Axiomatic Investigations of the Propositional Calculus of the "Principia Mathematica"1 Paul Bernays Translation of "Axiomatische Untersuchungen des Aussagen-Kalkuls der 'Principia Mathematica,' Mathematische Zeitschrift 25 (1926), 305–320, by Richard Zach. From Universal Logic: An Anthology. New York and Basel: pp. 43–58 (2012). §1. In Whitehead and Russell's "Principia Mathematica," the systematic development of the logical propositional calculus is carried out by starting with five propositional formulas as basic formulas ("primitive propositions"), from which every generally valid combination of propositions [Aussagenverknüpfung ], i.e., one which is correct for arbitrary values of the propositional variables occurring in it, can be obtained by substitutions and the application of a single formal rule of inference. This reduction of the theorems to the basic formulas is combined with a reduction of the logical connectives to the two operations: of negation ∼p (not p) and of disjunction p ∨ q (p or q; "or" in the sense of the latin "vel"), from which are formed by combination:2 1The content of this article is taken to a large extent from an unpublished Habilitationsschrift, which was submitted by the author to the Faculty of Mathematics and Natural Sciences at Göttingen in 1918. The questions on the possibility of replacing of formulas by inference rules investigated there has been left out of the present investigation. 2In order to avoid a proliferation of parentheses, dots shall be used as separating symbols, just as in the Princip. Math., where the rule is that more dots separate more strongly than fewer dots. This rule, however, applies only to the separation of symbols of the same kind. Whenever two different symbols from among ∼, ∨, ⊃ compete, in general the convention shall hold that for the separation of parts of the formula, ⊃ takes precedence over ∨ and ∼, as well as ∨ over 2 Paul Bernays the implication p⊃ q (if p, then q), as abbreviation for ∼p ∨ q and the conjunction3 p& q (p and q), as abbreviation for ∼(∼p ∨ ∼q). The five basic formulas are p ∨ p⊃ p,(Taut) q ⊃ p ∨ q,(Add) p ∨ q ⊃ q ∨ p,(Perm) p ∨ * q ∨ r ⊃ q ∨ * p ∨ r,(Assoc) q ⊃ r * ⊃ * p ∨ q ⊃ p ∨ r.(Sum) When replacing the implication by the defining expression, they read: ∼(p ∨ p) * ∨ p,(Taut) ∼q ∨ * p ∨ q,(Add) ∼(p ∨ q) * ∨ * q ∨ p,(Perm) ∼(p ∨ * q ∨ r) : ∨ : q ∨ * p ∨ r,(Assoc) ∼(∼q ∨ r) * ∨ :∼(p ∨ q) * ∨ * p ∨ r.(Sum) The rule of inference can be formulated as follows: From two formulas4 A, A⊃B (resp. ∼A ∨B) the formula B is to be obtained. The substitution rule says that any arbitrary formula can be substituted for a propositional variable. One may now easily convince oneself that each of the five basic formulas represents a generally valid combination of propositions when interpreted contentually. And that the application of the rules only yields formulas of this kind. At the same time, the system of basic formulas is complete in the sense that from it all generally valid combinations of propositions can be obtained with using the rules. This claim can even be strengthened as follows: ∼, as long as no other separation is indictated by parentheses. Accordingly, the formula p⊃∼q ∨ : p ∨ * q ∨ r * ⊃ * p⊃ p ∨ :∼q ∨ * q ∨ r has to be read in the same way as would be given without any further conventions by the following way of putting parentheses:( p⊃ ( (∼q) ∨ ( p ∨ (q ∨ r) ))) ⊃ ( p⊃ ( p ∨ ( (∼q) ∨ (q ∨ r) ))) . 