Saul Kripke was a philosopher and logician and emeritus professor at Princeton. He is best known for reintroducing modal concepts, such as necessity and possibility, with his landmark works Naming and Necessity and Identity and Necessity, as well as his popularizing Gottfried Leibniz's notion of "possible worlds" as a way of analyzing the concepts of 'a priori', 'analytic', and 'necessary'.

Kripke is enthusiastic about talk of "possible worlds," but says they should be interpreted as counterfactual possibilities in our "actual world." In the preface to the 1981 publication of Naming and Necessity, he says:

I will say something briefly about 'possible worlds'. (I hope to elaborate elsewhere.) In the present monograph I argued against those misuses of the concept that regard possible worlds as something like distant planets, like our own surroundings but somehow existing in a different dimension, or that lead to spurious problems of 'transworld identification'. Further, if one wishes to avoid the Weltangst and philosophical confusions that many philosophers have associated with the 'worlds' terminology, I recommended that 'possible state (or history) of the world', or 'counterfactual situation' might be better. One should even remind oneself that the 'worlds' terminology can often be replaced by modal talk—'It is possible that . . .'

(Naming and Necessity, p. 15)

By comparison with Kripke, the possible worlds of the actualist David Lewis are all deterministic worlds in which the only possibilities are actuals.

There are no counterfactual possibilities in Lewis' possible worlds.

Where for Leibniz, "truth in all possible worlds" is limited to logically possible and non-contradictory statements, Kripke applies his modal concepts to "quantifiable" material objects in the physical world, as had Rudolf Carnap and Ruth Barcan Marcus. Carnap distinguished the "object language" in what he called the "material mode" from a higher level "metalanguage" of logical syntax that he called the "formal mode." The formal mode studies the truth-functional analysis of propositions.

Kripke's metaphysical necessity claims that some empirical facts are necessarily true in all possible worlds, which is problematic for two reasons. First, truth in possible worlds describes the purely logical and mathematical aspects of any world. Second, there is no way to know anything empirical factual about merely possible worlds. However, Kripke's arguments for the "necessity of identity" suggests a connection between the fundamental properties of elementary physical and chemical systems that preserve their identity over time and his claims for necessary a posteriori truths.

Kripke also attacked the theory that proper names are descriptions, for examples bundles of properties, as espoused by Gottlob Frege and especially Bertrand Russell. In this case, Kripke is correct, his 1970 theory of names as "rigid designators" is a great improvement over the Russell theory of descriptions. But the best theory of what Willard Van Orman Quine called in 1943 "purely designative" references was the suggestion of arbitrary numeric "tags" by Ruth Barcan (later Marcus) in 1961, nearly a decade before Kripke's rigid designators.

The Frege-Russell theory of descriptions was also a theory of meaning. The meaning of a proper name was said to consist in all the properties attached to the named person. The Frege-Russell theory was also a theory of reference, of denotation, of terms that "pick out" or identify an individual, whether an inanimate object, a natural kind, or a human being.

Frege and Russell said that some of these properties can be substituted in statements for the name and preserve the truth value of the statements. For example, George Washington can be replaced by "the first president of the United States." But descriptive properties can be problematic.

Kripke's modal analysis of alternative possibilities shows that the first president of the United States might not have been Washington. Things might have been otherwise. Washington might have died in the Revolutionary War.

But Washington's proper name, given him as a child by his parents, told to family and friends and then to people widely through a chain of communications that grew worldwide, could only be a reference, a necessary reference, to this unique individual, an essential reference that identifies him more strongly than any accidental property.

Kripke says that proper names are "rigid designators" that only refer to the objects they designate. They contain none of the likely accidental properties that accrue to persons during their lifetimes, such as "first president." Rigidity of proper names describes their fixed, even necessary character, says Kripke, colorfully described as "true in all possible worlds." Kripke even claims to find truths that are "necessary a posteriori," presumably only "true" within a logical system or language framework, not a fact in the irreducibly contingent material world.

Kripke says that once an object is "baptized" with the first use (the origin) of its proper name, it more reliably denotes that individual than any other properties the individual might acquire during a lifetime.

But note that the name's privilege of being a "rigid designator" is only relative to its early date. So later names, descriptions, or other properties that became more widely known might also serve as a rigid designator. Any property that was established in the past is now unchangeable – "necessary ex post facto?" – even if it could have been otherwise. Mohammed Ali is today known to relatively few as Cassius Clay.

In her 1961 presentation on modalities at the Boston University Colloquium for the Philosophy of Science, Ruth Barcan Marcus suggested purely numerical "tags" to uniquely identify objects. This is today's globally unique identifier (GUID) that is used in transponding devices with a radio-frequency identifier (RFID) such as passports.

Kripke and Willard Van Orman Quine attended the Marcus presentation. Kripke was then a student at Harvard and he developed his idea of a rigid designator (stripped of any meaning) in the immediate following years. He presented his ideas in his 1970 lectures at Princeton, without mentioning Marcus' idea of "tags," perhaps having forgotten them.

Much of Quine's work has been devoted to the confusion when different descriptions are substituted in a statement that then alter the truth value of the statement. A famous example is replacing the number 9 with the number of planets, but most all Quine's conundrums share this substitution failure because the different descriptions do not purely and precisely "refer" (often called "referential opacity"). The Barcan Marcus "tags" and Kripke's rigid designators both solve Quine's problem.

The "number of planets" does not refer to the number 9 per se, but to the quantity of planets. Clarifying the reference is a question of what information philosophy calls "qualification before quantification." Qualification "picks out" the subset of the total information in an object that is relevant to a comparison with others, in what respect – qua – two objects are "identical." Qualification solves many paradoxes and puzzles of metaphysics and analytic philosophy.

In Quine's case, "number of planets" qua planets refers to how many planets there are in the solar system. The "number of planets" qua numerosity might refer to 9 (actually 8 today). But without this qualification, the reference is what Quine calls "opaque" and not useful for quantifying into propositions, including modal propositions.

