Logical Formula of the Day: self-distribution CCpCqrCCpqCpr.

Here's two links on condensed detachment.  If we have two formulas Cab, and "a", condensed detachment produces the most general formula "b".  For instance, if we have

1. CpCqp as "a" and
2. CCpCqrCCpqCpr as "Cab",

we could substitute "r" in CCpCqrCCpqCpr, and thus obtain CCpqCpp from a condensed detachment D2.1.  We could also substitute "r" with "Cqp", and "p" with "Cqp" in  CCpCqrCCpqCpr  with Cqp  obtaining CCCqpCqCqpCCCqppCCqpCqp, and then substitute "p" with "Cqp" in CpCqp obtaining CCqpCqCqp.  Thus, from 1. and 2. we could detach CCCqpqCCqpCqp as a theorem.  However, CCCqpqCCqpCqp is less general than CCpqCpp, since if we substitute "p" with "Cqp" in CCpqCpp, then we obtain CCCqpqCCqpCqp.  So, condensed detachment Da.b obtains a form from which you can obtain every other theorem detachable from a substitution instance of "a" as the major premise and "b" as the minor premise.

Let's look a little at the system {CCpCqrCCpqCpr}

axiom   1 CCpCqrCCpqCpr
D1.1     2 CCCpCqrCpqCCpCqrCpr
D1.2     3 CCCCpCqrCpqCpCqrCCCpCqrCpqCpr
D1.3     4 CCCCCpCqrCpqCpCqrCCpCqrCpqCCCCpCqrCpqCpCqrCpr

Every theorem of this system obtained via condensed detachment has three variables.  This follows, because the axiom has only three variables in the antecedent and the consequent.  Thus, the second theorem has only three variables in the antecedent and the consequent.

If we have CCpCqrCCpqCpr and CpCqp as axioms or theorems, then we have a deduction metatheorem for the system which says "if we have an assumption "a" which leads to a formula "b" under the scope of some set of assumptions "g", then Cab holds under the scope of the assumptions "g" also".  Or more concisely we can write:

Deduction Metatheorem: If {g, a} |- b, then {g}|-Cab.

Formula of the Day: Recursive Variable Prefixing CpCqp.

When proving that a logical system has a deduction metatheorem, CpCqp proves indispensable.   In fact, CpCqp along with CCpCqrCCpqCpr or CCpqCCpCqrCpr suffices to prove a deduction metatheorem.  Let's consider the system {CpCqp}.  For this system:

Lemma: Every theorem obtainable by condensed detachment for the system {CpCqp} has the form C0C1...CnCpCqp.

Demonstration: The first theorem obtainable by condensed detachment is CpCqCrq.  Now from {CpCqp, CpCqCrq} we can use either thesis as the minor premise, and either one as the major.  If CpCqp is the major, then we will detach Cab where "a" is a variable and "b" consists of the minor premise.  If CpCqCrq is the major premise, then we will obtain CpCqp.  More generally, if we have CpCqp as the major premise and use any theorem "b" as the minor premise, we will obtain Cab upon a condensed detachment.  If Cat, where "a" does not appear in "t" gets used as the major premise, then we will obtain "t" after a condensed detachment.  Thus, any time we use condensed detachment in this system we'll either obtain CpCqp or some theorem which has CpCqp as its last 5 letters.

The above lemma can get argued also via mathematical induction.

Corollary: Every thesis (theorem or axiom) obtainable via condensed detachment for the system {CpCqp} can serve as a sole axiom for a system with the same theses as {CpCqp}.

Demonstration: Since every axiom has form C0C1...CnCpCqp by the above lemma, we can obtain CpCqp as a theorem of the system.  Since CpCqp has C0C1...CnCpCqp as a theorem, it follows that the systems have the exact same theorems.

Consequently it follows that in addition to {CpCqp, CCpCqrCCpqCpr} sufficing to prove the deduction theorem *any* theorem obtainable from the system {CpCqp} solely by condensed detachment can take the place of CpCqp in {CpCqp, CCpCqrCCpqCpr} .

Formula of the Day: CδCpqδANpq.

"δ" in opposition to representing a propositional variable indicates a functorial variable of arity one.  In other words, it stands for any of the unary functions presumed.  Given the context of classical logic this means it stands for any of the functions belonging to {F, N, I, V} where F(0)=0, F(1)=0, N(0)=1, N(1)=0, I(0)=0, I(1)=1, V(0)=1, V(1).  Thus we can check that CδCpqδANpq constitutes a tautology by checking that each member of {CFCpqFANpq, CNCpqNANpq, CICpqIANpq, CVCpqVANpq} is a tautology in 2-valued logic.

In notation for substitution for δ, an apostrophe " ' " means that we substitute the argument that follows delta for the apostrophe.  For example, from 1 δp, 1 δ/C'q we substitute δ with C, then the argument that follows δ which is "p" in this case, and then "q" obtaining Cpq.  If we write 2 CsδCpCqr, 2 δ/C'' that means we substitute δ with C, then the argument of δ CpCqr, then the argument of δ CpCqr obtaining CsCCpCqrCpCqr.  If we write 3 CpCCpδqr, 3 δ/CC'r' then we write CC, the argument of δ q, r, and lastly the argument of δ "q" again in place of δq obtaining CpCCpCCqrqr.  δ/' just means we substitute δ with the argument of δ.
CδCpqδANpq can effectively take the place of a definition of the conditional "C" in a logical calculus.  Let's consider CδCpqδANpq in the following system:

axiom                     1 ApNp
axiom                     2 CδCpqδANpq
2 δ/CCδCpqδ'CδANpqδCpq    3 CCCδCpqδCpqCδANpqδCpqCCδCpqδANpqCδANpqδCpq
2 δ/Cδ'δCpq               4 CCδCpqδCpqCδANpqδCpq
D3.4                      5 CCδCpqδANpqCδANpqδCpq
D5.2                      6 CδANpqδCpq
6 δ/'                     7 CANpqCpq
D7.1                      8 CpNNp

Thus, double negation introduction CpNNp follows from these two axioms.  It also follows that in the presence of a definition corresponding to axiom 2, the law of the excluded middle ApNp allows us to derive the law of double negation introduction CpNNp.

Well-Formed Formula of the Day: ApNp

ApNp consists of a symbolic formulation of the law of the excluded middle "either a proposition is true or it is not true."  If we wanted to use at as an axiom, we might want to have as a deduction rule {ANpq, p}->q, but even then we'd probably want to commute ApNp and use ANpp instead.  Let's say we have ANpp as an axiom along with ANpANqp as an axiom.  We might proceed as follows.

axiom  1 ANpp
axiom  2 ANpANqp
D2.2    3 ANpANqANrq
D2.1    4 ANpANqq

ApNp can get proved from the axiom set {CpCqp, CCpCqrCCpqCpr, CCNpqCCNpNqp, CpApq, CpAqp}.  ApNp is unique among classical logic theorems which only have C, A, K, E, and N as connectives in them in that it has exactly 4 symbols.

Formula of the Day: EEEpqrEpEqr

EEEpqrEpEqr comes as a way of saying that logical equivalence associates.  Along with EEpqEqp can serve as one of the axioms of the equivalential calculus.  Equivalential calculus allows us to deduce all tautologies which only have equivalence symbols and variables in them.  The rule of inference for equivalential calculus comes as a modified form of detachment "from Eab, as well as "a", we may infer b" where "a" and "b" consist of meta-variables.  Or more compactly

{Eab, a} -> b

We can condensed such an E-detachment along with uniform substitution just like we can condense C-detachment with uniform substitution.  Taking EEEpqrEpEqr as an axiom, we can develop the system {EEEpqrEpEqr} as follows:

axiom 1 EEEpqrEpEqr
D1.1   2 EEpqErEpEqr
D1.2   3 EpEqErEpEqr
D2.1   4 EsEEEpqrEEpEqrs
D1.3   5 EpEqErEsEEpqErs
D1.4   6 EpEqEEErstEErEstEpq
D2.2   7 EpEEqrEEsEqErsp
D2.3   8 EsEpEEqErEpEqrs

The system with a single axiom which corresponds to each of these theorems has no last theorem under condensed detachment (unlike {Cpp} which has Cpp as its only theorem under condensed detachment).  For {EEpqErEpEqr} each theorem will have some form Eab, which implies that there exists a substitution which enables a detachment from the antecedent of Epq of EEpqErEpEqr with EEpqErEpEqr as the major premise, and any theorem of the system {EEpqErEpEqr}.  For the systems corresponding to theorems 3 through 8, each axiom serving as the major premise and any theorem serving as the minor premise enables a condensed detachment.  Does *every* theorem EEEpqrEpEqr correspond to a system with a sole axiom which has no last theorem under condensed detachment?

EEEpqrEpEqr seems to come as a theorem that requires a fair amount of work to prove in the natural deduction systems I've encountered and in non-equivalential systems of logic which use definitions to get equivalences into the system.

Formula of the Day: The law of commutation CCpCqrCqCpr.

If we look at CCpCqrCqCpr it turns out that we can obtain the consequent implying the antecedent just from substitution.  Let X p/a indicate that variable p gets substituted with a throughout formula X.

axiom  1 CCpCqrCqCpr
1 p/a    2 CCaCqrCqCar
2 q/p    3 CCaCprCpCar
3 a/q    4 CCqCprCpCqr

Or if we allow simultaneous substitution in a formula we can proceed as follows:

axiom       1 CCpCqrCqCpr
1 p/q, q/p  2 CCqCprCpCqr

The law of commutation, unlike the law of identity, can also serve as the sole axiom of a system under detachment, and from it as an axiom we can obtain well-formed formulas (or "formulas") which are not substitution instances of it (unlike Cpp as the sole axiom of a system).

axiom  1 CCpCqrCqCpr
D1.1    2 CpCCqCprCqr
D2.2    3 CCpCCqCCrCqsCrstCpt
D2.1    4 CCpCCCqCrsCrCqstCpt
D1.4    5 CpCCpCCCqCrsCrCqstt
D1.3    6 CpCCpCCqCCrCqsCrstt
D2.3    7 CCuCCCpCCqCCrCqsCrstCptvCuv
D2.7    8 CCwCCCuCCCpCCqCCrCqsCrstCptvCuvyCwy
D2.8    9 CCzCCCwCCCuCCCpCCqCCrCqsCrstCptvCuvyCwyaCza

The system {CCpCqrCqCpr} has no last most general theorem, as can get understood from theorem 2 CpCCqCprCqr.  D2.x will always yield a theorem, since any variable can get substituted by any theorem, and CCqCprCqr where p is any theorem is a well-formed formula.  Thus, D2.2 yields a theorem.  D2.D2.2 yields a theorem.  D2.D2.D2.2 yields a theorem and so on.  Each theorem has more symbols than the last theorem also, since the antecedent "p" appears in the consequent of CpCCqCprCqr, and since every theorem has more symbols than Cp, the detached theorem having the form CCqCprCqr will have more letters than the theorem that served as the major premise.  Theorems 5, and 6 also indicate that there exist subsystems of CCpCqrCqCpr which like the subsystem {CpCCqCprCqr} have no last most general theorem.

The law of commutation appeared in the first axiom system for propositional calculus ever proposed.  Gottlob Frege's system was {CpCqp, CCpCqrCCpqCpr, CCpCqrCqCpr, CNNpp, CpNNp, CCpqCNqNp}.  Jan Lukasiewicz figured out that the law of commutation could get derived from the first two axioms a few decades later.  Here's a proof:

axiom   1 CpCqp
axiom   2 CCpCqrCCpqCpr
D1.2     3 CpCCqCrsCCqrCqs
D2.3     4 CCpCqCrsCpCCqrCqs
D1.4     5 CpCCqCrCstCqCCrsCrt
D2.5     6 CCpCqCrCstCpCqCCrsCrt
D6.1     7 CCpCqrCsCCpqCpr
D4.7     8 CCpCqrCCsCpqCsCpr
D2.8     9 CCCpCqrCsCpqCCpCqrCsCpr
D1.1   10 CpCqCrq
D9.10 11 CCpCqrCqCpr

The law of commutation also appears as an axiom or theorem in many other systems of logic.  If one has it in a system S, and it does NOT need one of the axioms to get derived, then there may very well exist a way to figure out an alternative system A which has the same theorems as S.  For instance, we'll start by pointing out that CCpCqrCqCpr comes as derivable from {CpCqp, CCpCqrCCpqCpr} above.  Now since CCpqCNqNp and D [CCpCqrCCpqCpr].[CCpqCNqNp] exists it turns out that the theorems of Frege's system have another axiomization than {CpCqp, CCpCqrCCpqCpr, CNNpp, CpNNp, 5a-CCpqCNqNp}.  Namely, that axiomization is {CpCqp, CCpCqrCCpqCpr, CNNpp, CpNNp, 5b-CNpCCqpNq}.  To prove this we only need to write out that D[ CCpCqrCqCpr].5a yields 5b and D[CCpCqrCqCpr].5b yields 5a.

Also, let's suppose that we have CCpCqrCqCpr and Cpp as our axioms:

axiom   1  CCpCqrCqCpr
axiom   2  Cpp
D1.2     3  CpCCpqq
D3.2     4  CCCppqq
D1.4     5  CpCCCqqCprr
D1.5     6  CCCppCqrCqr

Notice that 6 is not the same formula as 4, but 6 is a more specialized (or less general) formula than 4 since we can obtain 6 from 4 by substitution only.

Whenever I refer to an axiom set, unless stated otherwise, I presume the rule of detachemnt {|-Cab, |-a} -> b, where "a" and "b" constitute metavariables, and uniform substitution for variables.  The notation Da.b stands for a condensed detachment with "a" as the major premise and "b" as the minor premise.

Logical Formula of the Day: Cpp

Since this consists of a new thing I plan to do I guess I'll start small with one of the shortest tautologies in existence.  Cpp sometimes appears as an axiom in some propositional calculi.  For instance, A. N. Prior's formal logic lists {Cpp, CCCpqrCCrpCsCtp} as a basis for implicational propositional calculus.  It also appeared as an axiom in Peirce's axiom set for C-0 (0 meaning "falsum") classical logic {Cpp, CCpCqrCqCpr, CCpqCCqrCpr, C0p, CCCpqpp}.  In a natural deduction setting you can prove it as follows:

hypothesis 1 | p
Ci 1-1        2   Cpp

where "Ci" stands for conditional introduction.  For positive implicational propositional calculus with the axiom set {CpCqp, CCpCqrCCpqCpr} we can prove it in two condensed detachments.

axiom 1 CpCqp
axiom 2 CCpCqrCCpqCpr
D2.1   3 CCpqCpp
D3.1   4 Cpp

In classical logic with the basis {CCpqCCqrCCpr, CCNppp, CpCNpq} we can prove it in two steps also.

axiom 1 CCpqCCqrCpr
axiom 2 CCNppp
axiom 3 CpCNpq
D1.3   4 CCCNpqrCpr
D4.2   5 Cpp

Here's another proof from another axiom set:

axiom 1 CCCpqrCqr
axiom 2 CCCpqpp
D 1.2  3 Cpp

Actually, let's say we fix CCCpqrCqr as the first axiom 1.  Now replace the first p in CCCpqpp with CCpqp.  This yields CCCCCpqpqpp.  D1. CCCCCpqpqpp yields Cpp.  In any system sufficient for the conditional of classical logic, the variable "p" can get replaced by "CCpqp" since CCCpqpp is a theorem and CpCCpqp is also a theorem.  Consequently, we can iterate the process of replacing a "p" in any formula obtained in this way with CCCpqpp as the starting point and still obtain a theorem in classical logic, provided that we do NOT replace the last two "p's".  Each theorem obtained in this way serving as the minor premise along with CCCpqrCqr as the major premise will yield Cpp after a condensed detachment.

One can also prove it in Wajsberg-Lukasiewicz three-valued from the following basis as follows:

axiom 1 CpCqp
axiom 2 CCpqCCqrCpr
axiom 3 CCCpNppp
axiom 4 CCNpNqCqp
D2.1   5 CCCqprCpr
D5.3   6 Cpp

Or Lukasiewicz infinite valued-logic:

axiom     1 CpCqp
axiom     2 CCpqCCqrCpr
axiom     3 CCNpNqCqp
axiom     4 CCCpqqCCqpp
D2.1       5 CCCpqrCqr
D2.2       6 CCCCqrCprsCCpqs
D6.6       7 CCpCqrCCsqCpCsr
D5.4       8 CpCCpqq
C8.7       9 CCpCqrCqCpr
D9.1     10 CpCqq
D10.10 11 Cpp

Or from double suffixing CCpqCCqrCpr and the double negation laws CpNNp, CNNpp.

axiom   1 CCpqCCqrCpr
axiom   2 CpNNp
axiom   3 CNNpp
D1.2     4 CCNNpqCpq
D4.3     5 Cpp

Cpp as a major premise will only give you back the minor premise that you used when doing a condensed detachment.  On the other hand, Cpp as a minor premise has some interesting uses.  For instance D[CCpCqrCqCpr.Cpp] yields "modus ponens" CpCCpqq.  D[CCpCqrCCpqCpr.Cpp] yields CCCpqpCCpqq.  Perhaps more interestingly D[CCNpNqCCNpqq.Cpp] yields the law of Clavius CCNppp.  Additionally, if we consider D[CCpqCCqrCpr.a], and D[CCpqCCqrCpr.a] exists (which is equivalent to saying that "a" is a conditional), then D[D[CCpqCCqrCpr.a]].[Cpp] yields "a".