Principia Mathematica propositional logic

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Principia Mathematica propositional logic
The purpose of this page is to prove the statements of Interface:Principia Mathematica propositional logic theorems from the Principia Mathematica[1] axioms for propositional logic.
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Interface:Principia Mathematica propositional logic theorems

We define some variables for well-formed formulas:

var (wff p q r s)

Principia Mathematica does not define the nullary connectives "the true", , and "the false", , so we define them here:

def (() (p  (¬ p)))
def (() (p  (¬ p)))

We shall now begin to derive the statements. Whitehead and Russell use a decimal numbering system of the form , where is a rational number with a small number of digits after the decimal point. Unless we give theorems our own name, we shall adopt their system for easier reference. Where we do use our own names, we sometimes give the decimal reference in a JHilbert comment.

Contents

Disjunction and implication[edit]

First steps[edit]

The modus ponens rule is an axiom of Principia Mathematica, so we don't need to prove it. The Perm axiom immediately gives rise to the disjunctive commutativity rule:

thm (swapDisjunction () ((H (p  q))) (q  p) (
        H
        p q Perm
        applyModusPonens
))

Likewise it will be convenient to have the Sum axiom as a rule:

thm (disjoinLL () ((H (p  q))) ((r  p)  (r  q)) (
        H
        p q r Sum
        applyModusPonens
))

Since Principia Mathematica defines implication, , as , the Add axiom yields the introduction of an antecedent (Whitehead and Russell call it "Simplification").

thm (AntecedentIntroduction () () (p  (q  p)) ( # *2.02
        p (¬ q) Add
))

It will be convenient to have this theorem as helper rule:

thm (introduceAntecedent () ((H p)) (q  p) (
        H
        p q AntecedentIntroduction
        applyModusPonens
))

Again due to the way implication is defined, Perm gives us our first transposition rule:

thm (*2.03 () () ((p  (¬ q))  (q  (¬ p))) (
        (¬ p) (¬ q) Perm
))

thm (transposeWithNegatedConsequent () ((H (p  (¬ q)))) (q  (¬ p)) (
        H
        p q *2.03
        applyModusPonens
))

Next, we prove a precursor to the commutative law of Conjunction, which will be very convenient until we get Peano's transportation principle. We call this theorem Comm in accordance with Whitehead and Russell.

thm (Comm () () ((p  (q  r))  (q  (p  r))) ( # *2.04
        (¬ p) (¬ q) r Assoc
))

thm (applyComm () ((H (p  (q  r)))) (q  (p  r)) (
        H
        p q r Comm
        applyModusPonens
))

This theorem enables us to prove two forms of the syllogism from the Sum axiom. The rule applySyllogism expresses the syllogism nature of these theorems (we have two implications and derive a third). But they also can be used to build up more complicated formulas, a pattern which is expressed by the addCommonAntecedent and addCommonConsequent rules, in which we just have one implication and derive a more involved formula.

thm (CommonAntecedentAddition () () ((q  r)  ((p  q)  (p  r))) ( # *2.05

        q r (¬ p) Sum
))

thm (addCommonAntecedent () ((H (q  r))) ((p  q)  (p  r)) (
        H
        q r p CommonAntecedentAddition
        applyModusPonens
))

thm (CommonConsequentAddition () () ((p  q)  ((q  r)  (p  r))) ( # *2.06

        q r p CommonAntecedentAddition
        (q  r) (p  q) (p  r) Comm
        applyModusPonens
))

thm (addCommonConsequent () ((H (p  q))) ((q  r)  (p  r)) (
        H
        p q r CommonConsequentAddition
        applyModusPonens
))

thm (applySyllogism () ((H1 (p  q)) (H2 (q  r))) (p  r) (
        H2
        H1
        p q r CommonConsequentAddition
        applyModusPonens
        applyModusPonens
))

The syllogism yields the "identity" Id, , and, by Perm, tertium non datur, which is, by our definition, just :

thm (Id () () (p  p) ( # *2.08

        p p Add
        p Taut
        applySyllogism
))
# Another name for the same theorem:
thm (ImplicationReflexivity () () (p  p) (
        p Id
))

thm (TertiumNonDatur () () (p  (¬ p)) ( # *2.11

        p Id
        swapDisjunction        
))

thm (True () () () (
        p TertiumNonDatur
))

Actually, for Whitehead and Russell use the permutation of our TertiumNonDatur as theirs.

With the TertiumNonDatur, we can tackle double negation:

thm (*2.12 () () (p  (¬ (¬ p))) (
        (¬ p) TertiumNonDatur
))

thm (*2.13 () () (p  (¬ (¬ (¬ p)))) (
        p TertiumNonDatur

        (¬ p) *2.12
        p disjoinLL

        applyModusPonens
))

thm (*2.14 () () ((¬ (¬ p))  p) (
        p *2.13
        swapDisjunction
))

thm (introduceDoubleNegation () ((H p)) (¬ (¬ p)) (
        H
        p *2.12
        applyModusPonens
))

thm (eliminateDoubleNegation () ((H (¬ (¬ p)))) p (
        H
        p *2.14
        applyModusPonens
))

Transposition[edit]

Combined with double negation, the transposition law we already have yields the remaining three:

thm (*2.15 () () (((¬ p)  q)  ((¬ q)  p)) (
        q *2.12
        (¬ p) addCommonAntecedent

We now have . We transpose the consequent using our transposition law

        (¬ p) (¬ q) *2.03
        applySyllogism

to get . All that remains to do now is to eliminate the double negation.

        p *2.14
        (¬ q) addCommonAntecedent
        applySyllogism
))

thm (transposeWithNegatedAntecedent () ((H ((¬ p)  q))) ((¬ q)  p) (
        H
        p q *2.15
        applyModusPonens
))

The proofs of the next two theorems work similar but are actually easier due to a more favourable distribution of negations:

thm (*2.16 () () ((p  q)  ((¬ q)  (¬ p))) (
        q *2.12
        p addCommonAntecedent
        p (¬ q) *2.03
        applySyllogism
))

thm (introduceTransposition () ((H (p  q))) ((¬ q)  (¬ p)) (
        H
        p q *2.16
        applyModusPonens
))

thm (*2.17 () () (((¬ q)  (¬ p))  (p  q)) (
        (¬ q) p *2.03
        q *2.14
        p addCommonAntecedent
        applySyllogism
))

thm (eliminateTransposition () ((H ((¬ q)  (¬ p)))) (p  q) (
        H
        q p *2.17
        applyModusPonens
))

Disjunction introduction[edit]

Next, we prove the disjunction introduction laws. Introduction from the left is just the Add axiom of principia, and a permutation yields the introduction from the right.

thm (DisjunctionLeftIntroduction () () (p  (q  p)) (
        p q Add
))

thm (introduceLeftDisjunction () ((H p)) (q  p) (
        H
        p q DisjunctionLeftIntroduction
        applyModusPonens
))

thm (DisjunctionRightIntroduction () () (p  (p  q)) ( # *2.2
        p q DisjunctionLeftIntroduction
        q p Perm
        applySyllogism
))

thm (introduceRightDisjunction () ((H p)) (p  q) (
        H
        p q DisjunctionRightIntroduction
        applyModusPonens
))

Modus ponens law[edit]

Next, we prove a version of the modus ponens law with Assoc:

thm (*2.25 () () (p  ((p  q)  q)) (
        (p  q) Id
        (¬ (p  q)) p q Assoc
        applyModusPonens
))

thm (*2.27 () () (p  ((p  q)  q)) (
        (¬ p) q *2.25
))

Disjunction associativity[edit]

We now "repair" the extra twist in the Assoc axiom to get the actual associativity rules for disjunction:

thm (*2.3 () () ((p  (q  r))  (p  (r  q))) (
        q r Perm

        p disjoinLL
))

thm (*2.31 () () ((p  (q  r))  ((p  q)  r)) (
        p q r *2.3
        p r q Assoc
        applySyllogism
        r (p  q) Perm
        applySyllogism
))

thm (groupDisjunctionLeft () ((H (p  (q  r)))) ((p  q)  r) (
        H
        p q r *2.31
        applyModusPonens
))

thm (*2.32 () () (((p  q)  r)  (p  (q  r))) (
        (p  q) r Perm
        r p q Assoc
        applySyllogism
        p r q *2.3
        applySyllogism
))

thm (groupDisjunctionRight () ((H ((p  q)  r))) (p  (q  r)) (
        H
        p q r *2.32
        applyModusPonens
))

Expression building with disjunctions[edit]

Now we prove three helpful theorems regarding expression building with disjunctions, that is, they are companions to the Sum axiom. All these proofs proceed in two steps, using a syllogism to alter the consequent of Sum.

thm (DisjunctionSummationLR () () ((q  r)  ((p  q)  (r  p))) ( # *2.36

As first step, we simply take the Sum axiom,

        q r p Sum

and prove as second step,

        p r Perm
        (p  q) addCommonAntecedent

so that the result follows via a syllogism:

        applySyllogism
))

thm (DisjunctionSummationRL () () ((q  r)  ((q  p)  (p  r))) ( # *2.37

As first step, we take the Sum axiom with transposed consequent, that is ,

        q r p Sum
        (p  q) (p  r) *2.16
        applySyllogism

As second step, we prove

        q p Perm
        introduceTransposition
        (¬ (p  r)) addCommonAntecedent

Combining these two steps, the result follows by transposing the consequent back.

