. Instead, Kant understood freedom as an unconditioned rather than a conditioned cause, in which knowledge and reason are factors in our purposes but where these purposes are not determined by physical causes. Randomness in quantum mechanics is cold comfort anyway, since quantum populations, despite individual indeterminacy, distribute just as deterministically as anything else. The individual doing something comletely new and unexpected, and jumping off the bell curve, does not compute.
Kant's theory was thus not something designed to accommodate quantum mechanics or the nature of possibility. But it ends up doing that better than anything else. And we can even push Plato's Forms in there, if need be, reduced from their paradoxical concreteness, but with their potential concreteness perhaps allowing for Plato's aesthetic needs.
Philosophy of Science, Physics
Oddly enough, P ⊃ Now, I have no objection to a philosophical critique of things in modern logic, since I have some such critique to offer myself. And, sometimes Brouwer's scruples seem sensible and revealing on certain issues about the foundations of mathematics. However, where the Intuitionists wish to abridge principles of logic that have been used since the Greeks (like indirect proof, responsible, for instance, for the Pythagorean proof that the square root of 2 -- The absurdity of the inference to
While modal logic can be built up as an axiomatic system, most modal logicians assume that necessity means logical necessity, and their systems fold in prinicples of "strict implication" by which conditionals, if necessary, mean that the consequent logically follows from the antecedent. This is more than is proper if a modal logic is to deal with necessity and the other modals in a properly general way, i.e. not all necessity is logical necessity, as I have had occasion to examine in detail elsewhere. Thus, we must hold conditionals to the truth functional meaning of "material implication," by which the ground of the connection between antecedent and consequent is indeterminate.
This raises some interesting questions about the application of a necessity operator, as in the considerations above, and now in other matters. Thus, an important theorem of modal logic is:
I would wonder if the following theorem would also be true:
If we have a conditional, which is true, then, would it be enough to have a necessary consequent for the antecedent to be necessary? Does the conditional itself need to be necessary?
This question can be examined by manipulating the propositions into forms whose truth can be judged more intuitively -- perhaps in tribute to L.E.J. Brouwer -- although this runs the danger, pace Brouwer, of producing a reducio ad absurdum. We can deal with the first theorem thus:
The Metaphysics of Possiblity, Note 1
P is regarded as true in some systems of modal logic. This is called the "Brouwerian axiom," part of the "Brouwerian system," so called after Intuitionist mathematician L.E.J. Brouwer [cf. A New Introduction to Modal Logic, G.E. Hughes & M.J. Cresswell, Routledge, 1996, 2005, pp.62 & 70]. Now, if P ⊃ P is true, this is no problem for Kant; but we get Brouwer's results mainly by prohibiting some ordinary rules of logic, such as the removal of double negations or the use of indirect proof. This is part of the program of Brouwer's Intuitionist school of mathematics.
2 -- is an irrational number), and where the device of truth tables demonstrates that the inferences preserve truth (salve veritate), which is actually all that matters in logical inference, then something has gone rather too far, to deceptive results.
∼P ⊃ ∼P above could only be prevented by the Brouwerian by prohibiting one of the logical rules that generates it. This could involve some problem with the transposition rule, perhaps that it involves removing a double negation; but then that rule can be written in various ways, including ways that could avoid removing double negations. The Brouwerian would need to pick and choose which versions of transposition would violate his scruples. This strikes me as ridiculous.
The Metaphysics of Possiblity, Note 2
(P ⊃ Q) ⊃ (P ⊃ Q).P ⊃ Q).
| 1. (P ⊃ Q) ⊃ (P ⊃ Q) | Premise |
| 2. ∼(P ⊃ Q) ⊃ ∼(P ⊃ Q) | Transposition |
| 3. ∼(∼P v Q) ⊃ ∼(∼P v Q) | Conditionals to Disjunctions |
| 4. (∼∼P & ∼Q) ⊃ (∼∼P & ∼Q) | De Morgan, Disjunctions to Conjunctions |
| 5. (P & ∼Q) ⊃ (P & ∼Q) | Double Negations removed |
| 6. (P & ∼∼Q) ⊃ (P & ∼∼Q) | ∼Q substituted for Q |
| 7. (P & Q) ⊃ (P & Q) | Double Negations removed |
This result seems unobjectionable. If P is necessary and Q is possible, then it is possible that (P & Q). Of course, this is stronger than it needs to be. (P & Q) ⊃ (P & Q) will also be intuitively correct.
The corresponding derivation for the second form of the theorem could look like this:
| 1. (P ⊃ Q) ⊃ (P ⊃ Q) | Premise |
| 2. ∼(P ⊃ Q) ⊃ ∼(P ⊃ Q) | Transposition |
| 3. ∼(∼P v Q) ⊃ ∼(∼P v Q) | Conditionals to Disjunctions |
| 4. (∼∼P & ∼Q) ⊃ (∼∼P & ∼Q) | De Morgan, Disjunctions to Conjunctions |
| 5. (P & ∼Q) ⊃ (P & ∼Q) | Double Negations removed |
| 6. (P & ∼∼Q) ⊃ (P & ∼∼Q) | ∼Q substituted for Q |
| 7. (P & Q) ⊃ (P & Q) | Double Negations removed* |
This result is not good. If P is necessary and Q is possible, then it will not necessarily be the case that (P & Q) is true. Since Q is only possible, it might be false, in which case (P & Q) would be false. So we have a reductio ad absurdum here of (P ⊃ Q) ⊃ (P ⊃ Q). I think this means that (P ⊃ Q) ⊃ (P ⊃ Q), after all, was the right idea.