Beyond Kripke's interest in names and unambiguous references to objects, he proposes a dramatic change in the meaning and the use of some core philosophical concepts in logic, language, and metaphysics – particularly 'a priori', 'analytic', 'necessary' and 'certainty,' normally a quantitative measure of the probability (provability?) of factual evidence.

Before I go any further into this problem, I want to talk about another distinction which will be important in the methodology of these talks. Philosophers have talked (and, of course, there has been considerable controversy in recent years over the meaningfulness of these notions) [about] various categories of truth, which are called 'a priori', 'analytic', 'necessary' — and sometimes even 'certain' is thrown into this batch. The terms are often used as if whether there are things answering to these concepts is an interesting question, but we might as well regard them all as meaning the same thing. Now, everyone remembers Kant (a bit) as making a distinction between 'a priori' and 'analytic'. So maybe this distinction is still made. In contemporary discussion very few people, if any, distinguish between the concepts of statements being a priori and their being necessary. At any rate I shall not use the terms 'a priori' and 'necessary' interchangeably here.

Consider what the traditional characterizations of such terms as 'a priori' and 'necessary' are. First the notion of a prioricity is a concept of epistemology. I guess the traditional characterization from Kant goes something like: a priori truths are those which can be known independently of any experience.

Nothing is known independently of any experience. But a priori truths can be shown independently of any experience, after the fact of learning them a posteriori.

This introduces another problem before we get off the ground, because there's another modality in the characterization of 'a priori', namely, it is supposed to be something which can be known independently of any experience. That means that in some sense it's possible (whether we do or do not in fact know it independently of any experience) to know this independently of any experience. And possible for whom? For God? For the Martians? Or just for people with minds like ours? To make this all clear might [involve] a host of problems all of its own about what sort of possibility is in question here. It might be best therefore,

A priori truths believed on the basis of a priori evidence seems to be a contradictio in adjecto?

instead of using the phrase 'a priori truth', to the extent that one uses it at all, to stick to the question of whether a particular person or knower knows something a priori or believes it true on the basis of a priori evidence.

(Naming and Necessity, pp.34-35)

I won't go further too much into the problems that might arise with the notion of a prioricity here. I will say that some philosophers somehow change the modality in this characterization from can to must.

Kripke is correct. Everything we learn initially is information that comes from experience. But once we build abstract systems like logic, we can then prove some things a priori, within the system. These are truths "in all possible worlds."

They think that if something belongs to the realm of a priori knowledge, it couldn't possibly be known empirically. This is just a mistake. Something may belong in the realm of such statements that can be known a priori but still may be known by particular people on the basis of experience. To give a really common sense example: anyone who has worked with a computing machine knows that the computing machine may give an answer to whether such and such a number is prime. No one has calculated or proved that the number is prime; but the machine has given the answer: this number is prime. We, then, if we believe that the number is prime, believe it on the basis of our knowledge of the laws of physics, the construction of the machine, and so on. We therefore do not believe this on the basis of pure a priori evidence. We believe it (if anything is a posteriori at all) on the basis of a posteriori evidence.

Again, once we invent logical and mathematical systems, we can then show some things a priori, within the system.

Nevertheless, maybe this could be known a priori by someone who made the requisite calculations. So 'can be known a priori' doesn't mean 'must be known a priori'.

(Naming and Necessity, p.35)

The second concept which is in question is that of necessity, Sometimes this is used in an epistemological way and might then just mean a priori. And of course, sometimes it is used in some (I hope) nonpejorative sense. We ask whether something might have been true, or might have been false. Well, if something is false, it's obviously not necessarily true. If it is true, might it have been otherwise? Is it possible that, in this respect, the world should have been different from the way it is?

No "fact about the world" is necessary, because the world might have been otherwise This is to accept the reality of alternative possibilities.

If the answer is 'no', then this fact about the world is a necessary one. If the answer is 'yes', then this fact about the world is a contingent one. This in and of itself has nothing to do with anyone's knowledge of anything. It's certainly a philosophical thesis, and not a matter of obvious definitional equivalence, either that everything a priori is necessary or that everything necessary is a priori. Both concepts may be vague.' That may be another problem. But at any rate they are dealing with two different domains, two different areas, the epistemological and the metaphysical.

(Naming and Necessity, pp.35-36)

Let's try to summarize Kripke's usages with a table...

a priori	analytic	necessary
epistemological, logical, mathematical true by logic	semantic formal mode, metalanguage true by meaning	metaphysical material mode, object language not true in world?

The terms 'necessary' and 'a priori', then, as applied to statements, are not obvious synonyms. There maybe a philosophical argument connecting them, perhaps even identifying them; but an argument is required, not simply the observation that the two terms are clearly interchangeable. (I will argue below that in fact they are not even coextensive—that necessary a posteriori truths, and probably contingent a priori truths, both exist.)

I think people have thought that these two things must mean the same for these reasons:

Running through all possible worlds in our heads is absurd. We are in the "actual world," and can know nothing about them, let alone statements in other-worldly languages!

First, if something not only happens to be true in the actual world but is also true in all possible worlds, then, of course, just by running through all the possible worlds in our heads, we ought to be able with enough effort to see, if a statement is necessary, that it is necessary, and thus know it a priori. But really this is not so obviously feasible at all.

Kripke is not correct. As Quine said, our system of logic is valid because it works in the world (synthetic, not analytic), but everything we learn initially is information that comes from looking at the world. Only after we build abstract systems like logic, can we then show/prove some things a priori, within the logical system.

What is true in all possible worlds, as Leibniz knew, is the abstract truth of logic and mathematics within a formal system

Second, I guess it's thought that, conversely, if something is known a priori it must be necessary, because it was known without looking at the world. If it depended on some contingent feature of the actual world, how could you know it without looking? Maybe the actual world is one of the possible worlds in which it would have been false. This depends on the thesis that there can't be a way of knowing about the actual world without looking that wouldn't be a way of knowing the same thing about every possible world. This involves problems of epistemology_and the nature of knowledge; and of course it is very vague as stated. But it is not really trivial either. More important than any particular example of something which is alleged to be necessary and not a priori or a priori and not necessary, is to see that the notions are different, that it's not trivial to argue on the basis of something's being something which maybe we can only know a posteriori, that it's not a necessary truth. It's not trivial, just because something is known in some sense a priori, that what is known is a necessary truth.