        applySyllogism
        (p  r) (q  p) *2.17
        applySyllogism
))

Finally, DisjunctionSummationRR follows directly from DisjunctionSummationRL in the same way.

thm (DisjunctionSummationRR () () ((q  r)  ((q  p)  (r  p))) ( # *2.38
        q r p DisjunctionSummationRL
        p r Perm
        (q  p) addCommonAntecedent
        applySyllogism
))

The rule forms of all three are:

thm (disjoinLR () ((H (p  q))) ((r  p)  (q  r)) (
        H
        p q r DisjunctionSummationLR
        applyModusPonens
))

thm (disjoinRL () ((H (p  q))) ((p  r)  (r  q)) (
        H
        p q r DisjunctionSummationRL
        applyModusPonens
))

thm (disjoinRR () ((H (p  q))) ((p  r)  (q  r)) (
        H
        p q r DisjunctionSummationRR
        applyModusPonens
))

Implication distribution theorem (if part)[edit]

Our next goal will be to prove the if part of the implication distribution theorem, . Given that implication is defined from disjunction and negation in Principia, it would seem more natural to use de Morgan's law for disjunction negation for that. However, we don't have that yet, and in fact, implication distribution is instrumental to prove the "principle of the factor" Fact below, which in turn is required for de Morgan's law. So what we shall do instead is to install equivalences such as , and in more or less deep subexpression in a long chain of theorems until we finally arrive at the desired result.

thm (*2.53 () () ((p  q)  ((¬ p)  q)) (

That is, we make from .

        p *2.12
        q disjoinRR
))

thm (*2.6 () () (((¬ p)  q)  ((p  q)  q)) (
        (¬ p) q q DisjunctionSummationRR

This gives us already the result, except that the last has been duplicated, . We remove the duplicate using the Taut axiom.

        q Taut
        (¬ ((¬ p)  q)) disjoinLL

        applySyllogism
))

thm (*2.62 () () ((p  q)  ((p  q)  q)) (
        p q *2.53
        p q *2.6
        applySyllogism
))

thm (*2.621 () () ((p  q)  ((p  q)  q)) (
        p q *2.62
        applyComm
))

thm (*2.73 () () ((p  q)  (((p  q)  r)  (q  r))) (
        p q *2.621
        (p  q) q r DisjunctionSummationRR
        applySyllogism
))

thm (*2.74 () () ((q  p)  (((p  q)  r)  (p  r))) (

This is the same as except for a permutation in the second antecedent. Since this permutation is nested four levels deep, some work is required. We begin with ,

        q p r *2.73

and prove the permutation :

        p q r *2.32
        p q r Assoc
        applySyllogism
        q p r *2.31
        applySyllogism

We can now install this permutation as an antecedent through a transposition. Then the theorem follows through a syllogism.

        introduceTransposition
        (p  r) disjoinRR
        applySyllogism
))

thm (*2.75 () () ((p  q)  ((p  (q  r))  (p  r))) (

This is , with replaced with , plus some of the usual transformations.

        p q Perm
        q p *2.53
        applySyllogism

This gives us , which we can connect with ,

        (¬ q) p r *2.74
        applySyllogism

so we get . The rest of the proof is now to effect the disjunctive associativity law in the second antecedent.

        p (¬ q) r *2.31
        introduceTransposition
        (p  r) disjoinRR

        applySyllogism
))

thm (*2.76 () () ((p  (q  r))  ((p  q)  (p  r))) (
        p q r *2.75
        applyComm
))

thm (*2.77 () () ((p  (q  r))  ((p  q)  (p  r))) (
        (¬ p) q r *2.76
))

thm (distributeAntecedent () ((H (p  (q  r)))) ((p  q)  (p  r)) (
        H
        p q r *2.77
        applyModusPonens
))

Theorem is now precisely the if part of the implicational distribution law.

Syllogism in the consequent[edit]

We prove a "syllogism in the consequent law", that is, , which is useful for syllogisms depending on a common hypothesis. We first show a modus ponens law for disjunctions.

thm (*2.8 () () ((q  r)  ((r  s)  (q  s))) (
        q r Perm
        r q *2.53
        applySyllogism

This gives us . The theorem follows now by a summation:

        (¬ r) q s DisjunctionSummationRR
        applySyllogism
))

Next, we prove a three terms summation law:

thm (*2.81 () () ((q  (r  s))  ((p  q)  ((p  r)  (p  s)))) (
        q (r  s) p Sum

This gives us , so we only need to distribute the over :

        p r s *2.76
        (p  q) introduceAntecedent
        distributeAntecedent
        applySyllogism
))

Combining and , we get

thm (*2.82 () () ((p  (q  r))  ((p  (r  s))  (p  (q  s)))) (
        q r s *2.8
        (q  r) (r  s) (q  s) p *2.81
        applyModusPonens
))

from which we can immediately derive our new syllogism law:

thm (SyllogismInConsequent () () ((p  (q  r))  ((p  (r  s))  (p  (q  s)))) ( # *2.83
        (¬ p) (¬ q) r s *2.82
))

thm (applySyllogismInConsequent () ((H1 (p  (q  r))) (H2 (p  (r  s)))) (p  (q  s)) (
        H2
        H1
        p q r s SyllogismInConsequent
        applyModusPonens
        applyModusPonens
))

Implication distribution theorem (only-if part)[edit]

Next we prove the only if part of the implicational distribution law (the if part of which we proved in ). We start with a simple consequence of and the converse of :

thm (*2.55 () () ((¬ p)  ((p  q)  q)) (
        p q *2.53
        applyComm
))

thm (*2.54 () () (((¬ p)  q)  (p  q)) (
        p *2.14
        q disjoinRR
))

The meat of our desired result is ( ). It differs only by being partly expressed in terms of disjunction instead of implication. Each of the following thm's is just an intermediate step in proving :

thm (*2.85-1 () () (((p  q)  r)  (q  r)) (
        q p Add
        r addCommonConsequent
))

thm (*2.85-unnamed1 () () ((¬ p)  (((p  q)  (p  r))  ((p  q)  r))) (
        p r *2.55
        (p  r) r (p  q) CommonAntecedentAddition
        applySyllogism
))

thm (*2.85-2 () () ((¬ p)  (((p  q)  (p  r))  (q  r))) (
        p q r *2.85-unnamed1

        p q r *2.85-1
        (¬ p) introduceAntecedent

        applySyllogismInConsequent
))

thm (*2.85-unnamed2 () () (((p  q)  (p  r))  ((¬ p)  (q  r))) (
        p q r *2.85-2
        applyComm
))

thm (*2.85 () () (((p  q)  (p  r))  (p  (q  r))) (
        p q r *2.85-unnamed2
        p (q  r) *2.54
        applySyllogism
))

From this follows the only if part of the implicational distribution law:

thm (*2.86 () () (((p  q)  (p  r))  (p  (q  r))) (
        (¬ p) q r *2.85
))

thm (collectAntecedent () ((H ((p  q)  (p  r)))) (p  (q  r)) (
        H
        p q r *2.86
        applyModusPonens
))

Conjunction[edit]

In this section we prove statements involving conjunction . Recall that conjunction is defined by . Our first theorem is the combination of two statements to a conjunction.

thm (ConjunctionRightIntroduction () () (p  (q  (p  q))) ( # *3.2
        ((¬ p)  (¬ q)) TertiumNonDatur

The trick is now to write as :

        groupDisjunctionRight
))

thm (introduceConjunction () ((H1 p) (H2 q)) (p  q) ( # *3.03
        H2
        H1
        p q ConjunctionRightIntroduction
        applyModusPonens
        applyModusPonens
))

We also provide a commuted version:

thm (ConjunctionLeftIntroduction () () (p  (q  (q  p))) ( # *3.21
        q p ConjunctionRightIntroduction
        applyComm
))

We can prove the commutative law for conjunction from Perm using transpositions:

thm (*3.22 () () ((p  q)  (q  p)) (
        (q  p) Id
        eliminateTransposition
        (¬ q) (¬ p) Perm
        applySyllogism
        introduceTransposition
))

thm (swapConjunction () ((H (p  q))) (q  p) (
        H
        p q *3.22
        applyModusPonens
))

Next, we prove the negation of :

thm (*3.24 () () (¬ (p  (¬ p))) (
        (¬ p) TertiumNonDatur
        ((¬ p)  (¬ (¬ p))) *2.12
        applyModusPonens
))

thm (NotFalse () () (¬ ()) (
        p *3.24
))

Conjunction elimination[edit]

Next, we prove the conjunction elimination theorems.

thm (ConjunctionRightElimination () () ((p  q)  p) ( # *3.26
        p q AntecedentIntroduction
        groupDisjunctionLeft

This gives us . All we need now is a double negation of the left bracket.