3In the Princip. Math. the symbol for conjunction is simply a dot. 4We shall use uppercase Fraktur letters as symbols used to indicate formulas of an indefinite form. Axiomatic Investigations of the Propositional Calculus 3 If A is any formula formed from the symbols introduced, then either it is derivable from the five basic formulas according to the rules, or any arbitrary formula B becomes derivable if A is added to the basic formulas. The justification of this claim shall be indicated briefly.5 It results from considering the conjunctive normal form. A conjunctive normal form is a formula which is formed from one or more "simple disjunctions" combined using & (conjunction), i.e., such disjunctions in which every disjunct is either a variable or a variable to which a negation is applied. For instance, p& (p ∨ ∼q ∨ r) is a conjunctive normal form. Now the familiar theorem holds, that every formula can be brought into a conjunctive normal form; i.e., we have a procedure to find, for any given formula A, a normal form N such that A⊃N as well as N⊃ A is derivable from the basic formulas. (As an aside, the determination of N for A is not unique.) One now sees immediately: If A is derivable, so is N, and conversely; furthermore, if N is derivable, then so is each one of the simple disjunctions occuring in A (as conjuncts). Our claim is thus reduced to the corresponding claim for simple disjunctions. For those, however, it follows immediately; for a simple disjunction either contains two disjuncts one of which is the negation of the other (such as p and ∼p); then it is derivable; or the disjunction does not contain two such disjuncts; then from it one can obtain, by substitution, a disjunction every disjunct of which is either the variable p or the double negation of p, i.e., ∼(∼p). Fom such a formula, however, the formula consisting only of the variable p is derivable, and from this one can obtain any arbitrary formula by substitution. This consideration at the same time yields a simple procedure to determine whether a given formula A represents a generally valid combination of propositions, or-and this amounts to the same thing-is derivable from the basic formulas: find a conjunctive normal form N of A. If in each simple disjunction occurring in N two disjuncts occur one of which is the negation of the other, then the given formula A is generally valid, otherwise it is not. This decision procedure solves the main problem of the propositional calculus completely, and if one only wanted to characterize the generally valid logical combinations of propositions, one would be able to obtain this result more directly than by the method of Principia Mathematica. The insight, that the listed five basic formulas suffice for the derivation of all generally valid combinations of propositions (using the rules), is nevertheless 5A slightly different proof can be found in the treatise by E. L. Post, "Introduction to a general theory of elementary propositions" (American Journal of Math. 43 (1921)). 