Another term used in philosophy is 'analytic'. Here it won't be too important to get any clearer about this in this talk. The common examples of analytic statements, nowadays, are like 'bachelors are unmarried'.

If gold was as a matter of fact not yellow (in another possible world), that would not make anything "false." False, and true, are attributes of statements within a logical or mathematical system.

Kant (someone just pointed out to me) gives as an example 'gold is a yellow metal', which seems to me an extraordinary one, because it's something I think that can turn out to be false. At any rate, let's just make it a matter of stipulation that an analytic statement is, in some sense, true by virtue of its meaning and true in all possible worlds by virtue of its meaning. Then something which is analytically true will be both necessary and a priori. (That's sort of stipulative.)

Certainty about an empirical fact is the limiting case of its probability approaching unity. A proof in a math book may be in fact incorrect, but when we say it is certain we mean it is true.

Another category I mentioned was that of certainty. Whatever certainty is, it's clearly not obviously the case that everything which is necessary is certain. Certainty is another epistemological notion. Something can be known, or at least rationally believed, a priori, without being quite certain. You've read a proof in the math book; and, though you think it's correct, maybe you've made a mistake. You often do make mistakes of this kind. You've made a computation, perhaps with an error.

There is one more question I want to go into in a preliminary way. Some philosophers have distinguished between essentialism, the belief in modality de re, and a mere advocacy of necessity, the belief in modality de dicto. Now, some people thing, something creating great additional problems, is whether we can say of any particular that it has necessary or contingent properties, even make the distinction between necessary and contingent properties.

This is a serious error by Kripke. Some things do depend on the way they are described. But the truths of logic and mathematics do not. The number 9 is odd by definition and yes odd in the logical systems of all possible worlds.

Look, it's only a statement or a state of affairs that can be either necessary or contingent! Whether a particular necessarily or contingently has a certain property depends on the way it's described. This is perhaps closely related to the view that the way we refer to particular things is by a description. What is Quine's famous example? If we consider the number 9, does it have the property of necessary oddness? Has that number got to be odd in all possible worlds? Certainly it's true in all possible worlds, let's say, it couldn't have been otherwise, that nine is odd. Of course, 9 could also be equally well picked out as the number of planets. It is not necessary, not true in all possible worlds, that the number of planets is odd. For example if there had been eight planets, the number of planets would not have been odd. And so it's thought:

The "inexorable process" would be strict determinism, an idea still believed by many philosophers. No fact in the world depends on our way of referring to it. Nixon's election does not depend on our calling him 'the man who won the election in 1968.' That fact is still contingent when we add the "necessary truth" that the man who won the election in 1968 won the election in 1968. Like any tautological statement, this tells us nothing about the world, about Nixon, or his election.

Was it necessary or contingent that Nixon won the election? (It might seem contingent, unless one has some view of some inexorable processes. . . .) But this is a contingent property of Nixon only relative to our referring to him as 'Nixon' (assuming 'Nixon' doesn't mean 'the man who won the election at such and such a time'). But if we designate Nixon as 'the man who won the election in 1968', then it will be a necessary truth, of course, that the man who won the election in 1968, won the election in 1968.

(Naming and Necessity, pp.38-40)

Using the popular example of "Hesperus is Phosphorus," the two ancient names for the planet Venus that appears as both the Evening star and the Morning star, Kripke claims that since the two names refer to the same thing, they are identical. But this seems extreme. They are only identical in some respect, namely qua referents to Venus.

In his 1892 essay, Sense and Reference, Gottlob Frege suggested if two names, 'a" and 'b', refer to the same object, they can be described as "identical" in some sense. Frege said:

Identity gives rise to challenging questions which are not altogether easy to answer. Is it a relation ? A relation between objects, or between names or signs of objects? In my Begriffsschrift I assumed the latter. The reasons which seem to favor this are the following: a = a and a = b are obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labeled analytic, while statements of the form a=b often contain very valuable extensions of our knowledge and cannot always be established a priori...

Now if we were to regard identity as a relation between that which the names "a" and "b" designate, it would seem that a = b could not differ from a = a (i.e., provided a=b is true).

Frege is saying that two names referring to the same thing can be in some respect "identical" because the thing they refer to is identical to itself.

A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing. What is intended to be said by a = b seems to be that the signs or names "a" and "b" designate the same thing, so that those signs themselves would be under discussion; a relation between them would be asserted... It would be mediated by the connection of each of the two signs with the same designated thing.

If we found "a = a" and "a = b" to have different cognitive values, the explanation is that for the purpose of knowledge, the sense of the sentence, viz., the thought expressed by it, is no less relevant than its referent, i.e., its truth value. If now a=b, then indeed the referent of "b" is the same as that of "a," and hence the truth value of "a = b" is the same as that of "a = a." In spite of this, the sense of "b" may differ from that of "a," and thereby the sense expressed in "a = b" differs from that of "a = a." In that case the two sentences do not have the same cognitive value.

(Sense and Reference, pp.209, 230)

Granted that someone who knows that Venus can appear on either side of the sun, Hesperus and Phosphorus refer to the same thing. But there is no way the names themselves (as words) are identical to one another. We must select a subset of the information contained in the two words and in factual, even scientific and empirical knowledge available, to pick out the fact that these words refer to the same object.

There are not two things (names) here that are identical to one another. Identical terms should be substitutable for one another in propositions and preserve the truth value. Hesperus and Phosphorus are two different words. They contain significantly different information.

One name describes a morning phenomenon. So, there is no truth to the statement "Phosphorus is the Evening Star." Phosphorus never appears in the evening. Circumlocutions are needed like "What we call Phosphorus is a planet that sometimes appears as Hesperus."

Part of the information content here is that we have two words referring to one thing. But each word provides different knowledge about the planet Venus, one telling that Venus sometimes appears to the East of the Sun, the other that it sometimes appears to the West. It is false that "The Morning Star IS The Evening Star." except in a limited sense.