        ((¬ p)  (¬ q)) p *2.53
        applyModusPonens
))

thm (eliminateRightConjunct () ((H (p  q))) p (
        H
        p q ConjunctionRightElimination
        applyModusPonens
))

thm (ConjunctionLeftElimination () () ((p  q)  q) ( # *3.27
        p q *3.22
        q p ConjunctionRightElimination
        applySyllogism
))

thm (eliminateLeftConjunct () ((H (p  q))) q (
        H
        p q ConjunctionLeftElimination
        applyModusPonens
))

Import and export[edit]

We now prove Peano's import and export principles. We begin with exportation:

thm (Exportation () () (((p  q)  r)  (p  (q  r))) ( # *3.3

To shift the bracket right, we combine transposition and the Comm axiom to get :

        ((¬ p)  (¬ q)) r *2.15
        (¬ r) p (¬ q) Comm
        applySyllogism

What remains to do now is to install a transposition of with :

        r q *2.17
        p introduceAntecedent distributeAntecedent
        applySyllogism
))

thm (export () ((H ((p  q)  r))) (p  (q  r)) (
        H
        p q r Exportation
        applyModusPonens
))

Importation is simpler as the conjunction is in the consequent this time:

thm (Importation () () ((p  (q  r))  ((p  q)  r)) ( # *3.31
        (¬ p) (¬ q) r *2.31
        ((¬ p)  (¬ q)) r *2.53
        applySyllogism
))

thm (import () ((H (p  (q  r)))) ((p  q)  r) (
        H
        p q r Importation
        applyModusPonens
))

We can use importation to prove syllogisms in conjunction form:

thm (ImplicationTransitivity () () (((p  q)  (q  r))  (p  r)) ( # *3.33
        p q r CommonConsequentAddition
        import
))

thm (*3.34 () () (((q  r)  (p  q))  (p  r)) (
       q r p CommonAntecedentAddition
       import
))

Import and export also give us another transposition theorem, (which we prove in several steps):

thm (*3.37-1 () () ((p  (q  r))  (p  ((¬ r)  (¬ q)))) (
        q r *2.16
        p addCommonAntecedent
))

thm (*3.37 () () (((p  q)  r)  ((p  (¬ r))  (¬ q))) (
        p q r Exportation
        p q r *3.37-1
        applySyllogism

        p (¬ r) (¬ q) Importation
        applySyllogism
))

Composition[edit]

Next, we prove Comp, the principle of composition,

thm (Comp () () (((p  q)  (p  r))  (p  (q  r))) ( # *3.43
        q r ConjunctionRightIntroduction
        p addCommonAntecedent
        p r (q  r) *2.77
        applySyllogism
        import
))

thm (composeConjunction () ((H1 (p  q)) (H2 (p  r))) (p  (q  r)) (
        H1 H2 introduceConjunction
        p q r Comp
        applyModusPonens
))

Disjunction composition[edit]

Conjunction composition has an analogue for disjunction ( ), which we prove in several steps:

thm (*3.44-1 () () ((((¬ q)  r)  (r  p))  ((q  p)  p)) (
        (¬ q) r p ImplicationTransitivity
        q p *2.6
        applySyllogism
))

thm (*3.44-2 () () (((¬ q)  r)  (((q  p)  (r  p))  p)) (
        q r p *3.44-1
        export

        (r  p) (q  p) p Comm
        applySyllogism

        (q  p) (r  p) p Importation
        applySyllogism
))

thm (*3.44 () () (((q  p)  (r  p))  ((q  r)  p)) (
        q r p *3.44-2

        applyComm

        q r *2.53
        p addCommonConsequent

        applySyllogism
))

thm (composeDisjunction () ((HQP (q  p)) (HRP (r  p)))
  ((q  r)  p) (
        HQP HRP introduceConjunction
        q p r *3.44
        applyModusPonens
))

Principle of the factor[edit]

Finally, we prove Peano's principle of the factor, called Fact by Whitehead and Russell. It complements the Sum axiom and its companion theorems. Two consequences are partial builder theorems for conjunction and disjunction.

thm (Fact () () ((p  q)  ((p  r)  (q  r))) ( # *3.45
        p q (¬ r) CommonConsequentAddition
        (q  (¬ r)) (p  (¬ r)) *2.16
        applySyllogism
))

thm (ConjunctionMultiplication () () (((p  r)  (q  s))  ((p  q)  (r  s))) ( # *3.47

We prove this theorem in two steps. First we use the left hand side of the antecedent to deduce :

        (p  r) (q  s) ConjunctionRightElimination
        p r q Fact
        applySyllogism
        r q *3.22
        (p  q) introduceAntecedent distributeAntecedent
        applySyllogism

Now we do the same with the right hand side of the antecedent to get . The theorem then follows from a syllogism in the consequent.

        (p  r) (q  s) ConjunctionLeftElimination
        q s r Fact
        applySyllogism
        s r *3.22
        (q  r) introduceAntecedent distributeAntecedent
        applySyllogism
        applySyllogismInConsequent
))

thm (conjoin () ((H1 (p  r)) (H2 (q  s))) ((p  q)  (r  s)) (
        H1 H2 introduceConjunction
        p r q s ConjunctionMultiplication
        applyModusPonens
))

thm (DisjunctionSummation () () (((p  r)  (q  s))  ((p  q)  (r  s))) ( # *3.48

This theorem follows exactly as ConjunctionMultiplication, except that we use the Sum type theorem DisjunctionSummationRL instead of Fact and .

        (p  r) (q  s) ConjunctionRightElimination
        p r q DisjunctionSummationRL
        applySyllogism
        (p  r) (q  s) ConjunctionLeftElimination
        q s r DisjunctionSummationRL
        applySyllogism
        applySyllogismInConsequent
))

thm (disjoin () ((H1 (p  r)) (H2 (q  s))) ((p  q)  (r  s)) (
        H1 H2 introduceConjunction
        p r q s DisjunctionSummation
        applyModusPonens
))

Biconditional[edit]

In Principia, the biconditional is defined simply as , so our combination and elimination theorems for conjunction immediately yield the corresponding introduction and elimination rules for the biconditional.

thm (BiconditionalForwardElimination () () ((p  q)  (q  p)) (
        (p  q) (q  p) ConjunctionLeftElimination
))

thm (BiconditionalReverseElimination () () ((p  q)  (p  q)) (
        (p  q) (q  p) ConjunctionRightElimination
))

thm (introduceBiconditionalFromImplications () ((H1 (p  q)) (H2 (q  p))) (p  q) (
        H1 H2 introduceConjunction
))

thm (eliminateBiconditionalForward () ((H (p  q))) (q  p) (
        H eliminateLeftConjunct
))

thm (eliminateBiconditionalReverse () ((H (p  q))) (p  q) (
        H eliminateRightConjunct
))

Negation[edit]

In order to prove the negation function theorem , we combine the two transposition laws and :

thm (NegationFunction () () ((p  q)  ((¬ p)  (¬ q))) ( # *4.11
        p q *2.16
        q p *2.16
        conjoin

This gives us , the right hand side of which must be permuted:

        ((¬ q)  (¬ p)) ((¬ p)  (¬ q)) *3.22
        applySyllogism

Now the same again for :

        p q *2.17
        q p *2.17
        conjoin
        (q  p) (p  q) *3.22
        applySyllogism
        introduceBiconditionalFromImplications
))

thm (addNegation () ((H (p  q))) ((¬ p)  (¬ q)) (
        H
        p q NegationFunction
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (removeNegation () ((H ((¬ p)  (¬ q)))) (p  q) (
        H
        p q NegationFunction
        eliminateBiconditionalForward
        applyModusPonens
))

Biconditionalized transposition laws[edit]

We proved various transposition laws for the conditional earlier. Now we provide some for the biconditional, starting with . The proof is a straightforward application of the conditional transposition laws, but has to prove each direction of the biconditionals.

thm (*4.12-forward () () ((p  (¬ q))  (q  (¬ p))) (
        p (¬ q) BiconditionalReverseElimination
        p q *2.03 applySyllogism
))

thm (*4.12-reverse () () ((p  (¬ q))  ((¬ p)  q)) (
        p (¬ q) BiconditionalForwardElimination
        q p *2.15 applySyllogism
))

thm (BiconditionalTranspositionWithNegatedRight () () ((p  (¬ q))  (q  (¬ p))) ( # *4.12
        p q *4.12-forward
        p q *4.12-reverse
        composeConjunction

        q p *4.12-forward
        q p *4.12-reverse
        composeConjunction

        introduceBiconditionalFromImplications
))

thm (transposeBiconditionalWithNegatedRight () ((H (p  (¬ q)))) (q  (¬ p)) (
        H
        p q BiconditionalTranspositionWithNegatedRight eliminateBiconditionalReverse
        applyModusPonens
))

Double negation[edit]

thm (DoubleNegation () () (p  (¬ (¬ p))) ( # *4.13
        p *2.12
        p *2.14
        introduceBiconditionalFromImplications
))

Some more deduction tools[edit]

The next two theorems, and , enable making some deductions involving biconditionals and conjunctions. They are similar to but extend it.

