Kripke, Casullo, One Metre, my two cents, and the Contingent A Priori

Abstract

Kripke provides what he believes is an example of a contingent truth knowable a priori. Kripke’s demonstration involves the naming of a definite length exemplified by a particular stick in Paris. His example, however, is but a token of a more generic type of example. That is, we can extrapolate from Kripke’s example a formula which can be used to orchestrate many other contingent truths that can be known a priori. Albert Casullo argues, however, that Kripke has failed to make his case. But Casullo’s argument stems from a fundamental misunderstanding of Kripke’s position; he neglects the subtleties inherent in Kripke’s argument, namely, the theses and chronology of (baptismal) reference fixing and rigid designation. Instead, Casullo’s argument fixates on Kripke’s sloppy wording, and neglects the fact that, on at least one plausible construal, Kripke’s argument succeeds. Having said that Kripke succeeds, I do acknowledge an intuitive stumbling block that many people cannot seem to navigate, namely, they cannot seem to understand the fact that the truth knowable a priori on Kripke’s example is not a necessary truth. I will show that this stumbling block stems from a confusion as to the scope of necessity. I will then show that the ‘stumbling block’ rests on an even more general confusion.

Traditionally, the notions of apriority and necessity were thought to be extensionally identical. That is, the set of all truths knowable a priori was thought to be one and the same as the set of all necessary truths. At least there had been no clear counterexamples to suggest otherwise. However, in Naming and Necessity, Saul Kripke challenges this traditional belief. He argues that not only are there necessary truths that can only be known a posteriori, but that some truths known a priori are not necessary truths. It is this second argument, found in A Priori Knowledge, Necessity, and Contingency, that I wish to examine more closely.

Even though they were thought to be extensionally identical, it was (and is) widely acknowledged that the notions of necessity and apriority pertain to different categories of truth. Apriority is an epistemological notion, whereas necessity is a metaphysical notion. The epistemological category includes the notions essentially pertaining to modes of knowledge; these include notions such as a priori and a posteriori. A priori knowledge is knowable through cognitive reflection alone, deriving its epistemic warrant independently of empirical experience, whereas a posteriori knowledge requires empirical evidence for its warrant and cannot be known independently of empirical experience. The metaphysical category includes the modal notions of possibility, necessity, contingency, and impossibility, the definitions of which can be clarified through possible world semantics. A proposition is possible if it is true in at least one possible world (possible closed state of affairs); a proposition is necessary if it is true in all possible worlds; a proposition is contingent if it is true in at least one possible world and false in at least one possible world (i.e., possible but not necessary); a proposition is impossible if it is false in all possible worlds.

Kripke’s thesis introduces the notion of reference fixing and the distinction between rigid and non-rigid designation. According to Kripke, a designator is used non-rigidly if, in some possible worlds, it designates (i.e., refers to) one thing and, in other possible worlds, it designates something else. [Kripke: 149] For example, in the actual world (i.e., the possible world in which we find ourselves), the phrase ‘the first President of the U.S.’ designates George Washington. However, in another (non-actual) possible world it may designate Thomas Jefferson (if in that world Thomas Jefferson was the first President of the U.S.). A designator is used rigidly if it designates the same thing in every possible world. [Kripke: 149] Non-rigid designators are descriptive references predicated upon contingent (i.e., accidental) properties of the thing in question. Rigid designators, on the other hand, are not descriptive (at all), but are simply references (name tags) that pick out specific things within possible worlds. [Kripke: 150-151] Kripke’s intuitive thesis is that names are rigid designators. [Kripke: 150]

However, before we can use a designation rigidly, the reference of the designation must be fixed. (i.e., the thing in question must be named). [Kripke: 152] Usually we adopt a previous fixed referent (for historical figures such as Aristotle, for instance), but we could fix the referent ourselves (when we christen a ship or a child, for instance). Fixing the referent is not to give intensional meaning, but only to secure an extensional referent. There are a number of ways to do this. But usually it is done by picking a possible world (almost always the actual world) and designating a thing in that world in virtue of some of the accidental properties held by that thing in that world. There are, of course, a number of ways to do this. We can do it ostensibly, by pointing at some thing (perhaps a newborn) and dubbing it ‘Nicole’, say, or we can use a definite description used either attributively or referentially. Moreover, the same reference could be fixed in different ways; by specifying a different accidental property (or set of such properties), for instance. Regardless however, after (and only after) the reference has been fixed (using accidental properties of the thing in question in a specific possible world), then (and only then) can the designation be used rigidly. For example, typically we use the term ‘water’ to designate that kind of stuff that we are acquainted with in this world (and call ‘water’). However, if we stipulate that ‘water’ is the term we wish to use to designate that same kind of stuff in all possible worlds (whether or not any token of that type of stuff exists in other worlds) then we are using ‘water’ as a rigid designator.