4 Paul Bernays significant in itself. In this light, the further question now arises whether the five basic formulas are mutually independent in the sense of formal derivability. This is indeed not the case, rather, the formula Assoc can be derived from the other four. Therefore, the system of basic formulas can be replaced by that of the four formulas Taut, Add, Perm, Sum (System 1). Of these four formulas, none are redundant, not even if one retains Assoc. These claims shall be proved below. Our investigation of dependencies is not exhausted by it, however. A number of further questions shall also be considered, to which this result naturally leads and which arise from two remarks. The first remark is that, in the event that Assoc is retained, the formula Perm becomes provable, if instead of Add : q ⊃ p ∨ q the formula Add∗ : p⊃ p ∨ q is taken as a basic formula. These two formulas are known to be mutually equivalent using Perm and Sum, i.e., one can, using Perm and Sum, pass from the formula Add to Add∗, and also back. It will thus become apparent that the following four formulas suffice as a system of basic formulas: Taut, Add∗, Assoc, Sum (System 2). This fact is not so surprising, since an exchange occurs in the formula Assoc. One will therefore ask if the formula Perm is still dependent if instead of Assoc the formula Assoc∗ : p ∨* q ∨ r ⊃ p ∨ q *∨ r, which expresses the associative character of disjunction more purely, is chosen as a basic formula. It can be shown that this is not the case; i.e., in system 2 the formula Assoc cannot be replaced by Assoc∗. It can, however, be replaced by Add and Assoc∗ together, so that the five formulas Taut, Add, Add∗, Assoc∗, Sum (System 3) suffice as a system of basic formulas. (That the formulas Assoc and Assoc∗ do not have the same inferential power when Perm is removed does of course not contradict the known fact that these two formulas are equivalent in the presence of Perm and Sum.) Axiomatic Investigations of the Propositional Calculus 5 The second remark is that in system 1, the formula Add can be replaced by the specialized formula Simp : q ⊃* p⊃ q resp. ∼q ∨* ∼p ∨ q, so that the four formulas Taut, Simp, Perm, Sum (System 4) suffice as basic formulas. By an analogous specialization as that of Add to Simp, the formula Comm : p⊃ * q ⊃ r :⊃ : q ⊃ * p⊃ r resp. ∼(∼p ∨ * ∼q ∨ r) ∨ :∼q ∨ * ∼p ∨ r results from Assoc, and the formula Syll : q ⊃ r * ⊃ : p⊃ q * ⊃ * p⊃ r resp. ∼(∼q ∨ r) ∨ :∼(∼p ∨ q) ∨ * ∼p ∨ r from Sum. Now the question arises whether Assoc might not also be replaced by Comm, or Sum by Syll . This question is decided in the negative. Neither Assoc in system 2, nor Assoc∗ in system 3 can be replaced by Comm, and Sum cannot be replaced by Syll in any of the systems 1, 2, 3, or 4. Furthermore, it is shown that the forumla Add cannot be replaced by Simp in system 3. Finally the relation of the formula Id : p⊃ p resp. ∼p ∨ p to the formulas of systems 1, 2, 3, 4 shall be considered. Id is distinguished by being the simplest generally valid combination of propositions. One would therefore hope to use it as a basic formula. However, it cannot replace any of the formulas in any of the systems 1, 2, 3, 4. The simplest derivation of Id is that from Taut ,Add ,Syll 6 Paul Bernays where one first obtains from Add by substitution the formula Id∗ : p⊃ p ∨ p resp. ∼p ∨ * p ∨ p. Without applying Sum, one can derive Id from Add and Assoc, even already from Simp and Comm; moreover from Taut ,Add∗,Assoc∗, and also from Add∗,Perm,Assoc∗. Carrying out these derivations, which are found easily, shall be left to the reader. In every one of these derivations either Add or Simp or Add∗ is used. We will show that without applying one of these formulas, the formula Id is no longer provable from the remaining formulas in systems 1, 2, 3, 4. Of course the formula Id is not a sufficient replacement for either of the formulas Add , Add∗. It will be shown that by adding Id to the formulas so that the four formulas Taut ,Perm,Assoc,Sum not even the formula Id∗ is provable, and moreover, that even if Id∗ is taken in addition to Id , the formulas Add and Add∗ as well as Simp remain unprovable. The claims put forward shall now be established, specifically, the claimed dependencies on the basis of which the formula systems 1, 2, 3, 4 are recognized to be sufficient systems of basic formulas will be proved in §2; in §3, proofs for the independence claims will be given, and it shall be shown that none of the systems 1, 2, 3, 4 contains a redundant formula. For the sake of convenience, let us collect the systems here again: System 1: Taut ,Add ,Perm,Sum, ,, 2 : Taut ,Add∗,Assoc,Sum, ,, 3 : Taut ,Add ,Add∗,Assoc∗,Sum, ,, 4 : Taut ,Simp,Perm,Sum. §2. By way of explanation of the derivations below, we start with a few remarks. If a formula is obtained by literal repetition or by substitution from a basic formula or a formula already derived, then the reference label of said formula is given on the left. A new reference label for a formula obtained is given on its right. Axiomatic Investigations of the Propositional Calculus 7 The application of the inference rule follows the schema A A⊃B B Furthermore, we will use-in order to give an abbreviated description of proofs- the schema A⊃B B⊃ C A⊃ C This is explained as follows: From the basic formula Sum one obtains, as mentioned already, the formula Syll : q ⊃ r * ⊃ : p⊃ q * ⊃ * p⊃ r by substitution. By using this formula twice in conjunction with the inference rule, one can derive the formula A⊃ C from the formulas A⊃B and B⊃ C. Therefore, we may proceed, wherever it is permitted to use the basic formula Sum, as if we had an inference rule according to which the formula A⊃ C can be obtained from A⊃B, B⊃ C. And this we shall do, in order to avoid unnecessary complexities. Reference labels introduced in a proof for derived formulas are given by numerals in brackets; these labels need to be fixed only within a proof. Furthermore let us remark that two formulas which result from one another by applying the abbreviation p⊃ q for ∼p ∨ q shall count as equivalent in our considerations. Passing from one of the ways of writing the formula to the other is indicated by "resp." We will now proceed to the proofs of the claimed dependencies. The theorems will be numbered in such a way that the number of each theorem coincides with that of the system of formulas which is established as complete by it.6 1. Derivation of Assoc from Taut , Add , Perm, Sum. (Add) r ⊃ p ∨ r (Sum) r ⊃ p ∨ r * ⊃ * q ∨ r ⊃ q ∨ * p ∨ r q ∨ r ⊃ q ∨ * p ∨ r 6Note that negation does not occur explicitly anywhere in the derivations 1, 2, 3. The dependencies 1, 2, 3 therefore still obtain if implication is considered as a primitive connective instead of as a combination of negation and disjunction. 8 Paul Bernays (Sum) q ∨ r ⊃ q ∨ * p ∨ r * ⊃ * p ∨ * q ∨ r ⊃ p ∨ : q ∨ * p ∨ r p ∨ * q ∨ r ⊃ p ∨ : q ∨ * p ∨ r (Perm) p ∨ : q ∨ * p ∨ r ⊃ q ∨ * p ∨ r :∨ p p ∨ * q ∨ r ⊃ q ∨ * p ∨ r :∨ p (1) (Add) p⊃ r ∨ p (Perm) r ∨ p⊃ p ∨ r p⊃ p ∨ r (Add) p ∨ r ⊃ q ∨ * p ∨ r p⊃ q ∨ * p ∨ r (Sum) p⊃ q ∨ * p ∨ r * ⊃ * q ∨ * p ∨ r :∨ p⊃ q ∨ * p ∨ r : ∨ : q ∨ * p ∨ r q ∨ * p ∨ r :∨ p⊃ q ∨ * p ∨ r : ∨ : q ∨ * p ∨ r (Taut) q ∨ * p ∨ r : ∨ : q ∨ * p ∨ r ⊃ q ∨ * p ∨ r q ∨ * p ∨ r :∨ p⊃ q ∨ * p ∨ r (2) (1) p ∨ * q ∨ r ⊃ q ∨ * p ∨ r :∨ p (2) q ∨ * p ∨ r :∨ p⊃ q ∨ * p ∨ r p ∨ * q ∨ r ⊃ q ∨ * p ∨ r 2. Derivation of Perm from Taut , Add∗, Assoc, Sum. (Taut) p ∨ p⊃ p (Sum) p ∨ p⊃ p * ⊃ * q ∨ * p ∨ p⊃ q ∨ p q ∨ * p ∨ p⊃ q ∨ p (1) (Add∗) q ⊃ q ∨ p (Sum) q ⊃ q ∨ p * ⊃ * p ∨ q ⊃ p ∨ * q ∨ p p ∨ q ⊃ p ∨ * q ∨ p (Assoc) p ∨ * q ∨ p⊃ q ∨ * p ∨ p p ∨ q ⊃ q ∨ * p ∨ p (1) q ∨ * p ∨ p ∨ p⊃ q ∨ p p ∨ q ⊃ q ∨ p 3. Derivation of Assoc from Taut , Add , Assoc∗, Sum. (Add) r ⊃ p ∨ r (Sum) r ⊃ p ∨ r * ⊃ * q ∨ r ⊃ q ∨ * p ∨ r q ∨ r ⊃ q ∨ * p ∨ r (Add) q ∨ * p ∨ r ⊃ r ∨ : q ∨ * p ∨ r q ∨ r ⊃ r ∨ : q ∨ * p ∨ r (Sum) q ∨ r ⊃ r ∨ : q ∨ * p ∨ r * ⊃ * p ∨ * q ∨ r ⊃ p ∨∴ r ∨ : q ∨ * p ∨ r p ∨ * q ∨ r ⊃ p ∨∴ r ∨ : q ∨ * p ∨ r (Assoc∗) p ∨∴ r ∨ : q ∨ * p ∨ r ⊃ p ∨ r * ∨ : q ∨ * p ∨ r p ∨ * q ∨ r ⊃ p ∨ r * ∨ : q ∨ * p ∨ r (Add) p ∨ r * ∨ : q ∨ * p ∨ r ⊃ q ∨∴ p ∨ r * ∨ : q ∨ * p ∨ r Axiomatic Investigations of the Propositional Calculus 9 p ∨ * q ∨ r ⊃ q ∨∴ p ∨ r * ∨ : q ∨ * p ∨ r (Assoc∗) q ∨∴ p ∨ r * ∨ : q ∨ * p ∨ r ⊃ q ∨ * p ∨ r : ∨ : q ∨ * p ∨ r p ∨ * q ∨ r ⊃ q ∨ * p ∨ r : ∨ : q ∨ * p ∨ r (Taut) q ∨ * p ∨ r : ∨ : q ∨ * p ∨ r ⊃ q ∨ * p ∨ r p ∨ * q ∨ r ⊃ q ∨ * p ∨ r 4. Derivation of Add from Taut , Simp, Perm, Sum. (Taut) ∼p ∨ ∼p⊃∼p (Sum) ∼p ∨ ∼p⊃∼p * ⊃ * ∼(∼p) ∨ * ∼p ∨ ∼p ∨ ∼p⊃∼(∼p) ∨ ∼p ∼(∼p) ∨ * ∼p ∨ ∼p⊃∼(∼p) ∨ ∼p (1) (Simp) ∼p⊃ * p⊃∼p resp. ∼(∼p) ∨ * ∼p ∨ ∼p (1) ∼(∼p) ∨ * ∼p ∨ ∼p⊃∼(∼p) ∨ ∼p ∼(∼p) ∨ ∼p (Perm) ∼(∼p) ∨ ∼p⊃∼p ∨ ∼(∼p) ∼p ∨ ∼(∼p) resp. p⊃∼(∼p) (Sum) p⊃∼(∼p) * ⊃ * q ∨ p⊃ q ∨ ∼(∼p) q ∨ p⊃ q ∨ ∼(∼p) (Perm) q ∨ ∼(∼p)⊃∼(∼p) ∨ q q ∨ p⊃∼(∼p) ∨ q resp. q ∨ p⊃ * ∼p⊃ q (2) (Taut) p ∨ p⊃ p resp. ∼(p ∨ p) ∨ p (Perm) ∼(p ∨ p) ∨ p⊃ p ∨ ∼(p ∨ p) p ∨ ∼(p ∨ p) (2) p ∨ ∼(p ∨ p)⊃ * ∼ ( ∼(p ∨ p) ) ⊃ p ∼ ( ∼(p ∨ p) ) ⊃ p (Sum) ∼ ( ∼(p ∨ p) ) ⊃ p * ⊃ * q ∨ ∼ ( ∼(p ∨ p) ) ⊃ q ∨ p q ∨ ∼ ( ∼(p ∨ p) ) ⊃ q ∨ p (3) (Perm) ∼ ( ∼(p ∨ p) ) ∨ q ⊃ q ∨ ∼ ( ∼(p ∨ p) ) (3) q ∨ ∼ ( ∼(p ∨ p) ) ⊃ q ∨ p ∼ ( ∼(p ∨ p) ) ∨ q ⊃ q ∨ p (Sum) ∼ ( ∼(p ∨ p) ) ∨ q ⊃ q ∨ p * ⊃ * ∼q ∨ * ∼ ( ∼(p ∨ p) ) ∨ q ⊃∼q ∨ * q ∨ p ∼q ∨ * ∼ ( ∼(p ∨ p) ) ∨ q ⊃∼q ∨ * q ∨ p (4) 10 Paul Bernays (Simp) q ⊃ * ∼(p ∨ p)⊃ q resp. ∼q ∨ * ∼ ( ∼(p ∨ p) ) ∨ q (4) ∼q ∨ * ∼ ( ∼(p ∨ p) ) ∨ q ⊃∼q ∨ * q ∨ p ∼q ∨ * q ∨ p resp. q ⊃ q ∨ p (Perm) q ∨ p⊃ p ∨ q q ⊃ p ∨ q §3. The task is now to prove the theorems claimed in §1 about independence-to which also the claims about non-replaceability belong. That a formula A is not dispensable within a system of formulas, resp., that it cannot be replaced by another formula B, is established if we show that some generally valid formula C cannot be derived from the formula system which remains after deleting the formula A, resp., which results by replacing the