The idea behind the proof of the converse of below is quite simple: becomes its own converse (modulo some double negation) when substituted with some negated terms. The rest of the proof is just getting rid of the double negation.

thm (converse-of-*3.37 () () (((p  (¬ r))  (¬ q))  ((p  q)  r)) (
        p (¬ r) (¬ q) *3.37
        # (((p ∧ (¬ r)) → (¬ q)) → ((p ∧ (¬ (¬ q))) → (¬ (¬ r))))


        p (¬ (¬ q)) (¬ (¬ r)) Exportation
        # ((p ∧ (¬ (¬ q))) → (¬ (¬ r)))) → (p → ((¬ (¬ q)) → (¬ (¬ r))))

        (¬ q) (¬ r) *2.17
        r q *2.17
        applySyllogism
        # ((¬ (¬ q)) → (¬ (¬ r))) → (q → r)

        p addCommonAntecedent
        # (p → ((¬ (¬ q)) → (¬ (¬ r)))) → (p → (q → r))


        applySyllogism

        p q r Importation
        applySyllogism

        applySyllogism
))

thm (*4.14 () () (((p  q)  r)  ((p  (¬ r))  (¬ q))) (
        p q r *3.37
        p r q converse-of-*3.37
        introduceBiconditionalFromImplications
))

The next theorem, , is similar. Until we have built up more of the biconditional machinery, it will be easier to prove each implication separately. The proof is a straightforward substitution together with a commutation of the initial . As with the previous proof, most of the length of the proof consists of building up formulas to handle things like removing deeply nested double negation, a process which will get (somewhat) easier later.

thm (*4.15-1 () () (((p  q)  r)  ((q  p)  r)) (
        q p *3.22
        r addCommonConsequent
))

thm (*4.15-2 () () (((p  q)  (¬ r))  ((q  (¬ (¬ r)))  (¬ p))) (
        p q (¬ r) *4.15-1
        q p (¬ r) *4.14 eliminateBiconditionalReverse
        applySyllogism
))

thm (*4.15-forward () () (((p  q)  (¬ r))  ((q  r)  (¬ p))) (
        p q r *4.15-2

        q (¬ (¬ r)) (¬ p) Exportation applySyllogism

        r *2.12
        (¬ p) addCommonConsequent

        q addCommonAntecedent

        applySyllogism

        q r (¬ p) Importation applySyllogism
))

thm (*4.15-2-reverse () () (((q  (¬ (¬ r)))  (¬ p))  ((p  q)  (¬ r))) (
        q p (¬ r) *4.14 eliminateBiconditionalForward
        q p (¬ r) *4.15-1
        applySyllogism
))

thm (*4.15-reverse () () (((q  r)  (¬ p))  ((p  q)  (¬ r))) (
        q r (¬ p) Exportation

        r *2.14
        (¬ p) addCommonConsequent

        q addCommonAntecedent

        applySyllogism

        q (¬ (¬ r)) (¬ p) Importation applySyllogism

        q r p *4.15-2-reverse
        applySyllogism
))

thm (*4.15 () () (((p  q)  (¬ r))  ((q  r)  (¬ p))) (
        p q r *4.15-forward
        q r p *4.15-reverse
        introduceBiconditionalFromImplications
))

Reflexive, symmetric, and transitive[edit]

The biconditional has these three properties (which correspond to those defining an equivalence relation).

thm (BiconditionalReflexivity () () (p  p) ( # *4.2

        p Id
        p Id
        introduceBiconditionalFromImplications
))

thm (BiconditionalSymmetryImplication () () ((p  q)  (q  p)) (
        (p  q) (q  p) *3.22
))

thm (BiconditionalSymmetry () () ((p  q)  (q  p)) ( # *4.21

        p q BiconditionalSymmetryImplication
        q p BiconditionalSymmetryImplication
        introduceBiconditionalFromImplications
))

thm (swapBiconditional () ((H (p  q))) (q  p) (
        H
        p q BiconditionalSymmetryImplication
        applyModusPonens
))

The proof of biconditional transitivity is somewhat more complicated. The antecedent contains essentially four disjunctions factors. Each of them has to be picked out and applied:

thm (BiconditionalTransitivity () () (((p  q)  (q  r))  (p  r)) ( # *4.22

First factor:

        (p  q) (q  r) ConjunctionRightElimination
        (p  q) (q  p) ConjunctionRightElimination
        applySyllogism

Third factor:

        (p  q) (q  r) ConjunctionLeftElimination
        (q  r) (r  q) ConjunctionRightElimination
        applySyllogism
        applySyllogismInConsequent

Fourth factor:

        (p  q) (q  r) ConjunctionLeftElimination
        (q  r) (r  q) ConjunctionLeftElimination
        applySyllogism

Second factor:

        (p  q) (q  r) ConjunctionRightElimination
        (p  q) (q  p) ConjunctionLeftElimination
        applySyllogism
        applySyllogismInConsequent
        composeConjunction
))

thm (applyBiconditionalTransitivity () ((H1 (p  q)) (H2 (q  r))) (p  r) (
        H1 H2 introduceConjunction
        p q r BiconditionalTransitivity
        applyModusPonens
))

Additional biconditional theorems[edit]

Another easy builder theorem:

thm (*4.37 () () ((p  q)  ((p  r)  (q  r))) (
        p q r DisjunctionSummationRR
        q p r DisjunctionSummationRR
        conjoin
))

thm (buildRightDisjunction () ((H (p  q))) ((p  r)  (q  r)) (
        H
        p q r *4.37
        applyModusPonens
))

More theorems stated using the biconditional[edit]

Some more theorems where we have proved implications in both directions, but can now express them using the biconditional:

thm (AntecedentDistribution () () ((p  (q  r))  ((p  q)  (p  r))) ( # *5.41

        p q r *2.77
        p q r *2.86
        introduceBiconditionalFromImplications
))

thm (Transposition () () ((p  q)  ((¬ q)  (¬ p))) ( # *4.1

        p q *2.16
        q p *2.17
        introduceBiconditionalFromImplications
))

thm (TranspositionWithNegatedAntecedent () () (((¬ p)  q)  ((¬ q)  p)) (
        p q *2.15
        q p *2.15
        introduceBiconditionalFromImplications
))

thm (TranspositionWithNegatedConsequent () () ((p  (¬ q))  (q  (¬ p))) (
        p q *2.03
        q p *2.03
        introduceBiconditionalFromImplications
))

thm (Transportation () () ((p  (q  r))  ((p  q)  r)) (
        p q r Importation
        p q r Exportation
        introduceBiconditionalFromImplications
))

Algebraic laws for disjunction and conjunction[edit]

Some of the theorems of the propositional calculus can be thought of as analagous to those of other algebras, showing properties such as commutivity and associativity. Although Whitehead and Russell think this concept was overemphasized in their day,[2] they do provide theorems which represent algebraic properties.

Idempotence[edit]

Idempotence for disjunction and conjunction are perhaps the most interesting, as they cause the biggest differences between this algebra and many other algebras:[3]

thm (DisjunctionIdempotence () () (p  (p  p)) ( # *4.25
        p p Add
        p Taut
        introduceBiconditionalFromImplications
))

To prove conjunction idempotence, we first catch up on a few basic implication theorems we haven't needed until now:

thm (*2.4 () () ((p  (p  q))  (p  q)) (
        p p q *2.31

That gives us and we just need to eliminate the extra :

        p Taut
        q disjoinRR

        applySyllogism
))

thm (*2.43 () () ((p  (p  q))  (p  q)) (
        (¬ p) q *2.4
))

thm (ConjunctionIdempotence () () (p  (p  p)) ( # *4.24
        p p ConjunctionRightIntroduction
        p (p  p) *2.43
        applyModusPonens

        p p ConjunctionRightElimination

        introduceBiconditionalFromImplications
))

Idempotence is also expressed in the following rules.

thm (cloneAsDisjunction () ((H p)) (p  p) (
        H
        p p Add
        applyModusPonens
))

thm (conflateDisjunction () ((H (p  p))) p (
        H
        p Taut
        applyModusPonens
))

thm (cloneAsConjunction () ((H p)) (p  p) (
        H
        p ConjunctionIdempotence eliminateBiconditionalReverse
        applyModusPonens
))

thm (conflateConjunction () ((H (p  p))) p (
        H
        p ConjunctionIdempotence eliminateBiconditionalForward
        applyModusPonens
))

Commutativity[edit]

We already have commutativity of disjunction and conjunction, but just need to express them using the biconditional:

thm (DisjunctionCommutativity () () ((p  q)  (q  p)) ( # *4.31

        p q Perm
        q p Perm
        introduceBiconditionalFromImplications
))

thm (ConjunctionCommutativity () () ((p  q)  (q  p)) ( # *4.3

        p q *3.22
        q p *3.22
        introduceBiconditionalFromImplications
))

Associativity[edit]

Both disjunction and conjunction are associative:

thm (DisjunctionAssociativity () () (((p  q)  r)  (p  (q  r))) ( # *4.33
        p q r *2.32
        p q r *2.31
        introduceBiconditionalFromImplications
))

The link between (which has some implications and negations) and the conjunctions in ConjunctionAssociativity may not be apparent, but follows from the definitions of conjunction and implication.

thm (ConjunctionAssociativity () () (((p  q)  r)  (p  (q  r))) ( # *4.32
        p q r *4.15
        addNegation

        (q  r) p ConjunctionCommutativity

        applyBiconditionalTransitivity
))

We already provided rules for associating disjunctions; here are the corresponding ones for conjunctions:

thm (groupConjunctionRight () ((H ((p  q)  r))) (p  (q  r)) (
        H
        p q r ConjunctionAssociativity
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (groupConjunctionLeft () ((H (p  (q  r)))) ((p  q)  r) (
        H
        p q r ConjunctionAssociativity
        eliminateBiconditionalForward
        applyModusPonens
))

Substitution and builders[edit]

If , then we want to be able to substitute for in a theorem to get a new theorem.[4] The mechanism which we are working towards, in Interface:Classical propositional calculus, is provided by addNegation, removeNegation, buildImplication, buildDisjunction, buildConjunction, and buildBiconditional. Those rules do not eliminate the need for a proof to build up the expressions embodying the substitution, but they reduce the process of constructing such a proof to a familiar (if perhaps tedious) pattern. We already proved addNegation and removeNegation, and we're now ready to prove the rest.