Most all statements of identity between two things should be paraphrased as "these two things are identical in some respect." They are only the same if we ignore their differences. We should say that Hesperus and Phosphorus are identical qua referents to the planet Venus

Gottfried Leibniz's famous law about the "identity of indiscernibles" can not be an absolute statement. The only absolute identity is self-identity. All things are identical only to themselves. Two indiscernibles are only indiscernible qua – in some respects. Numerically distinct objects are easily discerned to be two objects, in different places for example.

But any two things are similar if we ignore all their differences, just as they are different if we ignore their similarities. Exceptions are the identical and "indistinguishable" elementary particles of quantum physics, a deep problem for quantum mechanics and for metaphysics.

Hesperus and Phosphorus are identical only qua referents to a planet, and there is nothing necessary about this fact except that it began in the past and is now a convention and tradition, and as such Hesperus and Phosphorus are Kripke rigid designators.

But we cannot forget the obvious fact from linguistic theory, whether Peirce semiotics or Saussure semiology, that the names Hesperus and Phosphorus are arbitrary symbols, with no information in common with the planet Venus beyond our use of them as names, as designators. In ancient semitic languages, the planet was called Ishtar for centuries before Greeks invented Aphrodite and the Romans created the Latin name for the love goddess.

Given the fact that all human language terms are contingent and historically accidental, we must struggle to understand Kripke's claim for the names' necessity and it connection to identity.

In the physical and logical worlds, no entity can fail to be identical to itself. So we can speak of the necessity of identity of an entity to itself. But is this a tautology, empty of meaning, like A = A? Information philosophy maintains that the only strict identity is self-identity.

In recent years, modal logicians claim to prove the "necessity of identity" using Leibniz's Law – the "Identity of Indiscernibles."

This law claims that if x = y, then x and y must share all their properties, otherwise there would be a discernible difference. Now one of the properties of x is that x = x, so if y shares that property '= x" of x, we can say y = x. Necessarily, x = y. QED?

Our rule that the only identity is self-identity says that two numerically distinct things, x and y, cannot be identical because there is some difference in information between them – their "x-ness" and "y-ness."

Instead of claiming that y has x's property of being identical to x, information philosophy can say only that y has x's property of being self-identical, thus y = y. Necessarily, x ≠ y in at least one respect.

The necessity of identity in symbolic logic, first given by Ruth Barcan Marcus, is

Despite many such arguments in the philosophical literature over the past sixty or seventy years, this is a flawed argument. Numerically distinct objects can only be identical "in some respect," that is if they share qualities which we can selectively "pick out". We can say that a red house and a blue house are identical qua house. They are different qua color.

Here is Saul Kripke's argument against the possibility of contingent identity statements:

First, the law of the substitutivity of identity says that, for any objects x and y, if x is identical to y, then if x has a certain property F, so does y:

(1) (x)(y) [(x = y) ⊃ (Fx ⊃ Fy)]

On the other hand, every object surely is necessarily self-identical:

(2) (x) ◻(x = x)

But

(3) (x)(y) (x = y) ⊃ [◻(x = x) ⊃ ◻ (x = y)]

is a substitution instance of (1), the substitutivity law. From (2) and (3), we can conclude that, for every x and y, if x equals y, then, it is necessary that x equals y:

(4) (x)(y) ((x = y) ⊃ ◻ (x = y))

This is because the clause ◻(x = x) of the conditional drops out because it is known to be true.

This is an argument which has been stated many times in recent philosophy. Its conclusion, however, has often been regarded as highly paradoxical. For example, David Wiggins, in his paper, "Identity-Statements," says,

Now there undoubtedly exist contingent identity statements. Let a = b be one of them. From its simple truth and (5) [= (4) above] we can derive '◻{a = b)'. But how then can there be any contingent identity statements?

(Identity and Necessity, p. 136)

Where are Kripke's errors? First we must unpack his "indiscernibility of identicals." Instead of (x)(y) [(x = y) ⊃ (Fx ⊃ Fy)], we must say that we can clearly discern differences between x and y, their names and their numerical distinctness, unless we are actually talking about a single object using two different names, as with the Morning Star and Evening Star.

Kripke claims to prove the "necessity of identity" using the converse of Leibniz's Law – the "Identity of Indiscernibles."

His indiscernibility of identicals (1) claims that if x = y, then x and y must share all their properties, otherwise there would be a discernible difference. Now one of the properties of x is that x = x, so if y shares the property of '= x," we can say that y = x. Then, necessarily, x = y.

Our information philosophy rule that the only identity is self-identity says that two distinct things, x and y, cannot be identical because there is some difference in information between them. Instead of claiming that y has x's property of being identical to x, we should say only that y has x's property of being self-identical, thus y = y. Then x and y remain distinct in at least the properties "x = x" and "y = y." Arguments for the "necessity of identity" are seriously flawed, except when it is the tautological case of self-identity.

Kripke cites David Wiggins as saying in his 1967 "Identity-Statements"

Now there undoubtedly exist contingent identity-statements. Let a = b be one of them. From its simple truth and (5) [= (4) above] we can derive '◻ ( a = b)'.

Kripke goes on to state the specious argument about b sharing the property ("= a") of being identical to a (viz., "a = a." which information philosophy reads as merely self-identity!). It is not clear that self-identity is predicable of either a or b, like "is red". It feels more like Kant's "existence is not a predicate." And if self-identity is predicable of b, it should read "b = b." This is a monadic equivalence self-relation, not a dyadic relation with another object.

