Ordinary, everyday language is full of vagueness, ambiguity and unclarity. Science, which strives for precision, objectivity and exactness needs to utilize this compromising language in order to make headway at all, since, ultimately, ordinary language is all we have to go on. “To some degree, nevertheless, the scientist can enhance objectivity and diminish the interference of language, by his very choice of language (my emphasis)” (Quine 1957, 7). This is the goal canonical notation attempts to achieve; to reduce the ‘noise’ of ordinary language, with all its imperfections, unclarities and vague elements.34 A couple of qualifying remarks are in order: the purpose of devising a canonical notation in order to eliminate obscurity from scientific discourse has as its goal not a “practical language reform” (Quine 1957, 8), i.e. it is not supposed to replace ordinary language. Moreover, as was already hinted at in Section 3.2.2, canonical notation is neither supposed to constitute an analysis of ordinary discourse nor is it even demanded that it be synonymous with it.35 The only connection there need be between a piece of ordinary discourse and its canonical regimentation is that “[w]e fix on the particular functions of the unclear expression that make it worth troubling about, and then devise a substitute, clear and couched in terms to our liking, that fills these functions” (Quine 1960b; Hylton 2007, 258/259).
Canonical notation is supposed to present an idealized form of scientific language, a “philosoph- ical schematism” (Quine 1957, 9), such that, in principle, all science could be done in this way, but does not have to be: “Where the objective of a canonical notation is economy and clarity of elements, we need only show how the notation could be made to do the work of all the idioms to which we claim it to be adequate; we do not have to use it” (Quine 1960b, 161). In fact, intermediate notations of varying clarity and austerity might be easier and less laborious to use. Thus reassured “that we are in no way compromising our freedom, we can be uncompromising in our reductions” (Quine 1960b, 161).
Nevertheless, it should not be forgotten that canonical notation is deeply rooted in ordinary language, for “[s]cientific language is in any event a splinter of ordinary language, not a substitute” (Quine 1957, 9). In fact, we cannot do without ordinary language in the construction of canonical constructions, since those constructions are themselves explained in ordinary language. All of the vocabulary and grammatical constructions of a canonical notation will be ordinary (Quine 1960b, 159) and explaining a sentence in canonical notation amounts to saying how it can be expanded in ordinary or semi-ordinary language (Quine 1960b, 159): “Hence to paraphrase a sentence of ordinary language into logical symbols is virtually to paraphrase it into a special part still of ordinary or semi-ordinary language” (Quine 1960b, 159).36
However, canonical notation is merely to serve as a means of economic theory construction, not as a linguistic analysis or systematization of ordinary language37. Ordinary language is the starting, not, however, the end-point (Hylton 2007, 239), it is almost unavoidable that the canonical regimentation of a piece of ordinary discourse will at some point diverge from it: “We do not ask whether our reformed idiom constitutes a genuine semantical analysis, somehow, of the old idiom; we simply find ourselves ceasing to depend on the old idiom in our technical work” (Quine 1957, 11) and “[i]f the form of paraphrase happens incidentally to produce sense where the original suffered a truth-value gap and so was wanted for no purpose, we may just let the added cases turn out as they will” (Quine 1960b, 182). The purpose of a canonical notation is clarification and simplification of
34
Cf. (Quine 1985, 170): “I mean predicate logic not as the initial or inevitable pattern of human thought [...], but as the adapted form, for better or for worse, of scientific language.”
35
Cf. (Hylton 2007, 242).