Conjunction[edit]

The proof proceeds by expanding into four implications, rearranging them using associativity and commutativity, and applying ConjunctionMultiplication to each half.

First, the rearrangement we need is ConjunctionFunction-1, known as an4 in metamath.[5]

thm (ConjunctionFunction-1 () () (((p  q)  (r  s))  ((p  r)  (q  s))) (
        p q (r  s) ConjunctionAssociativity
        eliminateBiconditionalReverse

        p Id
        q r s ConjunctionAssociativity
        eliminateBiconditionalForward
        conjoin

        applySyllogism

        p Id
        q r ConjunctionCommutativity
        eliminateBiconditionalReverse
        s Id 
        conjoin
        conjoin

        applySyllogism

        p Id
        r q s ConjunctionAssociativity
        eliminateBiconditionalReverse
        conjoin

        applySyllogism

        p r (q  s) ConjunctionAssociativity
        eliminateBiconditionalForward

        applySyllogism
))

thm (ConjunctionFunction () () (((p  q)  (r  s))  ((p  r)  (q  s))) ( # *4.38
        (p  q) (q  p) (r  s) (s  r) ConjunctionFunction-1
        
        p q r s ConjunctionMultiplication
        q p s r ConjunctionMultiplication
        conjoin

        applySyllogism
))

thm (buildConjunction () (
  (H1 (p  q))
  (H2 (r  s)))
  ((p  r)  (q  s)) (
        H1 H2 introduceConjunction
        p q r s ConjunctionFunction
        applyModusPonens
))

Disjunction[edit]

This is just like the theorem for conjunction, except that we build on the partial builder theorem DisjunctionSummation instead of ConjunctionMultiplication:

thm (DisjunctionFunction () () (((p  q)  (r  s))  ((p  r)  (q  s))) ( # *4.39
        (p  q) (q  p) (r  s) (s  r) ConjunctionFunction-1
        
        p q r s DisjunctionSummation
        q p s r DisjunctionSummation
        conjoin

        applySyllogism
))

thm (buildDisjunction () (
  (H1 (p  q))
  (H2 (r  s)))
  ((p  r)  (q  s)) (
        H1 H2 introduceConjunction
        p q r s DisjunctionFunction
        applyModusPonens
))

Implication[edit]

The builder for implication is a simple consequence of the builder for disjunction together with the equivalence of and .

thm (ImplicationFunction () () (((p  q)  (r  s))  ((p  r)  (q  s))) (
        p q NegationFunction
        (r  s) BiconditionalReflexivity
        buildConjunction

        eliminateBiconditionalReverse

        (¬ p) (¬ q) r s DisjunctionFunction

        applySyllogism
))

thm (buildImplication () ((HPQ (p  q)) (HRS (r  s))) 
  ((p  r)  (q  s)) (
        HPQ HRS introduceConjunction
        p q r s ImplicationFunction
        applyModusPonens
))

Biconditional[edit]

To prove the biconditional builder, we need . As this is an equivalence of conjunctions, we'll get it with the conjunction builder. The equivalences needed to apply the conjunction builder will come from the implication builder (and conjunction commutativity in one of the two directions).

thm (BiconditionalFunction () () (((p  q)  (r  s))  ((p  r)  (q  s))) (
        p q r s ImplicationFunction

        (p  q) (r  s) *3.22
        r s p q ImplicationFunction
        applySyllogism

        composeConjunction

        (p  r) (q  s) (r  p) (s  q) 
        ConjunctionFunction

        applySyllogism
))

thm (buildBiconditional () ((HPQ (p  q)) (HRS (r  s))) ((p  r)  (q  s)) (
        HPQ HRS introduceConjunction
        p q r s BiconditionalFunction
        applyModusPonens
))

Unidirectional builders[edit]

The builders which we just proved start with biconditionals. If we only have implications, there is a similar set of builders (which, of course, only provide implications, not biconditionals, in the consequent). Here we summarize the ones we have already proved, and prove a few more.

Negation[edit]

The Transposition theorem, , is a unidirectional negation builder.

Disjunction[edit]

The disjunction summation theorem, , is the general form of the unidirectional disjunction builder.

We also provide convenience theorems for cases in which one of the implications is simply , and where there might be a commutation in one of the disjunctions. All of these are already proved (DisjunctionSummationLR, disjoinLR, and so on), so the only thing we need to do here is provide DisjunctionSummationLL as a new name for the Sum axiom.

thm (DisjunctionSummationLL () () ((p  q)  ((r  p)  (r  q))) (
        p q r Sum
))

Conjunction[edit]

The conjunction multiplication theorem, , is the general form of the unidirectional disjunction builder.

Here we add convenience theorems for cases in which one of the implications is simply , and where there might be a commutation in one of the conjunctions.

ConjunctionMultiplicationRR is just a new name for the Fact theorem; the rest could either be proved from Fact and commutativity, or as special cases of ConjunctionMultiplication.

thm (ConjunctionMultiplicationLL () () ((p  q)  ((r  p)  (r  q))) (
        r ImplicationReflexivity

        r r p q ConjunctionMultiplication
        export

        applyModusPonens
))

thm (ConjunctionMultiplicationLR () () ((p  q)  ((r  p)  (q  r))) (
        p q r ConjunctionMultiplicationLL

        r q ConjunctionCommutativity eliminateBiconditionalReverse
        (p  q) introduceAntecedent

        applySyllogismInConsequent
))

thm (ConjunctionMultiplicationRR () () ((p  q)  ((p  r)  (q  r))) (
        p q r Fact
))

thm (ConjunctionMultiplicationRL () () ((p  q)  ((p  r)  (r  q))) (
        p q r ConjunctionMultiplicationRR

        q r ConjunctionCommutativity eliminateBiconditionalReverse
        (p  q) introduceAntecedent

        applySyllogismInConsequent
))

thm (conjoinLL () ((H (p  q))) ((r  p)  (r  q)) (
        H
        p q r ConjunctionMultiplicationLL
        applyModusPonens
))

thm (conjoinLR () ((H (p  q))) ((r  p)  (q  r)) (
        H
        p q r ConjunctionMultiplicationLR
        applyModusPonens
))

thm (conjoinRL () ((H (p  q))) ((p  r)  (r  q)) (
        H
        p q r ConjunctionMultiplicationRL
        applyModusPonens
))

thm (conjoinRR () ((H (p  q))) ((p  r)  (q  r)) (
        H
        p q r ConjunctionMultiplicationRR
        applyModusPonens
))

Implication[edit]

The first two unidirectional implication builders came early on, CommonAntecedentAddition and CommonConsequentAddition.

The general unidirectional implication builder would be .[6]

Biconditional[edit]

There is no unidirectional builder for the biconditional.

Distributive laws[edit]

We prove two distributive laws. The first one, , is analogous to the distributive law in well-known algebras such as the real numbers, (if one thinks of conjunction as being like multiplication and disjunction as being like addition).

thm (ConjunctionLeftDistribution-forward () () 
  ((p  (q  r))  ((p  q)  (p  r))) (
        p q ConjunctionRightIntroduction
        p r ConjunctionRightIntroduction
        composeConjunction

        q (p  q) r (p  r) DisjunctionSummation
        applySyllogism

        import
))

thm (ConjunctionLeftDistribution-reverse () ()
  (((p  q)  (p  r))  (p  (q  r))) (
        p q ConjunctionRightElimination
        p r ConjunctionRightElimination
        composeDisjunction

        p q ConjunctionLeftElimination
        p r ConjunctionLeftElimination
        disjoin

        composeConjunction
))

thm (ConjunctionLeftDistribution () () ((p  (q  r))  ((p  q)  (p  r))) ( # *4.4
        p q r ConjunctionLeftDistribution-forward
        p q r ConjunctionLeftDistribution-reverse
        introduceBiconditionalFromImplications
))

The second distributive law, , has no analogue in ordinary algebra.[7]

thm (DisjunctionLeftDistribution-forward () () ((p  (q  r))  ((p  q)  (p  r))) (
        q r ConjunctionRightElimination
        p disjoinLL

        q r ConjunctionLeftElimination
        p disjoinLL

        composeConjunction
))

thm (DisjunctionLeftDistribution-reverse () () (((p  q)  (p  r))  (p  (q  r))) (
        p q *2.53
        p r *2.53
        conjoin

        (¬ p) q r Comp
        applySyllogism

        p (q  r) *2.54
        applySyllogism
))

thm (DisjunctionLeftDistribution () () ((p  (q  r))  ((p  q)  (p  r))) ( # *4.41
        p q r DisjunctionLeftDistribution-forward
        p q r DisjunctionLeftDistribution-reverse
        introduceBiconditionalFromImplications
))

We also supply commuted versions of both laws and some rules:

thm (DisjunctionRightDistribution () () (((p  q)  r)  ((p  r)  (q  r))) (
        (p  q) r DisjunctionCommutativity
        r p q DisjunctionLeftDistribution
        applyBiconditionalTransitivity

        r p DisjunctionCommutativity
        r q DisjunctionCommutativity
        buildConjunction

        applyBiconditionalTransitivity
))

thm (ConjunctionRightDistribution () () (((p  q)  r)  ((p  r)  (q  r))) (
        (p  q) r ConjunctionCommutativity
        r p q ConjunctionLeftDistribution
        applyBiconditionalTransitivity

        r p ConjunctionCommutativity
        r q ConjunctionCommutativity
        buildDisjunction

        applyBiconditionalTransitivity
))

thm (distributeLeftDisjunction () ((H (p  (q  r)))) ((p  q)  (p  r)) (
        H
        p q r DisjunctionLeftDistribution
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (collectLeftDisjunction () ((H ((p  q)  (p  r)))) (p  (q  r)) (
        H
        p q r DisjunctionLeftDistribution
        eliminateBiconditionalForward
        applyModusPonens
))

thm (distributeRightDisjunction () ((H ((p  q)  r))) ((p  r)  (q  r)) (
        H
        p q r DisjunctionRightDistribution
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (collectRightDisjunction () ((H ((p  r)  (q  r)))) ((p  q)  r) (
        H
        p q r DisjunctionRightDistribution
        eliminateBiconditionalForward
        applyModusPonens
))

thm (distributeLeftConjunction () ((H (p  (q  r)))) ((p  q)  (p  r)) (
        H
        p q r ConjunctionLeftDistribution
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (collectLeftConjunction () ((H ((p  q)  (p  r)))) (p  (q  r)) (
        H
        p q r ConjunctionLeftDistribution
        eliminateBiconditionalForward
        applyModusPonens
))

thm (distributeRightConjunction () ((H ((p  q)  r))) ((p  r)  (q  r)) (
        H
        p q r ConjunctionRightDistribution
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (collectRightConjunction () ((H ((p  r)  (q  r)))) ((p  q)  r) (
        H
        p q r ConjunctionRightDistribution
        eliminateBiconditionalForward
        applyModusPonens
))

De Morgan's laws[edit]

Since the definition of conjunction in Principia is based on De Morgan's laws, the laws themselves are not hard to prove.

thm (DeMorganPDP () () ((¬ (p  q))  ((¬ p)  (¬ q))) ( # *4.56

        p DoubleNegation
        q DoubleNegation
        buildDisjunction
        addNegation
))

thm (DeMorganPDN () () ((¬ (p  (¬ q)))  ((¬ p)  q)) ( # *4.54
        p DoubleNegation
        (¬ q) buildRightDisjunction
        addNegation
))

thm (DeMorganNDP () () ((¬ ((¬ p)  q))  (p  (¬ q))) ( # *4.52
        (¬ p) BiconditionalReflexivity
        q DoubleNegation
        buildDisjunction
        addNegation
))

thm (DeMorganNDN () () ((¬ ((¬ p)  (¬ q)))  (p  q)) ( # *4.5
        (¬ ((¬ p)  (¬ q))) BiconditionalReflexivity
))

thm (DeMorganPCP () () ((¬ (p  q))  ((¬ p)  (¬ q))) ( # *4.51
        ((¬ p)  (¬ q)) DoubleNegation
        swapBiconditional
))

thm (DeMorganPCN () () ((¬ (p  (¬ q)))  ((¬ p)  q)) ( # *4.53
        (¬ p) BiconditionalReflexivity
        q DoubleNegation
        buildDisjunction

        ((¬ p)  (¬ (¬ q))) DoubleNegation
        applyBiconditionalTransitivity
        swapBiconditional
))

thm (DeMorganNCP () () ((¬ ((¬ p)  q))  (p  (¬ q))) ( # *4.55
        p DoubleNegation
        (¬ q) BiconditionalReflexivity
        buildDisjunction

        ((¬ (¬ p))  (¬ q)) DoubleNegation
        applyBiconditionalTransitivity
        swapBiconditional
))

thm (DeMorganNCN () () ((¬ ((¬ p)  (¬ q)))  (p  q)) ( # *4.57

        p DoubleNegation
        q DoubleNegation
        buildDisjunction

        ((¬ (¬ p))  (¬ (¬ q))) DoubleNegation
        applyBiconditionalTransitivity
        swapBiconditional
))

thm (distributeNegationPDP () ((H (¬ (p  q)))) ((¬ p)  (¬ q)) (
        H
        p q DeMorganPDP eliminateBiconditionalReverse
        applyModusPonens
))

thm (distributeNegationPDN () ((H (¬ (p  (¬ q))))) ((¬ p)  q) (
        H
        p q DeMorganPDN eliminateBiconditionalReverse
        applyModusPonens
))

thm (distributeNegationNDP () ((H (¬ ((¬ p)  q)))) (p  (¬ q)) (
        H
        p q DeMorganNDP eliminateBiconditionalReverse
        applyModusPonens
))

thm (distributeNegationNDN () ((H (¬ ((¬ p)  (¬ q))))) (p  q) (
        H
        p q DeMorganNDN eliminateBiconditionalReverse
        applyModusPonens
))

thm (distributeNegationPCP () ((H (¬ (p  q)))) ((¬ p)  (¬ q)) (
        H
        p q DeMorganPCP eliminateBiconditionalReverse
        applyModusPonens
))

thm (distributeNegationPCN () ((H (¬ (p  (¬ q))))) ((¬ p)  q) (
        H
        p q DeMorganPCN eliminateBiconditionalReverse
        applyModusPonens
))

thm (distributeNegationNCP () ((H (¬ ((¬ p)  q)))) (p  (¬ q)) (
        H
        p q DeMorganNCP eliminateBiconditionalReverse
        applyModusPonens
))

thm (distributeNegationNCN () ((H (¬ ((¬ p)  (¬ q))))) (p  q) (
        H
        p q DeMorganNCN eliminateBiconditionalReverse
        applyModusPonens
))

thm (collectNegationPDP () ((H (p  q))) (¬ ((¬ p)  (¬ q))) (
        H
        p q DeMorganNCN eliminateBiconditionalForward
        applyModusPonens
))

thm (collectNegationPDN () ((H (p  (¬ q)))) (¬ ((¬ p)  q)) (
        H
        p q DeMorganNCP eliminateBiconditionalForward
        applyModusPonens
))

thm (collectNegationNDP () ((H ((¬ p)  q))) (¬ (p  (¬ q))) (
        H
        p q DeMorganPCN eliminateBiconditionalForward
        applyModusPonens
))

thm (collectNegationNDN () ((H ((¬ p)  (¬ q)))) (¬ (p  q)) (
        H
        p q DeMorganPCP eliminateBiconditionalForward
        applyModusPonens
))

thm (collectNegationPCP () ((H (p  q))) (¬ ((¬ p)  (¬ q))) (
        H
        p q DeMorganNDN eliminateBiconditionalForward
        applyModusPonens
))

thm (collectNegationPCN () ((H (p  (¬ q)))) (¬ ((¬ p)  q)) (
        H
        p q DeMorganNDP eliminateBiconditionalForward
        applyModusPonens
))

thm (collectNegationNCP () ((H ((¬ p)  q))) (¬ (p  (¬ q))) (
        H
        p q DeMorganPDN eliminateBiconditionalForward
        applyModusPonens
))

thm (collectNegationNCN () ((H ((¬ p)  (¬ q)))) (¬ (p  q)) (
        H
        p q DeMorganPDP eliminateBiconditionalForward
        applyModusPonens
))

Biconditional and conjunction[edit]

A true conjunct does not affect the truth of a proposition, or in symbols q → (p ↔ p ∧ q).