If x and y are the same things and we can talk about modal properties of an object at all, that is, in the usual parlance, we can speak of modality de re and an object necessarily having certain properties as such, then formula (1), I think, has to hold. Where x is any property at all, including a property involving modal operators, and if x and y are the same object and x had a certain property F, then y has to have the same property F. And this is so even if the property F is itself of the form of necessarily having some other property G, in particular that of necessarily being identical to a certain object. [viz., = x] Well, I will not discuss the formula (4) itself because by itself it does not assert, of any particular true statement of identity, that it is necessary. It does not say anything about statements at all. It says for every object x and object y, if x and y are the same object, then it is necessary that x and y are the same object. And this, I think, if we think about it (anyway, if someone does not think so, I will not argue for it here), really amounts to something very little different from the statement (2). Since x, by definition of identity, is the only object identical with x, "(y)(y = x ⊃ Fy)" seems to me to be little more than a garrulous way of saying 'Fx' and thus (x) (y)(y = x ⊃ Fx) says the same as (x)Fx no matter what 'F' is — in particular, even if 'F' stands for the property of necessary identity with x. So if x has this property (of necessary identity with x), trivially everything identical with x has it, as (4) asserts. But, from statement (4) one may apparently be able to deduce various particular statements of identity must be necessary and this is then supposed to be a very paradoxical consequence.

(Identity and Necessity, pp.137-138)

Among the "various particular statements" that must be necessary, Kripke develops one that is responsible for arguments that lead to necessitism and necessary beings.

Kripke has defined a different kind of necessity from that usually identified with the analytic and the a priori. He alters the traditional distinction between the necessary and the contingent.

Kripke calls his idea metaphysical necessity to distinguish it from epistemic necessity. Kripke further distinguishes analyticity and a prioricity from necessity. For him, analyticity is a semantic notion, a priori is epistemic, and his necessity is a metaphysical notion.

Analyticity covers everything known to be true or false by definition or meaning of the terms involved. This includes logical and mathematical truths, such as "A is A," and "7 + 5 = 12." He says, "an analytic statement is, in some sense, true by virtue of its meaning and true in all possible worlds by virtue of its meaning. Then something which is analytically true will be both necessary and a priori. (That's sort of stipulative.)" (Naming and Necessity, p.39).

Kripke claims to have necessary knowledge a posteriori. This argument goes against common sense as well as traditional deep thinking in science and philosophy. It is astonishing and dazzling that it has become so popular in modal logic and metaphysics.

What we know is that first, lecterns usually are not made of ice, they are usually made of wood. This looks like wood. It does not feel cold and it probably would if it were made of ice. Therefore, I conclude, probably this is not made of ice. Here my entire judgment is a posteriori. I could find out that an ingenious trick has been played upon me and that, in fact, this lectern is made of ice; but what I am saying is, given that it is in fact not made of ice, in fact is made of wood, one cannot imagine that under certain circumstances it could have been made of ice. So we have to say that though we cannot know a priori whether this table was made of ice or not, given that it is not made of ice, it is necessarily not made of ice.

P should be "The evidence suggests this table is not made of ice."

In other words, if P is the j statement that the lectern is not made of ice, one knows by a priori a philosophical analysis, some conditional of the form "if P, then necessarily P." If the table is not made of ice, it is necessarily not made of ice. On the other hand, then, we know by empirical investigation that P. the antecedent of the conditional, is true —that this table is not made of ice. We can conclude by modus ponens:

P ⊃ =◻P
P
______________
◻P

The conclusion should be "It is necessary that the evidence suggests this table is not made of ice."

The conclusion —'◻P' — is that it is necessary that the table not be made of ice, and this conclusion is known a posteriori, since one of the premises on which it is based is a posteriori. So, the notion of essential properties can be maintained only by distinguishing between the notions of a priori and necessary truth, and I do maintain it.

(Identity and Necessity, p.153)

Kripke's metaphysical necessity concerns empirical facts that are known to be the case by the nature of a physical object. This is based on the physical presumption that the way the world is, for example the laws of nature, could not have been otherwise. It may also be based on the fact that any event in the past is now fixed and so can be called metaphysically necessary – a sort of necessary ex post facto? In any case, Kripke believes that we discover the essential properties, the essence, of physical objects empirically (p.110).

Anything that has been empirically determined to be the case thus can be called metaphysically necessary or "necessary a posteriori," says Kripke.

Consider the modal claim 'Necessarily, water is H₂O.' It is said to follow from the empirical and a posteriori claim 'Water is H₂O' together with an a priori claim, such as 'If water is H₂O, then necessarily, water is H₂O' (p.128). But this seems dangerously like the redundancy in 'If water is H₂O, then it is true that water is H₂O'?

Kripke's other examples include: it is necessary that gold is necessarily a metal, that it is yellow, and has atomic number 79 (p.118). Lightning is necessarily an electrical discharge (p.132). "This table (pointing at a table in the room) is necessarily made of wood," if it was made of wood. Indeed, he says that the table was by metaphysical necessity made of the exact wood that it was made of.

We can take some of Kripke's "metaphysical necessity" examples with a metaphorical grain of salt (necessarily NaCl). This is because the physical world contains the possibility that the carpenter could have chosen a different piece of wood, or the table could have been made of ice (Kripke's cryptic alternative, p.114).

More dramatically, some prominent metaphysicians deny identity over time. They are "perdurantists," chief among them was perhaps David Lewis.

In this case, there is nothing illogical about the table being wood at one instant and ice at the next instant. It is physical science, and our information identity, that supports the idea of identity over time.

Kripke and Lewis are both famous for using the concept of possible worlds, but there are some extreme and very important differences between them. Kripke thinks that Lewis's idea has "encouraged philosophical pseudo-problems and misleading pictures." One major difference is that Lewis thinks of his super-infinity of possible worlds as actually existing in an infinite space-time continuum, where Kripke thinks his possible worlds are merely ways of talking about the alternative possibilities in our actual world. He says that ''possible worlds' are total 'ways the world might have been', or states or histories of the entire world, or 'counterfactual situations' might even be better.

I will say something briefly about 'possible worlds'. (I hope to elaborate elsewhere.) In the present monograph I argued against those misuses of the concept that regard possible worlds as something like distant planets, like our own surroundings but somehow existing in a different dimension, or that lead to spurious problems of 'transworld identification'. Further, if one wishes to avoid the Weltangst and philosophical confusions that many philosophers have associated with the 'worlds' terminology, I recommended that 'possible state (or history) of the world', or 'counterfactual situation' might be better. One should even remind oneself that the 'worlds' terminology can often be replaced by modal talk—'It is possible that . . .'