formula A by the formula B. Taking the established dependencies into account, one sees that the claimed independence claims are justified, provided the following independencies are shown: I. Taut cannot be derived from Add , Perm, Assoc, Sum; II. Sum ,, ,, ,, ,, Taut , Add , Add∗, Perm, Assoc, Assoc∗, Syll ; III. Simp ,, ,, ,, ,, Taut , Perm, Assoc, Sum, Id , Id∗; IV. Id∗ ,, ,, ,, ,, Taut , Perm, Assoc, Sum, Id ; V. Id ,, ,, ,, ,, Taut , Perm, Assoc, Sum; VI. Add ,, ,, ,, ,, Taut , Add∗, Simp, Assoc, Sum; VII. Add∗ ,, ,, ,, ,, Taut , Add , Assoc∗, Sum; VIII. Assoc∗ ,, ,, ,, ,, Taut , Add , Add∗, Comm, Sum; IX. Assoc∗ ,, ,, ,, ,, Taut , Add , Assoc, Sum. The proofs of these theorems will be given according to the usual method used for such investigations, viz., the method of exhibition: in each case, a finite group (in the extended sense of the word) is given, i.e., a finite totality of elements, for which the "disjunction" is defined as a two-place operation, and "negation" as a one-place operation, by giving the course-of-values (the operation in general need not be associative, uniquely invertible, nor commutative).7 Furthermore, a subtotality of "designated values" is singled out from the group. A formula is then called a "correct formula" with respect to the group under consideration, if it always yields a designated value for any substitution of elements of the group for the propositional variables. 7The elements will be designated by lowercase Greek letters. Axiomatic Investigations of the Propositional Calculus 11 From this it is first of all clear that every propositional formula which results from a correct formula by substitution is also correct. Furthermore, the groups will be set up in such a way that one obtains applying the inference rule to two correct formulas yields another correct formula. For this it is sufficient that the following condition B is satisfied: The disjunction ∼p ∨ q, which is formed from the negation of a designated value p and an arbitrary element q, has a designated value only if q is a designated value. If this condition is satisfied, it follows that every formula which is derivable from correct formulas is also corect. In order to prove that a formula F is independent of certain other formulas A, . . . , K, one only has to find a group which satisifes condition B and for which A, . . . , K are correct formulas while F is not a correct formula. For each of the independence theorems I through IX we will now give a corresponding group. They are specified as follows: First the elements are enumerated within parentheses, and the designated values between braces. Then negation and disjunction are are defined.8 (If a defining equation contains a propositional variable, this means that the equation shall hold for every value of the variable.) Subsequently the formulas that have to be verified as correct are listed. This verification is left to the reader; for the more complex formulas it can be abbreviated significantly by suitable case distinctions. Finally, a specific substitution of values is given for the formula to be shown independent, under which the formula yields a non-designated value, which shows that it is not a correct formula. In order to facilitate the verification, the following should