Before we prove that statement itself, we prove two equivalences involving implications and conjunctions. We will be using them in the proof of the result stated above.

thm (*4.7 () () ((p  q)  (p  (p  q))) (

We start with the forward implication. We stick p → p on the proof stack, and then start with (p → p) ∧ (p → q) → (p → p ∧ q)

        p ImplicationReflexivity
        p p q Comp

Exporting and detaching p → p finishes the forward implication.

        export
        applyModusPonens

The reverse direction is even easier.

        p q ConjunctionLeftElimination
        p addCommonAntecedent

        introduceBiconditionalFromImplications
))

thm (*4.71 () () ((p  q)  (p  (p  q))) (

We first stick two things on the stack for later use.

        p q *4.7

        p q ConjunctionRightElimination

Now we start with (p ∧ q → p) → ((p → p ∧ q) → (p ↔ p ∧ q)), and then detach the antecedent (a theorem) to give (p → p ∧ q) → (p ↔ p ∧ q)

        ((p  q)  p) (p  (p  q)) ConjunctionLeftIntroduction
        applyModusPonens

The converse of that statement, (p ↔ p ∧ q) → (p → p ∧ q), is even simpler. So combining the two, we get (p → p ∧ q) ↔ (p ↔ p ∧ q)

        p (p  q) BiconditionalReverseElimination
        introduceBiconditionalFromImplications

The only thing left is to combine with (p → q) ↔ (p → p ∧ q) which we left on the proof stack.

        applyBiconditionalTransitivity
))

thm (BiconditionalConjunct () () (q  (p  (p  q))) ( # *4.73
        q p AntecedentIntroduction

        p q *4.71
        eliminateBiconditionalReverse

        applySyllogism
))

Biconditionalized composition[edit]

We earlier proved composition laws for disjunction and conjunction:

The converses, while less interesting, are also true, and we prove them now.

thm (DisjunctionComposition () () 
  (((p  r)  (q  r))  ((p  q)  r)) ( # *4.77

        p r q *3.44

        p q DisjunctionRightIntroduction
        r addCommonConsequent

        q p DisjunctionLeftIntroduction
        r addCommonConsequent

        composeConjunction

        introduceBiconditionalFromImplications
))

thm (ConjunctionComposition () () 
  (((p  q)  (p  r))  (p  (q  r))) ( # *4.76

        p q r Comp

        q r ConjunctionRightElimination
        p addCommonAntecedent

        q r ConjunctionLeftElimination
        p addCommonAntecedent

        composeConjunction

        introduceBiconditionalFromImplications
))

thm (extractLeftDisjunction () ((H ((p  q)  r))) (p  r) (
        p q DisjunctionRightIntroduction
        H
        applySyllogism
))

thm (extractRightDisjunction () ((H ((p  q)  r))) (q  r) (
        q p DisjunctionLeftIntroduction
        H
        applySyllogism
))

thm (extractLeftConjunction () ((H (p  (q  r)))) (p  q) (
        H
        q r ConjunctionRightElimination
        applySyllogism
))

thm (extractRightConjunction () ((H (p  (q  r)))) (p  r) (
        H
        q r ConjunctionLeftElimination
        applySyllogism
))

Weakening of biconditional to disjunction[edit]

We've already dealt with weakening the biconditional to an implication. This section just has the same theorems, or slight variations thereof, phrased in terms of disjunction instead of implication.

thm (BiconditionalDisjunctionLeftElimination () () ((p  q)  (p  (¬ q))) (
        p q BiconditionalForwardElimination
        (¬ q) p DisjunctionCommutativity eliminateBiconditionalReverse
        applySyllogism
))

thm (BiconditionalDisjunctionRightElimination () () ((p  q)  ((¬ p)  q)) (
        p q BiconditionalReverseElimination
))

thm (eliminateLeftBiconditionalDisjunction () ((H (p  q))) (p  (¬ q)) (
        H
        p q BiconditionalDisjunctionLeftElimination
        applyModusPonens
))

thm (eliminateRightBiconditionalDisjunction () ((H (p  q))) ((¬ p)  q) (
        H
        p q BiconditionalDisjunctionRightElimination
        applyModusPonens
))

Case Elimination[edit]

Proofs often show that one of several cases must apply, and then prove the desired proposition for each case. Here we provide one form of this, where there are two cases: and . Note that in principle, it is always possible to reduce the handling of multiple cases to repeated handling of two cases.

# one direction of *4.83; much like *2.61
thm (CaseElimination () () (((p  q)  ((¬ p)  q))  q) (
        p q *2.6
        applyComm
        import
))

thm (eliminateCases () ((HP (p  q)) (HNOTP ((¬ p)  q))) q (
        HP HNOTP introduceConjunction
        p q CaseElimination
        applyModusPonens
))

Modus ponens and modus tollens[edit]

Now that we have import, we can derive the version of the modus ponens law which Interface:Classical propositional calculus expects:

thm (ModusPonens () () ((p  (p  q))  q) ( # *3.35
        p q *2.27
        import
))

Modus tollens is just a combination of modus ponens and transposition.

thm (ModusTollens () () (((¬ q)  (p  q))  (¬ p)) (
        p q Transposition eliminateBiconditionalReverse
        (¬ q) conjoinLL

        (¬ q) (¬ p) ModusPonens

        applySyllogism
))

thm (applyModusTollens () ((H1 (¬ q)) (H2 (p  q))) (¬ p) (
        H1

        H2
        introduceTransposition

        applyModusPonens
))

Tautology and contradiction[edit]

Interface:Classical propositional calculus gives the name Tautology to (p ∨ (¬ p)) ↔ (⊤) and Contradiction to (p ∧ (¬ p)) ↔ (⊥). They are somewhat more subtle than they appear (and in particular are not just trivial consequences of our definitions of and ), because the variable which appears in the definition of or is not the same as the one in the Tautology or Contradiction theorem. We therefore prove them as consequences of the notion that two true statements are equivalent ( ) or that two false statements are equivalent ( ).

thm (*3.4 () () ((p  q)  (p  q)) (
        p q ConjunctionLeftElimination
        q p AntecedentIntroduction
        applySyllogism
))

thm (TruthBiconditional () () ((p  q)  (p  q)) ( # *5.1

        p q *3.4

        p q *3.22
        q p *3.4
        applySyllogism

        composeConjunction
))

thm (Tautology () () ((p  (¬ p))  ()) (
        p TertiumNonDatur
        q TertiumNonDatur
        introduceConjunction

        (p  (¬ p)) (q  (¬ q)) TruthBiconditional
        applyModusPonens
))

thm (FalsityBiconditional () () (((¬ p)  (¬ q))  (p  q)) ( # *5.21

        (¬ p) (¬ q) TruthBiconditional
        p q NegationFunction eliminateBiconditionalForward
        applySyllogism
))

thm (Contradiction () () ((p  (¬ p))  ()) (
        p *3.24
        q *3.24
        introduceConjunction

        (p  (¬ p)) (q  (¬ q)) FalsityBiconditional
        applyModusPonens
))

Along the same lines is .

thm (NegationImplication-forward () () ((¬ p)  (p  ())) (
       p Contradiction eliminateBiconditionalReverse
       export
       applyComm
))

thm (NegationImplication-reverse () () ((p  ())  (¬ p)) (
        p () Transposition eliminateBiconditionalReverse

        NotFalse
        (¬ ()) (¬ p) ModusPonens export
        applyModusPonens

        applySyllogism
))

thm (NegationImplication () () ((¬ p)  (p  ())) (
        p NegationImplication-forward
        p NegationImplication-reverse
        introduceBiconditionalFromImplications
))

thm (convertFromNegationToImplication () ((H (¬ p))) (p  ()) (
        H
        p NegationImplication
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (convertToNegationFromImplication () ((H (p  ()))) (¬ p) (
        H
        p NegationImplication
        eliminateBiconditionalForward
        applyModusPonens
))

Relationships between connectives[edit]

Here we express implication in terms of disjunction, biconditional in terms of implication, etc.

Biconditional and implications[edit]

These are all straightforward because we define the biconditional as a conjunction of two implications.

thm (BiconditionalImplication () () ((p  q)  ((p  q)  (q  p))) (
        (p  q) BiconditionalReflexivity
))

thm (convertFromBiconditionalToImplications () ((H (p  q))) ((p  q)  (q  p)) (
        H
))

thm (convertToBiconditionalFromImplications () ((H ((p  q)  (q  p)))) (p  q) (
        H
))

Biconditional as disjunction of two conjunctions[edit]

One way of looking at is "both p and q are true, or neither are true", or in symbols, . We prove this via a fairly long string of simpler propositions.

thm (Abs () () ((p  (¬ p))  (¬ p)) ( # *2.01
        (¬ p) Taut
))

thm (*4.61 () () ((¬ (p  q))  (p  (¬ q))) (
        p q DeMorganNDP
))

thm (*4.82-forward () () (((p  q)  (p  (¬ q)))  (¬ p)) (
        q *3.24

        p q (¬ q) ConjunctionComposition
        p (q  (¬ q)) Transposition
        applyBiconditionalTransitivity
        eliminateBiconditionalReverse

        applyComm

        applyModusPonens
))

thm (*5.15 () () ((p  q)  (p  (¬ q))) (
        p q *4.61 eliminateBiconditionalReverse
        p (¬ q) TruthBiconditional
        applySyllogism