36Cf. also (Quine 1960b, 205): “our own language and [...] the canonical part of it”. 37
theory, if that requires departures from ordinary language, accidental or substantial, so be it; as long as these “sweeping” artificialities and departures simplify theory they are to be accepted on grounds of economy.38
The extensive procedure of how to devise a canonical notation, abstracting from and simplifying ordinary discourse wherever possible, even breaking with ordinary grammar and overstretching the naturalness of certain ordinary language expressions can be found in Chapters 3 - 5 of Word and Object. We will here focus on some of the essential steps and provide but a sketch of the original procedure to elucidate how the result became to be the language of FOL.39 Before going into the details a general remark is in order; the process of devising a canonical notation cuts in two ways: an upward-movement and a downward-movement, so to say. Quine starts, on the one hand, with ordinary language vocabulary and constructions and abstracts away from these, eliminating vaguenesses and bringing stylistically different constructions into a common canonical form. This process is upwards, as it starts with ordinary language and abstracts away from it. Sometimes, however, these canonical constructions happen to encompass more than just the cases they were abstracted from; they might turn out to be able to do the work of other natural language constructions which were not in the first place thought to have much in common with others, subsumed under the same canonical form. This is a welcome side effect and the downward direction of canonical notation: canonical constructions are reapplied to ordinary language constructions, unifying them with other, dissimilar constructions in a canonical form.40
Sentences of science are supposed to be true eternally, independent of occasion of utterance (Quine 1960b, 227), so in a first stepindicator words, such as ‘I’, ‘you’, ‘this’, ‘that’, ‘here’, ‘there’, ‘now’, ‘then’, etc. are banned from a canonical notation, thereby rendering truth invariant with respect to speaker and occasion. Their function need not be lost, but can be emulated through mention of concrete persons, places and times in canonical notation (Quine 1957, 8/9). Further- more, tense is dropped from words of ordinary language once they are incorporated into a canonical regimentation, we might keep their grammatical forms, but treat all terms as temporally neutral.41 Again, nothing is lost by this move – time and tense can be emulated by adopting a four-dimensional view of time through explicit mention of space-time coordinates or ordering relations (saying that something was before of after something else), – but much is gained in terms of simplicity and economy: logical theory applying to sentences of canonical notation now need not take into account different temporal dimensions of sentences, thereby greatly simplifying its rules of inference.42 Plu- rals in ordinary language also introduce unnecessary complications by modifying terms of language through plural endings and can easily be dispensed with. Instead of ‘I hear lions’ we now say ‘I hear a lion other than a lion that I hear’ (Quine 1960b, 118), a construction, however unnatural, that bears the great advantage of eliminating plural endings in favor of singular terms, identity and
38Cf. (Quine 1960b, 158): “Simplification of theory is a central motive of the sweeping artificialities of notation in
modern logic.”
39
For a more extensive treatment and a minute justification and elaboration of the process and steps leading to this result the reader is referred to the relevant chapters of (Quine 1960b).
40
An example of this downward movement would be the streamlining of all expressions into the ‘such-that’ idiom, which hopelessly violates grammatical and stylistic rules of many ordinary language expressions, but greatly simplifies canonical forms (Quine 1960b, Chapter 5).
41Cf. (Quine 1960b, 170): “ordinary language shows a tiresome bias in its treatment of time. [...] This bias is of
itself an inelegance, or break of theoretical simplicity.”
42
We can now say that frompandp→qwe can inferq rather than having to formulate this rule in a way that it takes the temporal relations betweenpandqinto account: “The letter ‘p’ standing for any sentence, turns up twice in each of these rules; and clearly the rules are unsound if the sentence which we put for ‘p’ is capable of being true in one of its occurrences and false in the other. But to formulate logical laws in such a way as not to depend thus upon the assumption of fixed truth and falsity would be decidedly awkward and complicated, and wholly unrewarding” (Quine 1957, 9).
Chapter 4. Canonical Notation 55
cardinality statements43, all of which will become standard components of a canonical notation, eliminating the need for extra devices to emulate plurals (minimality).
We saw in Section 3.2.1 (sentence (*)) that disambiguation is facilitated by holding singular terms (definite and indefinite) to subject position and reformulating the sentence containing them by means of the ‘such-that’ paraphrase. Given the clarifying and harmless nature of such linguistic move, it should be adopted as standard in a canonical notation in order to eliminate scope ambiguities. Moreover, the introduction of indexed pronouns, or even better variables, allowed for more reliable and accurate reference tracking, greatly simplifying and sharpening grammatical constructions and should therefore also be implemented in a canonical notation. All of these modifications have, so far, been upward-constructions: we began with natural language phenomena and found means of streamlining, standardizing and simplifying their grammar and construction. The last move, the introduction of variables, has, however, a downward effect: it broadens our notion of sentence as we can now distinguish between two types of sentences, open and closed sentences, i.e. those with free pronouns and those whose variables are all bound, in such a way that they all refer back to an object denoted by an indefinite or definite term at the beginning of the sentence (Quine 1957, 11). While closed sentences behave as before (they have a definite truth-value), open sentences share much more commonalities with predicates in that they have an extension, a set of objects of which they are true, rather than a determinate truth-value. This unintended side effect of introducing variables for pronouns and singular terms in canonical notation is not at all undesirable, in fact it proves the predicational completeness that we located in the relative clause of natural language: everything that can be said about an object can be said by predicating a predicate (= an open sentence) of it.