Our original choice of terms is contingent. Water could have been, and is (in other languages), referred to by other terms. The term chosen as the referent is not of critical importance. For when we assert A = B, we do not assert that the names or their meanings are identical, rather, we assert that the thing which ‘A’ names is identical to the thing which ‘B’ names. Although the name is contingently chosen, once chosen, it fixes a reference across possible worlds. Furthermore, it does not matter that different people with different idiolects or in different discussions (contexts) may use different names (or different symbols) to fix a reference, as long as those within the same idiolect (or discussion) fix the reference the same way (or, more commonly, adopt the same previously fixed reference).

Kripke’s metre bar example is meant to demonstrate how to orchestrate a contingent truth knowable a priori. To begin, we assume that there is yet no standard used to determine the metric system. We then consider a person, Amanda, say, who stipulates that the length of stick S (a definite length of a particular stick at a particular place in Paris in the actual world) at time t₀ (whatever that length may be, i.e., whatever the linear extent or proportion that length may be) shall be named ‘one metre’. She has named that length (whatever its linear extent) by using accidental properties of that length, namely, that that length is exemplified by that stick (at that time, etc.). Amanda designates a definite length, but she does not know what the linear extent or proportion of that length is, only that that stick exemplifies that length at t₀. And she knows that that length is the length of stick S at t₀ because that length is determined by the length of stick S at t₀ (in the actual world). Amanda could have just as well called the length ‘Bob’ or ‘Joe’ or ‘Billy Bob Joe’ or ‘abc’ or ‘$%#’ or ‘the length formerly known as Billy’, but decided on ‘one metre’. That is, ‘one metre’, here, has absolutely no descriptive force (of accidental properties) whatsoever — it is only a name. We thus have a statement ‘One metre is the length of stick S at t₀’, in which ‘one metre’ is a rigid designation (a name) and ‘the length of stick S at t₀’ is a non-rigid designation (a description).

Now, Kripke claims that there is an intuitive difference between the phrase ‘one metre’ (a name) and the phrase ‘the length of stick S at t₀’ (a definite description). [Kripke: 152] The phrase ‘one metre’ (not a description, but a name tag), once fixed, is meant to rigidly designate a particular thing (in this case, a length) in all possible worlds. That is, the length named ‘one metre’ is the same length in all possible worlds (regardless of whether anything exemplifies it in those worlds). [Kripke: 152] On the other hand, the phrase ‘the length of stick S at t₀’ (a description, not a name) does not designate anything rigidly. Instead, it is a descriptive predication, where a predicate is just a set of properties that may be contingently applied to (identified with) different individuals in different worlds. In some possible worlds, stick S has a length different than it has in the actual world. Thus, in different worlds, stick S will (at t₀) exemplify lengths different than the length named ‘one metre’. [Kripke: 152] In fact, for instance, in some possible worlds the length of stick S at t₀ is twice as long as the length dubbed ‘one metre’.

Thus, though the statement includes two designators, only one, the name ‘one metre’ is a rigid designator. The other, ‘the length of stick S at t₀’, a description of accidental properties (that differ in some possible worlds), is not a rigid designator. [Kripke: 153]

The difference between an identity statement with two rigid designators and a statement with one rigid designator and one non-rigid designator is the difference between identity and identification. Identity is commonly formally explicated conditionally, a = b ® " P(Pa ® Pb), asserting the indiscernibility of identicals. That is, the identity of individuals referenced by a and b implies that they share of all of their properties P. An example of identity is: plutonium = Pu; that is, plutonium is (identical to) Pu. This is different, however, than identifying something with a set of properties. That is, plutonium is not a property of Pu (nor is Pu a property of plutonium). Instead, Pu is plutonium. They are one and the same thing, referenced by two different names. An identification, on the other hand, is when an aggregation of properties, P, is held by individual a (in possible world W), such that a can be identified by P (in possible world W). An example of identification is: Plutonium is the most lethal poison known to man. That is, Plutonium is (identified as) the most lethal poison known to man. ‘Plutonium’ is a fixed reference, whereas ‘the most lethal poison known to man’ is a contingent description of plutonium ¾ a physically contingent (accidental) property of plutonium.