'Possible worlds' are little more than the miniworlds of school probability blown large. It is true that there are problems in the general notion not involved in the miniature version. The miniature worlds are tightly controlled, both as to the objects involved (two dice), the relevant properties (number on face shown), and (thus) the relevant idea of possibility. 'Possible worlds' are total 'ways the world might have been', or states or histories of the entire world. To think of the totality of all of them involves much more idealization, and more mind-boggling questions, than the less ambitious elementary school analogue. Certainly the philosopher of 'possible worlds' must take care that his technical apparatus not push him to ask questions whose meaningfulness is not supported by our original intuitions of possibility that gave the apparatus its point. Further, in practice we cannot describe a complete counterfactual course of events and have no need to do so.

Consider a miniworld where an agent considers action A, B, and C and does A, then B and C were only actual possibles

A practical description of the extent to which the 'counterfactual situation' differs in the relevant way from the actual facts is sufficient; the 'counterfactual situation' could be thought of as a miniworld or a ministate, restricted to features of the world relevant to the problem at hand...

There is nothing wrong in principle with taking these [possible worlds], for philosophical or for technical purposes, as (abstract) entities—the innocence of the grammar school analogue should allay any anxieties on that score. (Indeed the general notion of 'sample space' that forms the basis of modern probability theory is just that of such a space of possible worlds.) However, we should avoid the pitfalls that seem much more tempting to philosophers with their grand worlds than to schoolchildren with their modest versions. There are no special grounds to suppose that possible worlds must be given qualitatively, or that there need be any genuine problem of 'transworld identification'—the fact that larger and more complex states are involved than in the case of the dice makes no difference to this point. The 'actual world'—better, the actual state, or history of the world —should not be confused with the enormous scattered object that surrounds us. The latter might also have been called 'the (actual) world', but it is not the relevant object here. Thus the possible but not actual worlds are not phantom duplicates of the 'world' in this other sense. Perhaps such confusions would have been less likely but for the terminological accident that 'possible worlds' rather than 'possible states', or 'histories', of the world, or 'counterfactual situations' had been used. Certainly they would have been avoided had philosophers adhered to the common practices of schoolchildren and probabilists.

(Naming and Necessity, pp.15-20)

When thinking about different possibilities in the actual world, e.g., what if Nixon had lost the 1968 presidential election and Humphrey won it, Nixon in Kripke's alternative possible world is the same individual, differing only in the property of losing the election. All of Kripke's possible worlds are different ways our actual world might have been.

By contrast, David Lewis describes a Nixon in an alternate world as not the same individual, but a "counterpart" of Nixon who has the same bundle of properties as the actual Nixon, with the exception of the election loss. This raises the troubling problem of a "trans-world individual." Clearly no matter how similar, individuals in two different worlds are not identical.

I wish at this point to introduce something which I need in the methodology of discussing the theory of names that I'm talking about. We need the notion of 'identity across possible worlds' as it's usually and, as I think, somewhat misleadingly called.

(Misleadingly, because the phrase suggests that there is a special problem of 'transworld identification", that we cannot trivially stipulate whom or what we are talking about when we imagine another possible world. The term 'possible world' may also mislead; perhaps it suggests the 'foreign country' picture. I have sometimes used 'counterfactual situation' in the text; Michael Slote has suggested that 'possible state (or history) of the world' might be less misleading than 'possible world'. It is better still, to avoid confusion, not to say, 'In some possible world, Humphrey would have won' but rather, simply, 'Humphrey might have won'. The apparatus of possible words has (I hope) been very useful as far as the set-theoretic model-theory of quantified modal logic is concerned, but has encouraged philosophical pseudo-problems and misleading pictures.)

One of the intuitive theses I will maintain in these talks is that names are rigid designators. Certainly they seem to satisfy the intuitive test mentioned above: although someone other than the U.S. President in 1970 might have been the U.S. President in 1970 (e.g., Humphrey might have), no one other than Nixon might have been Nixon. In the same way, a designator rigidly designates a certain object if it designates that object wherever the object exists; if, in addition, the object is a necessary existent, the designator can be called strongly rigid. For example, 'the President of the U.S. in 1970' designates a certain man, Nixon; but someone else (e.g., Humphrey) might have been the President in 1970, and Nixon might not have; so this designator is not rigid.

In these lectures, I will argue, intuitively, that proper names are rigid designators, for although the man (Nixon) might not have been the President, it is not the case that he might not have been Nixon (though he might not have been called 'Nixon'). Those who have argued that to make sense of the notion of rigid designator, we must antecedently make sense of 'criteria of transworld identity' have precisely reversed the cart and the horse; it is because we can refer (rigidly) to Nixon, and stipulate that we are speaking of what might have happened to him (under certain circumstances), that 'transworld identifications' are unproblematic in such cases.

(Of course I don't imply that language contains a name for every object Demonstratives can be used as rigid designators, and free variables can be used as rigid designators of unspecified objects. Of course when we specify a counterfactual situation, we do not describe the whole possible world, but only the portion which interests us.)

(Naming and Necessity, pp.47-49)

It is critical to note that metaphysicians proposing possible worlds are for the most part materialists and determinists who do not believe in the existence, as abstract entities, of counterfactual and ontological possibilities in our world.

First, metaphysicians "index" our world as the "actual world." They are actualists who say that the only possibilities have always been whatever actually happened. This is Dan Dennett's position, for example, not that far from the original actualist, Diodorus Cronus.

Moreover, all of their infinite number of possible worlds are governed by deterministic laws of nature. This means that there are also no real possibilities in any of their possible worlds, only actualities there as well.

Now this is quite ironic, since the invention of possible worlds was proposed as a superior way of talking about counterfactual possibilities in our world.

Since information philosophy defends the existence of alternative possibilities leading to different futures, we can adopt a form of modal discourse to describe these possibilities as possible future worlds for our to-be-actualized world.

It turns out there is an infinity of such possible future worlds. The infinity is not as large as the absurdly extravagant number in David Lewis's possible worlds, which have counterparts for each and every living person with every imaginable difference in each of our counterparts, each counterpart in its own unique world.

Thus there are Lewisian worlds in which your counterpart is a butcher, baker, candlestick maker, and every other known occupation. There are possible worlds in which your counterpart eats every possible breakfast food, drives every possible car, and lives in every block on every street in every city or town in the entire word.

This extravagance is of course part of Lewis's appeal. It makes Hugh Everett's "many worlds" of quantum mechanics (which split the universe in two when a physicist makes a quantum measurement) minuscule, indeed quite parsimonious, by comparison.

Specifically, when an Everett universe splits into two, it doubles the matter and energy in the new universe(s) – an extreme violation of the principle of the conservation of matter/energy – and it also doubles the information. Apart from that absurdity, the two universes differ by only one bit of information, for example, whether the electron spin measured up or down in the quantum measurement.

Similarly, for every Lewisian universe, the change of one bit of information implies one other possible universe in which all the infinite number of other bits stay exactly the same. But Lewis imagines that every single bit in the universe may be changed at any time, an order of physical infinities that rivals the greatest number that Georg Cantor ever imagined. Is David Lewis ontologically committed to such a number?

Although Kripke does not seem to have said anything specific about the problem of free will, his view of "possible worlds" may be sympathetic to human freedom, since he describes the worlds as "ways the world might have been."

In our two-stage model of free will, we can describe the alternative possibilities for action generated by an agent in the first stage as "possible worlds." They are "counterfactual situations" in Kripke's sense, involving a single individual. Suppose the agent is considering three different courses of action. During the second stage of evaluation and deliberation only one of the three options (each a "possible world") will become actualized.

The agent is the same individual of interest in these three possible worlds. There are no Lewisian "counterparts." There is no problem of "transworld identification."

Note that these five possible worlds are extremely close to one another, "nearby" in the sense of their total information content. We can focus on the "miniworld" of the five options and hold the rest of the universe constant. As Kripke described it, "the 'counterfactual situation' could be thought of as a miniworld or a ministate, restricted to features of the world relevant to the problem at hand."

Quantification over the information in each world shows that the difference between them is very small number of bits, especially when compared to the typical examples given in possible worlds cases. In the case of Humphrey winning the election, millions of persons would have to have done something different. Such worlds are hardly "nearby" one another

For typical cases of a free decision, the possible worlds require only small differences in the mind of a single person. Kripke argued against the identity of mind and body (or brain), and in this example it would only be the thoughts of the agent that pick out the possible world that will be actualized.

Our thoughts are free. Our actions are willed by an adequately determined evaluation and decision process, not one that was pre-determined by the mechanical laws of nature acting on our material bodies.

Following Kripke, we can build a model structure M as an ordered triple <G, K, R>. K is the set of all "possible worlds," G is the "actual world," R is a reflexive relation on K, and G ∈ K.

If H₁, H₂, and H₃ are three possible worlds in K, H₁RH₂ says that H₂ is "possible relative to" or "accessible from" H₁, that every proposition true in H₂ is possible in H₁.

Indeed, the H worlds and the actual world G are all mutually accessible and each of these is possible relative to itself, since R is reflexive.

Now the model system M assigns to each atomic formula (propositional variable) P a truth-value of T or F in each world H ∈ K.

Let us define the worlds H₁, H₂, and H₃ as identical to the real world G in all respects except the following statements describing actions of a graduating college student Alice deciding on her next step.

In H₁, the proposition "Alice accepts admission to Harvard Medical School" is true.

In H₂, the proposition "Alice accepts admission to MIT" is true.

In H₃, the proposition "Alice postpones her decision and takes a 'gap year'" is true.

At about the same time, in the actual world G, the statement "Alice considers graduate school" is true.

Note that the abstract information that corresponds to the three possible worlds H is embodied physically in the matter (the neurons of Alice's brain) in the actual world and in the three possible worlds. There is no issue with the "transworld identity" of Alice as there would be with Lewis's modal realism," because all these possible worlds are in the same spatio-temporal domain. The four statements are true in all possible worlds.

The metaphysical question is which of the three possible worlds becomes the new actual world, say at time t. What is the fundamental structure of reality that supports the simultaneous existence of alternative possibilities?

Just before time t, we can interpret the semantics of the model structure M as saying that the above statements were "merely possible" thoughts about future action in Alice's mind.

Note also that just after the decision at time t, the three possible applications remain in Alice's Experience Recorder and Reproducer as memories.

In the future of world H₁, Alice's research discovers the genetic signals used in messaging by cancer cells and cancer is eliminated. Several hundred million lives are saved (extended) in Alice's lifetime.

In the future of world H₂, Alice engineers the miniaturization of nuclear weapons so they are small enough to be delivered by tiny drones. One is stolen from AFB by a terrorist and flown to X where millions of lives are lost. Alice kills herself the next day.

In the future of world H₃, a mature Alice returns to school, completes her Ph.D. in Philosophy at Princeton and writes a book on Free Will and Moral Responsibility.

Kripke is well known for his "metaphysical necessity" and the "necessary a posteriori."

Broadly speaking, modern philosophy has been a search for truth, for a priori, analytic, certain, necessary, and provable truth. For many philosophers, a priori, analytic, and necessary, have been more or less synonymous.

But all these concepts are mere ideas, invented by humans, some aspects of which have been discovered to be independent of the minds that invented them, notably formal logic and mathematics. Logic and mathematics are systems of thought, inside which the concept of demonstrable (apodeictic) truth is useful, but with limits set by Kurt Gödel's incompleteness theorem. The truths of logic and mathematics appear to exist "outside of space and time." We call them a priori because their proofs are independent of experience, although they were abstracted empirically from concrete human experiences.

Analyticity is the idea that some statements, some propositions in the form of sentences, can be true by the definitions or meanings of the words in the sentences. This is correct, though limited by verbal difficulties such as Russell's paradox and numerous other puzzles and paradoxes. Analytic language philosophers claim to connect our words with objects, material things, and thereby tell us something about the world. Some modal logicians, inspired by Kripke, claim that words that are names of things are necessary a posteriori, "true in all possible worlds." But this is nonsense, because we invented all those words and worlds. They are mere ideas.

Perhaps the deepest of all these philosophical ideas is necessity. Information philosophy can now tell us that there is no such thing as absolute necessity. There is of course an adequate determinism in the macroscopic world that explains the appearance of deterministic laws of nature, of cause and effect, for example. This is because macroscopic objects consist of vast numbers of atoms and their individual random quantum events average out. But there is no metaphysical necessity. At the fundamental microscopic level of material reality, there is an irreducible contingency and indeterminacy. Everything that we know, everything we can say, is fundamentally empirical, based on factual evidence, the analysis of experiences that have been recorded in human minds.

So information philosophy is not what we can logically know about the world, nor what we can analytically say about the world, nor what is necessarily the case in the world. There is nothing that is the case that is necessary and perfectly determined by logic, by language, or by the physical laws of nature. Our world and its future are open and contingent, with possibilities that are the source of human freedom.

For the most part, philosophers and scientists do not believe in possibilities, despite their invented "possible worlds," which are on inspection merely multiple "actual worlds." This is because they cannot accept the idea of ontological chance. They hope to show that the appearance of chance is the result of human ignorance, that chance is merely an epistemic phenomenon.

Now chance, like truth, is just another idea, just some more information. But what an idea! In a self-referential virtuous circle, it turns out that without the real possibilities that result from ontological chance, there can be no new information. Information philosophy offers cosmological and biological evidence for the creation of new information in the universe. So it follows that chance is real, fortunately something that we can keep under control. We are biological beings that have evolved, thanks to chance, from primitive single-cell communicating information structures to multi-cellular organisms whose defining aspect is the creation and communication of information.

The theory of communication of information is the foundation of our "information age." To understand how we know things is to understand how knowledge represents the material world of embodied "information structures" in the mental world of immaterial ideas.

All knowledge starts with the recording of experiences. The experiences of thinking, perceiving, knowing, feeling, desiring, deciding, and acting may be bracketed by philosophers as "mental" phenomena, but they are no less real than other "physical" phenomena. They are themselves physical phenomena.
They are just not material things.

Information philosophy defines human knowledge as immaterial information in a mind, or embodied in an external artifact that is an information structure (e.g., a book), part of the sum of all human knowledge. Information in the mind about something in the external world is a proper subset of the information in the external object. It is isomorphic to a small part of the total information in or about the object. The information in living things, artifacts, and especially machines, consists of much more than the material components and their arrangement (positions over time). It also consists of all the information processing (e.g., messaging) that goes on inside the thing as it realizes its entelechy or telos, its internal or external purpose.

All science begins with information gathered from experimental observations, which are mental phenomena. Observations are experiences recorded in minds. So all knowledge of the physical world rests on the mental. All scientific knowledge is information shared among the minds of a community of inquirers. As such science is a collection of thoughts in thinkers, immaterial and mental, some might say fundamental. Recall Descartes' argument that the experience of thinking is that which for him is the most certain.

Despite the absence of any absolute physical necessity about what there is (ontology), information philosophy can and does embrace Kripke's metaphysical necessity. We take this to be his proof of the necessity of identity, first suggested by Ruth Barcan Marcus using Leibniz's Law of the Identity of Indiscernibles and its converse, the indiscernibility of identicals.

It is metaphysically necessary, both logically and in terms of an information analysis, that everything is identical to itself. Self-identity is a necessary truth. If you exist, you do not exist necessarily, as Timothy Williamson claims, but you are necessarily self-identical.

If you exist, you are very nearly identical to yourself a moment ago. But because your information content is a strong function of time, you (t) ≠ you (t + 1). This will make the perdurantists happy, but the change in information is a tiny fraction of your total, so endurance theorists are closer to the truth in the problem of persistence.

Kripke's claims for the necessary a posteriori in some natural kinds can be viewed from the information standpoint. Fundamental elements like gold and even water = H₂O molecules have internal information content that is constant over time when they are isolated from other particles. These very simplest entities endure in the sense of information constancy. The electron and the proton appear to have infinite lifetimes, nearly perfect identity over time.

Information philosophy adds another essential modal element to philosophy that we can perhaps glimpse in Kripke's writings and borrow his unique version of metaphysicality to describe it. Based on the physical reality of ontological chance, information philosophy defends the existence of what we can call "metaphysical possibility"

We could describe it in the popular jargon of metaphysically possible worlds, but that would associate it with the extravagant ideas of David Lewis and Hugh Everett III, and Kripke has spoken out clearly against their extravagance. His words also hint at the location of the metaphysical possibilities of information philosophy: Our metaphysical possibilitiy is the very opposite of Lewisian extravagance - but it is essential to the philosophy of mind.

Barcan, R. C. (1946). "A functional calculus of first order based on strict implication." The Journal of Symbolic Logic, 11(01), 1-16.
Barcan, R. C. (1946). "The deduction theorem in a functional calculus of first order based on strict implication." The Journal of Symbolic Logic, 11(04), 115-118.
Barcan, R. C. (1947). "The identity of individuals in a strict functional calculus of second order." The Journal of Symbolic Logic, 12(01), 12-15.
Kripke, Saul. 1971. "Identity and Necessity." In Munitz 1971, 135-164.
Kripke, Saul. 1981. "Naming and Necessity." Blackwell Publishing.
Marcus, R. B. (1961). Modalities and intensional languages. Synthése, 13(4), 303-322.
Munitz, Milton, ed. 1971. Identity and Individuation. New York: New York University Press.
Quine, W. V. 1943. "Notes on Existence and Necessity." The Journal of Philosophy, 40 (5) p.113
Quine, W. V. 1980. From a Logical Point of View, 2d ed. Cambridge, MA: Harvard University Press.
Wiggins, David. 2001. Sameness and Substance Renewed. Cambridge University Press.
Williamson, T. (2002). "Necessary existents." Royal Institute of Philosophy Supplement, 51, 233-251. Chicago

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