be remarked: If the formula Id is a correct formula for a group, then if disjunction is commutative, Perm is also a correct formula; if disjunction is associative, Assoc∗ is also a correct formula; if disjunction is commutative as well as associative, Assoc is also a correct formula; if p ∨ p = p (for every value of p), then Taut is a correct formula; For each of the groups, condition B is satisfied. This has to be established for each group separately, but we will not mention it every time. Group I. (α, β, γ); {α}. ∼α = β, ∼β = α, ∼γ = γ, α ∨ p = p ∨ α = α, β ∨ p = p ∨ β = p, γ ∨ γ = α The disjunction so defined can be represented arithmetically as multiplication of the three congruence classes 0, 1, 2 mod 4. 8Here the equality sign is used in the sense of definitional equality. 12 Paul Bernays Id , Add , Perm, Assoc, Sum are correct formulas; by contrast, Taut is not a correct formula, since ∼(γ ∨ γ) ∨ γ = ∼α ∨ γ = β ∨ γ = γ Group II. (α, β, γ, δ); {α}. ∼α = β, ∼β = α, ∼γ = δ, ∼δ = α α ∨ p = p ∨ α = α, β ∨ p = p ∨ β = p, p ∨ p = p, γ ∨ δ = δ ∨ γ = α The disjunction can be represented arithmetically as multiplication of the congruence classes 0, 1, 3, 4 mod 6. Id , Taut , Add , Add∗, Perm, Assoc, Assoc∗, Syll are correct formulas; but not Sum, since ∼(∼δ ∨ β) ∨ * ∼(γ ∨ δ) ∨ (γ ∨ β) = ∼(α ∨ β) ∨ * ∼α ∨ γ = ∼α ∨ * β ∨ γ = β ∨ γ = γ. Group III. (α, β, γ, δ); {α, γ}. ∼α = β, ∼β = α, ∼γ = δ, ∼δ = γ α ∨ p = p ∨ α = α, β ∨ p = p ∨ β = β for p 6= α, p ∨ p = p, γ ∨ δ = δ ∨ γ = γ Id , Id∗, Taut , Perm, Assoc, Sum are correct formulas; but not Simp (and consequently neither is Add), since ∼γ ∨ * ∼α ∨ γ = δ ∨ * β ∨ γ = δ ∨ β = β. Group IV. This group differs from group III only in that γ ∨ γ = β is specified: Axiomatic Investigations of the Propositional Calculus 13 (α, β, γ, δ); {α, γ}. ∼α = β, ∼β = α, ∼γ = δ, ∼δ = γ α ∨ p = p ∨ α = α, β ∨ p = p ∨ β = β for p 6= α, γ ∨ γ = β, γ ∨ δ = δ ∨ γ = γ δ ∨ δ = δ. Id , Taut , Perm, Assoc, Sum are correct formulas; but not Id∗, since ∼γ ∨ * γ ∨ γ = δ ∨ β = β. Group V. (α, β, γ); {α}. ∼α = β, ∼β = α, ∼γ = γ α ∨ p = p ∨ α = α, β ∨ β = β ∨ γ = γ ∨ β = γ ∨ γ = β. Taut , Perm, Assoc, Sum are correct formulas; but not Id , since ∼γ ∨ γ = γ ∨ γ = β Group VI. (α, β, γ); {α}. ∼α = β, ∼β = α, ∼γ = γ α ∨ p = α, β ∨ p = p, γ ∨ p = γ. Id , Taut , Add∗, Simp, Assoc∗, Comm, Sum are correct formulas; but not Add (and consequently neither is Assoc), since ∼α ∨ * γ ∨ α = β ∨ γ = γ. Group VII. (α, β, γ); {α, γ}. ∼α = β, ∼β = α, ∼γ = β α ∨ p = α, β ∨ p = γ ∨ p = p. Id , Taut , Add , Assoc∗, Sum are correct formulas; but not Add∗, since ∼γ ∨ * γ ∨ β = β ∨ β = β. 14 Paul Bernays Group VIII. (α, β, γ, δ); {α}. ∼α = β, ∼β = α, ∼γ = α, ∼δ = γ α ∨ p = p ∨ α = α, β ∨ p = p ∨ β = p, p ∨ p = p, γ ∨ δ = α, δ ∨ γ = δ. Id , Taut , Add , Add∗, Comm, Sum are correct formulas; but not Perm (and consequently neither are Assoc and Assoc∗), since ∼(γ ∨ δ) ∨ * δ ∨ γ = ∼α ∨ δ = β ∨ δ = δ. Group IX. (α, β, γ, δ); {α, γ}. ∼α = β, ∼β = α, ∼γ = δ, ∼δ = γ α ∨ p = p ∨ α = α, β ∨ p = δ ∨ p = p, γ ∨ β = β, γ ∨ γ = γ ∨ δ = γ. Id , Taut , Add , Assoc, Sum are correct formulas; but not Assoc∗ (and consequently neither are Add∗ and Perm), since ∼(γ ∨ * β ∨ δ) ∨ : γ ∨ β * ∨ δ = ∼(γ ∨ δ) ∨ * β ∨ δ = ∼γ ∨ δ = δ ∨ δ = δ. (received April 7, 1925.) Paul Bernays