Kripke’s example is an identification: ‘One metre’ is a fixed reference (a rigid designator), whereas ‘the length of stick S at t₀’ is a non-rigid designator — a contingent property of (the length named) one metre. The description is true of the length named ‘one metre’ in some possible worlds (including, trivially, the actual world), but it is not true in every possible world. There are numerous (perhaps an infinite number of) reasons for this (all of which are exemplified in one possible world or another). Kripke alludes to one possible reason (exemplified in at least one possible world), namely, that in one non-actual possible world, W, say, heat had been applied to this stick S at t₀, such that at t₀ (in W) stick S would not have been the same length as the length named ‘one metre’ across possible worlds. [Kripke: 152, 153] Thus, since the statement ‘One metre is the length of stick S at t₀’ is not an identity between two rigid designators, it is a contingent truth (and not a necessary truth). [Kripke: 153]

In the following diagram, for instance, the four possible worlds are consistent with Amanda’s statement ‘One metre is the length of stick S at t₀’. For Amanda’s description (that fixes the referent) is only applicable to the state of her (actual) world. It just so happens that in some other possible worlds (including possible world Y) the length of stick S at t₀ is the same length as the length named ‘one metre’. The point is, however, that in some possible worlds (including possible worlds X and Z) the length of stick S at t₀ is not the same length as the length named ‘one metre’. It follows that ‘One metre is the length of stick S at t₀’ is not a necessary truth.

However, since Amanda instigated the baptism (or christening) of the name (i.e., she stipulated or fixed the reference as she pleased), she can know a priori that in the actual world (the world in which she fixed the reference) that one metre is the length of stick S at t₀. She can know this regardless of the linear extent or proportions of that length, for she can know this without knowing the extent or proportions of that length. All she has to know is that she named a length ‘one metre’ in virtue of it being the length of stick S at t₀. And since, trivially, the length of stick S at t₀ is the length of stick S at t₀, she can know, a priori, that one metre is the length of stick S at t₀. According to Kripke, "If [Amanda] used stick S to fix the reference of the term ‘one metre’, then as a result to this kind of ‘definition’ (which is not an abbreviative or synonymous definition), [she] knows automatically, without further investigation, that S is one metre long." [Kripke: 153]

Again, since the statement ‘One metre is the length of stick S at t₀’ is not a statement of identity between two rigid designators (but an identification of one rigid designator with one non-rigid designator), it is a contingent truth. And the fact that it is contingent, coupled with the fact that Amanda can know that the statement is true a priori, provides us with a contingent truth knowable a priori. [Kripke: 153]

One of the difficulties that some people seem to have with Kripke’s demonstration stems from their fixation on ‘one metre’ (used as a name and not as a description) as having some descriptive value (of accidental properties). But it doesn’t. To give a rigid designation, such as a name, any descriptive value (of accidental properties) — at all — would be to misunderstand not only Kripke’s thesis but the distinction between various possible worlds. Nevertheless, it may be helpful for those with this fixation to go through Kripke’s demonstration substituting a more familiar name such as ‘Bob’ in place of ‘one metre’. That is, ‘Bob is the length of stick S at t₀’. Try it. Similarly, to understand Kripke’s example, we must construe the ‘length of stick S at t₀’ to be a description and not a name.

Why Kripke’s argument was documented as it was (as an excerpt of an oral presentation, not as an original written document) is baffling. I wish to grant right away that Kripke’s argument could be more precise. There are parts that are sloppily worded that can mislead. For instance, Kripke should not have let it be said that, "there is a certain length which he wants to mark out. He marks it out by an accidental property, namely that there is a stick of that length." [Kripke: 152] For this could be construed such that it implies that the person in question had a certain length in mind, and he looked for something that exemplifies that length. This route, however, ultimately fails. Regardless however, no matter how many construals there are on which Kripke’s argument fails, as long as there is one (one!) construal on which Kripke’s argument succeeds (in showing us one contingent truth knowable a priori), then Kripke succeeds.

In Kripke on the A Priori and the Necessary, Albert Casullo objects to Kripke’s conclusion that we can orchestrate contingent truths knowable a priori, claiming that Kripke has failed to make his case. According to Casullo, to evaluate Kripke’s argument we must distinguish between two possible interpretations of the statement ‘One metre is the length of stick S at t₀’, corresponding to whether the definite description ‘the length of stick S at t₀’ is used attributively or referentially. [Casullo: 165] Casullo says "If one uses the definite description attributively in introducing ‘one metre’ by means of [the statement ‘One metre is the length of stick S at t₀’], then one is using ‘one metre’ as the name of the length of S at t₀ whatever it may be. The term is not being introduced as the name of a particular length which the speaker has singled out but as the name of whatever length happens to satisfy the definite description." [Casullo: 166] Ok so far, but then the wheels fly off. For Casullo then says, "This method of introducing the term results in what Kripke calls an ‘abbreviative definition’, for the speaker is using the term ‘one metre’ as an abbreviation for the phrase ‘the length of S at t₀’." [Casullo: 166]

Why Casullo claims this is puzzling, for Kripke says explicitly that "if he used stick S to fix the reference of the term ‘one metre’, then as a result of this kind of ‘definition’ (which is not an abbreviative or synonymous definition) …" [Kripke: 153; emphasis mine] Indeed, to enhance the mystery, Casullo includes this very quote in his own paper! [Casullo: 165] Casullo then continues, inevitably down the wrong path, "As a result of this definition, the proposition expressed by the sentence ‘The length of S at t₀ is one metre’ is a necessary one, true solely in virtue of the terms used in expressing it." [Casullo: 166]

You will note that Casullo has yet to distinguish (or even mention Kripke’s distinction) between rigid and non-rigid designation, even though it is critical to Kripke’s argument. A person won’t understand Kripke’s claim unless they understand and adhere to that distinction. Casullo’s confusion is exhibited later, as well, when he claims that the sentence ‘This length is one metre’ expresses what Kripke has in mind [Casullo: 167], where both ‘this’ and ‘one metre’ are used as rigid designators. This identity, because it is an identity between two rigid designators, if true, is a necessary truth (according to Kripke’s other thesis in Naming and Necessity). Hence, clearly, this is not what Kripke had in mind. For Kripke says, again, explicitly, that "The reason [that this is not a necessary truth is] that one designator (‘one metre’) is rigid and the other designator (‘the length of S at t₀’) is not." [Kripke: 153] Among the problems that Casullo exhibits in his inability to grasp Kripke’s thesis is an aversion to Kripke’s theory of reference, in which new terminology and new distinctions are introduced — both critical to Kripke’s claim. Yet, though they are critical to Kripke’s argument (and required to understand Kripke’s argument), Casullo fails to even mention them until the later stages of his paper (page 169), almost as an afterthought, and only after being told to do so. [Casullo: 169] Casullo’s misconstrual of Kripke’s argument is such that, even if Casullo shows, on his construal, that Kripke ought not have meant this or that, this does not alter that fact that on other, more plausible (more understanding) construals, Kripke’s argument may genuinely succeed. And keying only on the route that fails — even with some scholastic arm waving — will not change nor dampen the success of other, better, construals.

