One reason H. P. Grice dislikes Peirce's stroke is that Witters doesn't!

Wittgenstein’s Tractatus and the Sheffer Stroke

In his Tractatus Logico-Philosophicus (1922, 5.1311, 6.001) Wittgenstein extolled the significance of the Sheffer functions, hinting that discovery of the functions vindicates some of the seminal claims he was raising in this famous text. It is not clear that Wittgenstein knew that there are two binary functions with the same property of being functionally complete. Wittgenstein’s connective symbol may appear, at first blush, to be the same symbol as NOR, which is the connective used by Sheffer himself in his alternative axiomatization of Huntington’s system. Wittgenstein’s connective has been mistaken as such even by Bertrand Russell, but this is a mistake. Wittgenstein uses a rather eccentric function, known in the literature as the N-operator, which has attracted attention and even led to disputes. Although this is not the place to enter into details, a few words are in order about Wittgenstein’s N-operator which is not the sentential NOR operator although it is inspired by it. A technical study of the subject is given by Soames (1983; see also Geach, 1981.)

Wittgenstein’s N-operator is defined over an open-ended set of propositional variables. Because the language that is needed is that of first-order or predicate logic, a propositional variable atom is a predicate symbol, of any arity n, accompanied by n individual constants all of which have as specified referents members of the universe of discourse (or domain.) It is an open problem for Wittgenstein’s language (whose grammar specification is rudimentary) that the domain set may or may not have a denumerably infinite number of subjects. Assuming a finitary domain for this brief excursion, and bear in mind that whatever fixes are available to address problems with Wittgenstein’s operator, are not efficient in the case of an infinite domain. Consider a grammar that comprises symbol letters for 22, individual constants, predicate (non-logical) constants, and the operator symbol. (These are not Wittgenstein’s symbols. Instead he legislates:

  

where the circumflex hints at the recursive mode of defining what expressions are grammatically correct. He uses “ξ” instead of “z” for molecular, not necessarily atomic, well-formed formulas.)

  

Then, application of the N-operator is, by definition, to negate all atomic propositions

  

in the set. This means that the N-operator can be defined through the following logical equivalences (insofar as the additional symbols are allowed in the metalanguage). The symbols for the existential and universal quantifier are “” and “”. These are missing from Wittgenstein’s language which is more parsimonious; but, as will be seen, Wittgenstein’s language, constructed on the N-operator, is expressively incomplete! Take, as example, the case of a unary predicate constant:

  

One could then proceed to iterated applications of the N-operator, which will now give a clue as to how Wittgenstein’s operator is expressively incomplete.

  

The symbolic language cannot sort out more than one individual variable within the scope of another variable. It can express a formula like the following:

  

But the language lacks the resources to express formulas like the following, for which differentiation of individual variables within scopes is required:

  

  

Interestingly, the language also lacks resources for expressing  . As Soames shows (1985), the defect can be remedied by adopting some additional symbolic convention that permits differentiation of individual variables within scopes. Thus, ironically, Wittgenstein’s constructed analogue to a Sheffer function, his N-operator, lacks expressive completeness. The set  dispenses with the need for other connective symbols, and also for quantifier symbols (of which Wittgenstein thinks are defined through inclusive disjunction or conjunction, again disregarding the prospect of an infinite domain); yet, the language cannot express all constructible formulas of first-order logic. It was Moses Schöfinkel, the originator of combinatorial logic, (Bimbo, 2010) who constructed a functionally complete language for first-order logic using one Sheffer function.

To conclude, consider the discussion of functional completeness, as touted by Wittgenstein in the Tractatus, putting aside the vicissitudes of his symbolic language. Although Wittgenstein claimed that the main subject of his Tractatus is ethical, the work examines a plethora of philosophical and logical subjects. An oft-discussed overriding objective of the work is to demarcate the limits of language; what cannot be expressed by language can be “shown,” as Wittgenstein famously claimed. The present subject fits within the Tractatus’ discussion of the nature of propositional logic and its relationship to the task of elucidation of meaning. (See Wittgenstein.)

Bursting into the scene on the heels of advances in modern logic made by Frege and Russell, the Tractatus is remarkable for its contributions to the philosophical discussion of the new logic as an instrument for clarification of logical meaning. Wittgenstein later abandoned the work’s objective of constructing an ideal formal language that would be “isomorphic” to the world of empirically ascertainable facts; he also moved away from a version of the Correspondence Theory of Truth that seems to be underpinning the Tractatus.