At this point we see that a sentence in canonical notation will have the form of a predication: there is an object referred to by singular, definite or indefinite terms of which something is predi- cated via the remainder of the (open) sentence. This general structure affords another downward movement in that no limitations are imposed on the internal complexity that unanalyzable pred- icates featuring in the open sentences that are predicated of an object possess: “Where canonical notation is cut off, leaving unanalyzed components, will usually vary with one’s purpose; what re- mains unanalyzed has the form of a general term in predicative position” (Quine 1960b, 174). The level of fine-grainedness of analysis and the primitive predicates to be used in the construction of canonical sentences do not need to be natural kind terms, metaphysical primitives or fundamental relations, they completely depend on what the purpose of the paraphrase is. What canonical no- tation institutes is not which predicates are to denote the basic properties and relations or reality, but that “the only canonical position of a general term is predicative position, whatever the terms uncanonical substructure” (Quine 1960b, 175). It draws no distinction which terms to treat as complex and which as simple; all it does is to unify the position in which a term can occur within a canonical construction.44
A final (and crucial) further simplification involves the reduction of all terms that stand in sub- ject position of the pre-canonical constructions outlined above touniversal andexistential quantifi- cations ∀x and ∃x.45 The reduction proceeds through different stages: names/singular terms are
43
Cf. (Quine 1960b, 118) and (Quine 1982, 231ff).
44Cf. (Quine 1960b, 160): “There are regimented notations for constructions and for certain of the component
terms, but no inventory of allowable terms, nor even a distinction between terms to regard as simple and terms whose structure is to be exhibited in canonical constructions. Embedded in canonical notation in the role of logically simple components there may be terms of ordinary language without limit of verbal complexity. A maxim of shallow analysisprevails: expose no more logical structure than seems useful for the deduction or other inquiry at hand.”
45Strictly speaking only one of them is required, as each can be defined in terms of the other, viz. ∃ ≡ ¬∀¬and
∀ ≡ ¬∃¬. However, such saving would only be notational and not conceptual in nature and for convenience we will therefore retain both.
reduced to definite descriptions (see next section), definite descriptions are reduced in the famil- iar Russellian fashion and indefinite (singular) terms are captured by means of “every” and “some”, which, in turn, can be expressed through “everything” and “something” respectively, expressions that are symbolized by the universal and existential quantifiers (cf. Section 3.2.1). We will turn toward the issue of paraphrasing names in the next section and neglect it here. Definite terms of the form ‘the F’, where ‘F’ is any description, can be disposed with in favor of existential quantifications in sentences containing them, thereby lifting the denotational burden from these expressions and the existential import from sentences containing them, as is well known since Russell (Russell 2005). On Russell’s analysis, a sentence such as, e.g., “The present king of France is bald” becomes “Someone is such that he1 is king of France and he1 is bald and for anyone such that he2 is king of France he2 is identical to him1”. This way definite descriptions give way to existential quantification, predication and uniqueness claims and we can contextually define definite descriptions ι(x)(ψ(x))(thex, such thatψ(x), i.e. ‘theψ’) in sentences in which they occur (ϕ) the following way:
ϕ( ι(x)(ψ(x)))
becomes
∃x(ψ(x)∧ ∀y(ψ(y)→y=x)∧ϕ(x))46
without the need to assume a separate operator for definite terms.
Moreover, the such that idiom already allows us to reduce all indefinite singular terms to lo- cutions of the form ‘every F’ and ‘some F’ (possibly plus negation) (Quine 1960b, 162) with ‘F’ standing for some general term, however complex. These, in turn, give way to ‘everything’ and ‘something’ through the following transformations:47
(a) Every F isφ.
(b) Some F is φ.
become
(a’) Every F is an object x, such that φ(x).
(b’) Some F is an objectx, such that φ(x).
which then become
(a”) Everything is an objectx, such that (ifF(x) thenφ(x)).
(b”) Something is an objectx, such that (F(x)and φ(x)).
Expressions of the form ‘Everything/Something (is an object x, such that)’ are, however, precisely what is symbolized by the universal and existential quantifier∀x/∃x(see Section 3.2.1) and we have therefore disposed with the entire class of indefinite (and definite) terms in favor of the two quan- tificational expressions of universal and existential quantifications (of which we, strictly speaking, only require one).48
46Cf. (Forster 2003, 17). 47
Cf. (Quine 1960b, 162).
48For lack of space and to avoid unnecessary complications we here ignore class-abstraction and attribute-
abstraction operators, which, if needed, might require the introduction of additional primitive operators or, depending on the situation, allow reduction within the context of our regimentation (Quine 1960b, pp. 164).
Chapter 4. Canonical Notation 57
The proposals for paraphrase into canonical notation outlined above have the purpose of simpli- fying and clarifying theory – resolving issues and eliminating misunderstandings that might pertain to the formulation of such theory in a non-canonical idiom. It has the capability of dissolving problems by showing that they are purely a matter of the way we speak about things:
“Philosophy is in large part concerned [...] with what science could get along with, could be reconstructed by means of, as distinct from what science has historically made use of. If certain problems of ontology, say, or modality, or causality, or contrary-to-fact conditionals, which arise in ordinary language, turn out not to arise in science as recon- stituted with the help of formal logic, then those problems have in an important sense been solved: they have been shown not to be implicated in any necessary foundation of science.” (Quine 1953a, 151, my emphasis)49
This does not mean that canonical notation by itself offers solutions to long-standing problems, it merely helps us clarify and simplify theory, thereby showing us that some disputes (such as the one about Pegasus’ existence) might have been merely verbal. The task of regimenting a theory into canonical notation and thereby resolving such problems, however, still remains and is not unproblematic: “By developing our logical theory strictly for sentences in a convenient canonical form we achieve the best division of labor: on the one hand there is theoretical deduction, and on the other there is the work of paraphrasing ordinary language into the theory” (Quine 1960b, 159).
Singular Terms Eliminated
One inessential item of canonical notation whose paraphrase enables an attractive solution to the problem of non-existent entities is the proper name. The problems pertaining to non-referring names are well-known. Consider, for example, the sentence “Pegasus does not exist”. Here one first appears to be referring to an object (Pegasus) to then deny its existence – a surely contradictory circumstance. Quine is able to circumvent these issues by treating names as singular descriptions. Every name a is associated with a (unique) predicatePa (e.g. ‘pegasizes’) and a sentence such as “Pegasus flies” is then paraphrased as F( ι(x)(Pa(x))), i.e. as ∃x(Pax∧ ∀y(Pay → y = x)∧F x), treating Pegasus as a definite description rather than a standard a singular term which would have received the standard paraphrase ofF a. As Kaplan remarked, names are first Quinized into definite descriptions and definite descriptions are then Russelled away.50
A similar procedure can be applied to functions, treating them as relations of special kind, namely those whose first-coordinate only occurs once. All singular terms then are paraphrasable in this fashion and therefore unnecessary ‘notational ballast’ in our most austere scheme of canonical notation. Everything we can do with them we can achieve without them, their function can be em- ulated by the devices already present and they are therefore dispensable. Moreover, this treatment of names or singular terms in general has two nice side-effects. It closes truth-value gaps in that it assigns a sentence such as “Pegasus flies” the truth-value false, rather than leaving it meaning- or truth-valueless due to the failure of reference connected with the singular term ‘Pegasus’ (Quine 1964a), thereby solving the issues and puzzles associated with negative existential statements.
Furthermore, it provides us with a robust criterion enabling us to tell when a name refers. A name a refers precisely then when it is able to feature in an affirmed existentially quantified identity∃x(x=a). This latter way of confirming that a name refers also lets us see that predicates uniquely applying to the object we wish to refer to are readily at hand, simply treat ‘=a’ (=Pa)
49
As quoted in (Hylton 2007, 243/244).
50
as general term. The abstraction thus becomes a little less artificial. This method of paraphrase is