According to Kripke, the metaphysical status of the statement ‘One metre is the length of stick S at t₀’ (hereafter designated ‘B’) will be a contingent fact about the world. There are some people, however, who think that B is a necessary truth. Most of these people, at least the one’s I wish to satisfy, think that B is a necessary truth given the reference fixing. Casullo, for instance, says, "Hence, if the term ‘one metre’ is introduced by means of a sentence which uses the definite description attributively, both propositions — that expressed by the sentence ‘The length of S at t₀ is one metre’ and that expressed by the sentence ‘S is one metre long at t₀’ — are necessary and a priori." [Casullo: 166] These people, including Casullo, though non-fatalists, are, nonetheless, confused. B may follow of necessity from the reference fixing, but this does not make it a necessary truth. That is, the scope of the necessity is not limited to the statement, B, but instead, to the conditional, the antecedent of which consists of the reference fixing (among other things), and consequence of which is the statement, B. Let us grant that it is the case that, necessarily, if we fix the reference in the way Kripke specifies, then (it is true that) ‘One metre is the length of stick S at t₀’. But it is not the case, nor does it follow, that if we fix the reference in the way Kripke specifies, then, necessarily, ‘One metre is the length of stick S at t₀’. That is, formally (where ‘"’ designates necessity, ‘A’ designates the antecedent, and ‘B’ continues to designate the statement ‘One metre is the length of stick S at t₀’), we can grant that "(A®B), but we do not thereby have (A®"B). Yet, the claim of apriority concerns knowing B (i.e., ‘One metre is the length of stick S at t₀’), not the conditional. The only way for B to be necessary (in addition to being knowable a priori) is if the fatalist position is correct.

Kripke, however, will not even grant that the conditional is necessary. [Kripke: 152, 153] And he doesn’t have to. I granted it merely to demonstrate the modal scope confusion. But doing so belied an additional confusion. We must not confuse necessary truths (truths which could not possibly have been false) with immutable contingent truths (truths that could have been false, but that have been brought about true, and are thus fixed true for all time). The truth-value of a statement cannot change (this goes for all statements, whether or not they are necessary or contingent). Once brought about (whether true or false), the truth-value is fixed for all time. For instance, if you read my last paragraph, then the fact that you did cannot be altered. That truth (though surely contingent) will be true for all time. This is not to say, of course, that you were effectively caused or logically forced or determined to have read that paragraph. If a statement about your action specifies an event at time t, then it is your action at t that brings about the statement’s truth-value, not vice versa. The point is that, once bought about, the truth-value of a statement cannot be changed. However, this is not the same as being necessary. Necessary truths are those that could not be false, no matter what — there is no such possibility, even prior to that truth-value being brought about by the circumstances expressed by the statement.

Given Amanda’s actions, the truth-value of the statement ‘One metre is the length of stick S at time t₀’ is true. Amanda’s action brought it about. And this truth-value cannot be altered, it is fixed true for all time. This holds even if, later, at time t_n, say, one metre is then taken to be the length of stick R. This fact does not alter the fact that, at time t₀, one metre was the length of stick S. However, the fact that this truth cannot be altered does not make it a necessary truth. Surely, there was a possibility that Amanda never fixed the reference, or named a different length, or used a different name, etc. — all of which would be exemplified in other (non-actual) possible worlds. Even if, thereafter, it is true that one metre is the length of stick S at t₀ in the actual world, this does not entail that one metre is the length of stick S at t₀ in some non-actual world W. In short, we must not confuse necessity with omnitemporality.

So far I have shown that, according to Kripke’s notions of reference fixing and rigid/non-rigid designation, we must distinguish between identities and identifications. Identifications are contingent truths. But in orchestrating an identification, we can know a priori that we made such an identification. Thus, we have a formula for orchestrating contingent truths knowable a priori.

It is important to understand that Kripke’s metre bar example is but one token example of a more general type, and it is the generic type — the formula — that is important. The actual history as to how the metric standard actually came about would only be a fatal objection if Kripke’s example was not a token of a more generic type. However, we can extrapolate from Kripke’s example a number of further examples of the same type — some even more plausible and convincing than Kripke’s. To close, I wish to leave you with a very short example that is different than Kripke’s, but tokened from Kripke’s generic type.

For instance, blindfolded and facing a group of people, Amanda says "I can’t see you, but whomever I am pointing to [i.e., ostensibly designating] now [i.e., at time t₀] I hereby name ‘one metre’." Here, Amanda does not know which person it is that she pointed to at t₀. But she does know that that person (whomever they may be) she named ‘one metre’. Thus, she knows, without empirical investigation, the truth-value of the statement ‘One metre is the person that Amanda pointed to at t₀’. And since this statement is not a necessary truth — she could have pointed to others, or declined to play along, etc. — Amanda knows a contingent truth a priori.