        (p  q) (p  (¬ q)) *2.54
        applyModusPonens

At this point we have . We'll come back to that, after we prove :

        q p *4.61 eliminateBiconditionalReverse
        q (¬ p) TruthBiconditional
        applySyllogism

        q p BiconditionalTranspositionWithNegatedRight eliminateBiconditionalReverse
        applySyllogism

        (q  p) (p  (¬ q)) *2.54
        applyModusPonens

Now we have and on the proof stack. It is now enough to join those with a conjunction and apply the distributive law:

        introduceConjunction
        collectRightDisjunction
))

thm (*5.16-1 () () (((p  q)  (p  (¬ q)))  (¬ p)) (
        p q BiconditionalReverseElimination
        (p  (¬ q)) conjoinRR
        p q *4.82-forward
        applySyllogism
))

thm (*5.16-2 () () (((p  q)  (p  (¬ q)))  (¬ q)) (
        p q BiconditionalForwardElimination
        (p  (¬ q)) conjoinRR

        q p (¬ q) ImplicationTransitivity
        applySyllogism

        q Abs
        applySyllogism
))

thm (*5.16-3 () () (((p  q)  (p  (¬ q)))  (¬ ((¬ q)  p))) (
        p q *5.16-2
        p q *5.16-1
        composeConjunction

At this point we have , and we just need to apply DeMorgan's law and introduce double negation to get .

        q p DeMorganPDP eliminateBiconditionalForward
        applySyllogism

        q DoubleNegation
        p BiconditionalReflexivity
        buildDisjunction
        addNegation
        eliminateBiconditionalReverse

        applySyllogism
))

thm (*5.16 () () (¬ ((p  q)  (p  (¬ q)))) (
        p q *5.16-3
        export

We're much closer than it may appear (because JHilbert automatically applies definitions), but other than two applications of DeMorgan's law, it is just the definitions of implication and biconditional: from the definition of implication yields

        (p  (¬ q)) ((¬ q)  p) DeMorganPCP
        eliminateBiconditionalForward applySyllogism

Now we have and hence by the definition of the biconditional. A final appeal to the definition of implication yields .

        collectNegationNDN
))

thm (*5.17-1 () () ((p  q)  ((¬ q)  p)) (
        p q DisjunctionCommutativity

        q DoubleNegation
        p BiconditionalReflexivity
        buildDisjunction

        applyBiconditionalTransitivity
))

thm (*5.17-2 () () ((¬ (p  q))  (p  (¬ q))) (
        p q DeMorganPCP
))

thm (*5.17 () () (((p  q)  (¬ (p  q)))  (p  (¬ q))) (
        p q *5.17-1
        p q *5.17-2
        buildConjunction

        (¬ q) p BiconditionalSymmetry

        applyBiconditionalTransitivity
))

One way to see is as a relationship between the biconditional and the exclusive or. One way to express an exclusive or is , and seen this way, states that the biconditional is the negation of the exclusive or:

thm (*5.18 () () ((p  q)  (¬ (p  (¬ q)))) (
        p q *5.15
        p q *5.16
        introduceConjunction

        (p  q) (p  (¬ q)) *5.17
        eliminateBiconditionalReverse applyModusPonens
))

thm (*5.22 () () ((¬ (p  q))  ((p  (¬ q))  (q  (¬ p)))) (
        (p  q) (q  p) DeMorganPCP

At this point we have (applying the definition of implication) and we need (applying the definition of conjunction) . So we just need to fix the double negation.

        (¬ p) BiconditionalReflexivity
        q DoubleNegation
        buildDisjunction
        addNegation

        (¬ q) BiconditionalReflexivity
        p DoubleNegation
        buildDisjunction
        addNegation

        buildDisjunction

        applyBiconditionalTransitivity
))

thm (BiconditionalDisjunction () () ((p  q)  ((p  q)  ((¬ p)  (¬ q)))) ( # *5.23
        p q *5.18
        p (¬ q) *5.22
        applyBiconditionalTransitivity

We now have , so we just need to fix the double negation and commute the second conjunction.

        p BiconditionalReflexivity
        q DoubleNegation swapBiconditional
        buildConjunction

        (¬ q) (¬ p) ConjunctionCommutativity

        buildDisjunction

        applyBiconditionalTransitivity
))

thm (convertFromBiconditionalToDisjunction () ((H (p  q))) ((p  q)  ((¬ p)  (¬ q))) (
        H
        p q BiconditionalDisjunction
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (convertToBiconditionalFromDisjunction () ((H ((p  q)  ((¬ p)  (¬ q))))) (p  q) (
        H
        p q BiconditionalDisjunction
        eliminateBiconditionalForward
        applyModusPonens
))

Biconditional as conjunction of two disjunctions[edit]

This one follows immediately from our definitions of biconditional and implication, and commutativity.

thm (BiconditionalConjunction () () ((p  q)  (((¬ p)  q)  (p  (¬ q)))) (
        ((¬ p)  q) BiconditionalReflexivity
        (¬ q) p DisjunctionCommutativity
        buildConjunction
))

thm (convertFromBiconditionalToConjunction () ((H (p  q))) (((¬ p)  q)  (p  (¬ q))) (
        H
        p q BiconditionalConjunction
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (convertToBiconditionalFromConjunction () ((H (((¬ p)  q)  (p  (¬ q))))) (p  q) (
        H
        p q BiconditionalConjunction
        eliminateBiconditionalForward
        applyModusPonens
))

thm (introduceBiconditionalFromDisjunctions () ((HPQ ((¬ p)  q)) (HQP (p  (¬ q)))) (p  q) (
        HPQ HQP introduceConjunction
        p q BiconditionalConjunction
        eliminateBiconditionalForward
        applyModusPonens
))

Implications and disjunctions[edit]

The relationship between implication and disjunction is just our definition of implication, or an easy consequence thereof.

thm (ImplicationDisjunction () () ((p  q)  ((¬ p)  q)) ( # *4.6

        (p  q) BiconditionalReflexivity
))

thm (DisjunctionImplication () () ((p  q)  ((¬ p)  q)) ( # *4.64

        p DoubleNegation
        q BiconditionalReflexivity
        buildDisjunction
))

thm (convertFromImplicationToDisjunction () ((H (p  q))) ((¬ p)  q) (
        H
))

thm (convertToImplicationFromDisjunction () ((H ((¬ p)  q))) (p  q) (
        H
))

thm (convertFromDisjunctionToImplication () ((H (p  q))) ((¬ p)  q) (
        H
        p q DisjunctionImplication
        eliminateBiconditionalReverse
        applyModusPonens
))

thm (convertToDisjunctionFromImplication () ((H ((¬ p)  q))) (p  q) (
        H
        p q DisjunctionImplication
        eliminateBiconditionalForward
        applyModusPonens
))

Implication distribution over biconditional[edit]

Antecedent distribution says that we can distribute the antecedent in a formula of the form p → (q → r). Here we prove a similar result for p → (q ↔ r).

thm (ImplicationDistributionOverBiconditional () () ((p  (q  r))  ((p  q)  (p  r))) ( # *5.74

We split (p → q) ↔ (p → r) into two implications:

        (p  q) (p  r) BiconditionalImplication

Then we apply AntecedentDistribution to each one,

        p q r AntecedentDistribution swapBiconditional
        p r q AntecedentDistribution swapBiconditional

and combine them.

        buildConjunction

The left hand side from buildConjunction was ((p → q) → (p → r)) ∧ ((p → r) → (p → q)), so we are ready to apply transitivity there.

The right hand side from buildConjunction was (p → (q → r)) ∧ (p → (r → q)), which we first transform to p → ((q → r) ∧ (r → q)),

        p (q  r) (r  q) ConjunctionComposition
        applyBiconditionalTransitivity

and then to p → (q ↔ r).

        p BiconditionalReflexivity
        q r BiconditionalImplication swapBiconditional
        buildImplication
        applyBiconditionalTransitivity

We now have our desired result except the two sides are interchanged.

        swapBiconditional
))

thm (distributeImplicationOverBiconditional () ((H (p  (q  r)))) ((p  q)  (p  r)) (
        H
        p q r ImplicationDistributionOverBiconditional eliminateBiconditionalReverse
        applyModusPonens
))

That's it! We're ready to export our theorems to Interface:Principia Mathematica propositional logic theorems. That interface also requires us to define the alias formula for wffs.

kindbind (wff formula)
export (CLASSICAL Interface:Principia_Mathematica_propositional_logic_theorems () ())

We also export Interface:Law of the excluded middle, just to emphasize that the law of the excluded middle is a theorem of classical propositional logic.

export (EXCLUDED_MIDDLE Interface:Law_of_the_excluded_middle (CLASSICAL) ())

References[edit]

  1. A. Whitehead, B. Russell, Principia Mathematica, Cambridge University Press, 1910.
  2. Whitehead and Russell, loc. cit., p. 120
  3. Whitehead and Russell, loc. cit., p. 121
  4. Whitehead and Russell, loc. cit., p. 120
  5. an4 in set.mm, metamath.org, accessed February 15, 2010
  6. imim12i in set.mm, metamath.org, accessed February 15, 2010
  7. Whitehead and Russell, loc. cit., p. 124

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