Since that theorem is the equivalent to something that, above, I said was "stronger than necessary," namely (P & Q) ⊃ (P & Q), I might wonder what the weaker formula, (P & Q) ⊃ (P & Q), would look like with some manipulation.
| 1. (P & Q) ⊃ (P & Q) | Premise |
| 2. ∼(P & Q) ⊃ ∼(P & Q) | Transposition |
| 3. ∼(P & Q) ⊃ ∼(P & Q) | Possible to Necessary |
| 4. (∼P v ∼Q) ⊃ (∼P v ∼Q) | De Morgan, Conjunctions to Disjunctions |
| 5. (P ⊃ ∼Q) ⊃ (P ⊃ ∼Q) | Disjunctions to Conditionals |
| 6. (P ⊃ ∼Q) ⊃ (P ⊃ ∼Q) | Possible to Necesasry |
| 7. (P ⊃ ∼∼Q) ⊃ (P ⊃ ∼∼Q) | ∼Q substituted for Q |
| 8. (P ⊃ Q) ⊃ (P ⊃ Q) | Double Negations removed |
This is an interesting result. If (P ⊃ Q) is necessarily true, then it doesn't matter if P is only possibly true; for even if P is false, (P ⊃ Q) is be necessarily true if Q is necessarily true. So this theorem passes the intuitive test. This makes an intriguing complement to the original theorem, (P ⊃ Q) ⊃ (P ⊃ Q). And in all these derivations, we can go back the other way, so they are equivalences and not just conditionals.
If (P ⊃ Q) ⊃ (P ⊃ Q) is a reasonable principle, it is not clear why (P ⊃ Q) ⊃ (P ⊃ Q) would not be also. If Q is necessarily true, then (P ⊃ Q) would be necessarily true. So let's if an absurdity follows from it.
| 1. (P ⊃ Q) ⊃ (P ⊃ Q) | Premise |
| 2. ∼(P ⊃ Q) ⊃ ∼(P ⊃ Q) | Transposition |
| 3. ∼(∼P v Q) ⊃ ∼(∼P v Q) | Conditionals to Disjunctions |
| 4. (∼∼P & ∼Q) ⊃ (∼∼P & ∼Q) | De Morgan, Disjunctions to Conjunctions |
| 5. (P & ∼Q) ⊃ (P & ∼Q) | Double Negations removed |
| 6. (P & ∼Q) ⊃ (P & ∼Q) | Necessary to Possible |
| 7. (P & ∼∼Q) ⊃ (P & ∼∼Q) | ∼Q substituted for Q |
| 7. (P & Q) ⊃ (P & Q) | Double Negations removed |
This looks like a reasonable result. If P is true and Q is possible, then (P & Q) is possible. Thus, the strong theorem (P ⊃ Q) ⊃ (P ⊃ Q) and be reduced to (P ⊃ Q) ⊃ (P ⊃ Q) and then even to (P ⊃ Q) ⊃ (P ⊃ Q). This turns on the principle that a conditional can be true even if the antecedent is false. A necessarily true consequent will make the conditional necessarily true.
What about the opposite? What if Q is necessary, does this means that the conditional will then be necessary? As in, (P ⊃ Q) ⊃ (P ⊃ Q)? Let's see what we get:
| 1. (P ⊃ Q) ⊃ (P ⊃ Q) | Premise |
| 2. ∼(P ⊃ Q) ⊃ ∼(P ⊃ Q) | Transposition |
| 3. ∼(P ⊃ Q) ⊃ ∼(P ⊃ Q) | Necessary to Possible |
| 4. ∼(∼P v Q) ⊃ ∼(∼P v Q) | Conditional to Disjunction |
| 5. (∼∼P & ∼Q) ⊃ (∼∼P & ∼Q) | De Morgan, Disjunction to Conjunction |
| 6. (P & ∼Q) ⊃ (P & ∼Q) | Double Negations removed |
| 7. (P & ∼Q) ⊃ (P & ∼Q) | Necessary to Possible |
| 8. (P & ∼∼Q) ⊃ (P & ∼∼Q) | ∼Q substituted for Q |
| 9. (P & Q) ⊃ (P & Q) | Double Negations removed* |
This is not a good result. If (P & Q) is possible, we cannot infer that P. So a necessary consequent does not make a conditional necessary. Does it at least make it true? Let's see:
| 1. (P ⊃ Q) ⊃ (P ⊃ Q) | Premise |
| 2. ∼(P ⊃ Q) ⊃ ∼(P ⊃ Q) | Transposition |
| 3. ∼(∼P v Q) ⊃ ∼(∼P v Q) | Conditional to Disjunction |
| 4. (∼∼P & ∼Q) ⊃ (∼∼P & ∼Q) | De Morgan, Disjunction to Conjunction |
| 5. (P & ∼Q) ⊃ (P & ∼Q) | Double Negations removed |
| 6. (P & ∼Q) ⊃ (P & ∼Q) | Necessary to Possible |
| 7. (P & ∼∼Q) ⊃ (P & ∼∼Q) | ∼Q substituted for Q |
| 8. (P & Q) ⊃ (P & Q) | Double Negations removed |
Yes it does. If (P & Q) is true, then we can infer that P. We can even infer (Q ⊃ Q), which we already know. Since these derivations all use only Replacement rules, we can always state things like this as equivalances: (P ⊃ Q) ⊃ (P ⊃ Q) ≡ (P & Q) ⊃ (P & Q).