In the Tractatus, Wittgenstein explains that the logic of our theories about the world is not itself to be sought in the world. Let us assume that “A” symbolizes the proposition expressed by the sentence “snow is white” and “B” symbolizes the proposition “snow is a kind of precipitation.” Let us also assume for our present purposes that the truth or falsehood of propositions A and B are to be established by referring to empirical facts. It so happens in this example that both propositions, expressed by the two English sentences, are true in our actual, empirically accessible, world. Now form the compound proposition “A and B.” This new proposition must be true because both its component propositions are true. This is evident because the meaning of “and.” But how is this known? The empirical world itself does not come to our assistance. We know this regardless of empirical experience: what we know is that any compound proposition of the logical form “p and q” has to be true if, and only if, both of its components, the individual or atomic propositions p and q, are true. Thus, given p and q, the conclusion “p and q” follows validly: it is logically impossible to have a case in which the given premises are all true but the conclusion is false. Nevertheless, the logical meaning of any conjunctive proposition of the logical form “p and q” is identical with its truth conditions which comprises the determinate relations between truth value assignments to the components (whether p and q are true or false) the functionally determined truth value of the whole conjunction. Thus, the empirical fact that the conjunctive sentence is true in our actual world is irrelevant from the standpoint of the logical meaning (the truth conditions) of the logical form exemplified by the sentence “snow is white and snow is a form of precipitation.” The valuation dependency

  

is one of four logically possible combinations which comprise the truth conditions of the conjunctive logical form:

  

The actual world is not logically privileged, and Wittgenstein’s conceit that an isomorphic mapping can be accomplished, which would produce an ideal language of comprehensive applicability, was bound to be frustrated. Disregarding this rather metaphysical aspect, which Wittgenstein later also disregarded, the Tractatus contains an astute understanding and analysis of the formal logical instrument that has arisen out of modern mathematical developments. Wittgenstein’s contribution to the discussion of functional completeness fit under this aspect of the work.

Wittgenstein makes the point that “internal,” or “structural,” features of propositional forms account for truth preservation from the joint premises to the conclusion of a valid argument form. It is structural features that account, for instance, for the equivalence of logical meaning between any two propositions. This means that the propositions have forms that receive the same truth values for the same valuations (truth value assignments to their components.) Cases or valuations (also called interpretations and models) are determined by assigning truth values, true and false, to all the components of a propositional form. Wittgenstein uses the term “truth grounds” and “(logically) possible worlds” when referring to truth value assignments or valuations. Wittgenstein says that “these relations are internal and they exist as soon as, and by the very fact, that the propositions exist.” (1922, 5.13) The next thesis in Wittgenstein’s text (5.1311) is the one in which he uses his N-operator. The point made there is now presented roughly: having briefly examined the complications that arise out of Wittgenstein’s definition of an N-operator, one adjusts, instead, to a propositional language, pretending that Wittgenstein actually used the NOR function to make his case. Nothing is lost in this way because the point is to illustrate Wittgenstein’s remarks on the significance of functionally complete operators rather than to pursue further any details attaching to the N-operator itself.

Consider a valid argument form:

  

The usual name of this valid argument form is Disjunctive Syllogism. This is not a string of propositional forms; it is a schema, and so is something like a recipe for how to proceed correctly when drawing inferences. Wittgenstein makes the point that conventions of symbolism may create the wrong impression that there is no internal, structural connection running through all propositional forms; that there is something newly productive introduced by the multiple (connective) symbols. This, however, would be wrong. The accidental fact that many different symbols are used is what is misleading. Moreover, Wittgenstein has philosophical objections to working from the semantic side of constructing logical systems, and this has consequences for the subject under discussion. Wittgenstein considers semantic attempts to be nonsensical: for instance, to specify the referent of conjunction, in order to obtain a working semantics, commits one to the nonsense of speaking about extra-empirical items and, indeed, about things that cannot be talked about. This way of thinking shows certain underlying philosophical assumptions, which lie beyond this article’s scope, but the problem that arises is this: The construction of a logical system is to be understood as a matter of specifying formal-grammatical rules for concatenating and transforming the available symbolic resources of the system. Because of this, the failure of the grammatical or syntactical setup to show perspicuously what happens in logical operations is serious. Hence, it is imperative to show solely by manipulating the symbolic resources that there is an internal structural connection that relates all possible transformations. This is accomplished by using only one functionally complete operator symbol. This is the reason Wittgenstein extols the “discovery”. Even if one opts to multiply connective symbols, because of the greater simplicity and even intuitive appeal gained in that way, it is still crucial to be able to show that only one connective symbol suffices. Indeed, as is known from the above study of functional completeness, one could have opted for eliminating all but one connective symbol, one of the Sheffer functions. Consider further how the claim is to be made that single-connective symbolism reveals something deeper about logic itself.

Logical properties are structural features of the forms: thus, one can have tautologous, contradictory, and indeterminate (also called contingent and indefinite) propositional forms. All tautologies would have to have the same referent which, in the Fregean analysis, is the truth value true. If semantic referents are rejected, however, that leaves the grammatical means for showing the collapse of all tautologies, namely that they all have the logical meaning. The same is the case with all contradictory logical forms; they check as false for all logically possible assignments of values to their components. The remaining structural type, the contingent propositional form, is basically not logic’s business! This is indicated by the convention of assigning both truth values to a single propositional variable to generate two cases: these are two logically possible worlds if one is to semantically model the setup. The proposition can logically be true in one case and false in another; as a proposition it must be one or the other and it is not logically possible for it to be both true and false. Notice then that the two logical possibilities (p-T and p-F) have the same status. It does not matter if one of those, for an interpretation of the propositional symbol, happens to be the actual world. Logic, not depending on the workings of the empirical world, is attuned to characteristics that are invariable across all possible cases: this means, tautologies, which are true in all logically possible cases, and contradictions, which are false in all logically possible cases. The validity of the inferential schema above guarantees, for two-valued logic, that the following is a propositional tautology:

  

Once again, the proliferation of symbols obscures the facts about the internal structural simplicity of logic. All compound propositional forms are internally connected because they result from elementary propositional forms by means of connectives. The logic is determined by how the logical connectives are defined. Starting with elementary (also called individual or atomic) propositions, one always proceeds by combining them by means of connectives: the compounds generated are in every case dependent for their meanings (truth and falsehood) on the meanings (truth and falsehood) of their components. If one were to proceed in the opposite direction, from compounds toward the elementary propositions, there would be a decomposition of the compound propositions; the process would terminate with the elementary propositions. This is possible because all the connectives are truth-functional connectives. Thus, if, for instance, “p and q” is given as true, one can dissolve this into “p is true” and “q is true” given the definition of “and.” Once again, one sees that propositional forms are related with each other and, ultimately, they are related to two basic propositions, the true and the false, out of which any complex can be generated by using truth-functional connectives. This also shows that nothing in the logic of propositions can ever be arbitrary.

The symbolism that uses multiple connective symbols obscures this. A stronger point can be made: Something is wrong with a notational idiom, a symbolism, that fails to capture the identity of logical meanings (logical equivalence). For instance, consider the following two logically equivalent expressions or formulas, which are well-formed, it is assumed, in the idiom or notation (and represented here in the symbolically enriched metalanguage):

  

Even though the expressions are logically equivalent, the grammatically correct formulas representing them are not the same! This may be considered a radical notational or formal-grammatical defect. It gets even worse. There is a view that formalism is fundamentally a matter of systematic and specified manipulation of symbolic resources. Consequently, the defect faced in this case goes all the way to the roots of the most basic task of all: how to construct a faithful symbolic system relative to a given purpose. In that case, it would appear that the correct way to construct a formal system is exclusively through its minimally functionally complete sets of operators. If one has to switch to alternative idioms that have redundant operators in them (operators that can be defined by the other operators in the system), that would have to be justified by pleading such a reason as expediency or convenience.

The symbolic notation of a formal language idiom that uses only one connective symbol would remove this notational illusion, or, to make the stronger case, would remedy the deep formal-grammatical defect: then it could be perspicuously shown that all that is had is an unfolding of internal connections that run across propositional forms. Wittgenstein proceeds to write the above argument schema by using one single connective symbol which allows elimination of the symbols for disjunction and negation in order to make “the inner connection” obvious. (The contemporary symbol for the connective is the one used by Wittgenstein, which is NOR.)

  

To do this, replace “” by “” (thus eliminating the inclusive-disjunction symbol) and “” by “” (thus eliminating the negation symbol). The NOR symbol is used to effect both eliminations. As a result, we have the schema shown above, in which only one connective symbol is used. Of course, one could have used the NAND or Sheffer Stroke function to effect the same elimination, in which case the result would be:

  

Moreover, when multiple logical connectives are used in the construction of a formal system, an impression of arbitrariness may be created. Why, one may ask, is one set of logical connectives used instead of another set? The right answer is that nothing depends on which connectives are used because all the propositional formulas are internally related in strict, non-arbitrary fashion, and the construction ultimately depends on the basic building blocks and connectives. To illustrate this point, construct a formal system of the standard propositional logic by using as its set of connectives either

  

. Basically, it amounts to the same thing whichever one is used. This is not immediately obvious regarding the plurality of connectives seen above. But now consider how all of the connectives in these sets are definable in terms of the connective in . Thus,  can be replaced by ; and  can be replaced by ; and  can be replaced by . This fact makes clear that nothing depends on arbitrary choices about the connectives used. This discovery can be used as proof that there is a strict internal connection that runs through all the expressive resources.

Wittgenstein even points out (5.42) that having connectives in the formal system, which are interdefinable, means that they should not be properly regarded as “primitives.”
Now one can revisit the subject of the triviality of tautologies (and of logical contradictions), which is another subject on which Wittgenstein touches. There is one tautology, and a contradiction is the negation of the tautology (for the standard definition of negation.) Of course, negation itself can be expressed in terms of a Sheffer function. The ultimately perspicuous manifestation of the inner structural inter-connectedness of all logical propositions can be shown insofar as all valid tautologies can be derived from one single axiom that uses a single connective symbol. Rules of transformation and inference can be specified, to be applied to the axiom schema, to generate all valid tautologies. This is indeed possible, as French logician Jean Nicod (1917) demonstrated by constructing producing a one-postulate axiomatization of the standard propositional logic. Nicod’s postulate, written with metalinguistic symbols for writing a schema, is:

  

An alternative and equivalent formulation of the Nicod Postulate, which avoids having any sub-formulas of the postulate schema being tautologous, is the following. (Notably, in the original formulation, the sub-formula  is tautologous.)

  

The Nicod Postulate can be deployed as the single axiom in a formal system whose only rule of inference is given by the following rule schema: