This blog will be based on a logical system developed by Polish logician Stanislaw Jakowski. It was published in the journal *Studia Logica: An International Journal for Symbolic Logic, *T. 24 (1969), pp. 143-1960

Implication (–> or C), Conjunction (& or K), Disjunction (v or A), Equivalence (<–> or E), Negation (~ or N, Possibility (<> or M), Necessity ([] or L), and Variables (P or p, Q or q, R or r).

P = <>P or p = Mp

<>P = ~[]~P or Mp = NLNp

Discussive Implication (D): P–>Q = <>P–>Q or Dpq = CMpq

Discussive Equivalence (T): P<–>Q = (<>P–>Q) & (<>Q–><>P) or Tpq = KCMpqCMqMp

D1: P**–>**P = Dpp

D2: (P<–>Q) <–> (Q<–>P) = TTpqTqp

D3: (P–>Q) –> ((Q–>P)–>(P<–>Q)) = DDpqDDqpTpq

D4: ~(P&~P) = NKpNp [Law of Contradiction]

D5: (P&~P) –>Q = DKpNpq [Conjunction Law Overfilling]

D6: (P&Q) –>P = DKpqp

D7: P –> (P&Q) = DpKpq

D8: (P&Q) <–> (Q&P) = TKpqKqp

D9: (P&(Q&R)) <–> ((P&Q)&R) = TKpKqrKKpqr

D10: (P–>(Q–>R)) –> ((P&Q)–>R) = DDpDqrDKpqr [law importation]

D11: ((P–>Q)&(P–>R)) <–> (P–>(Q&R)) = TKDpqDprDpKqr

D12: ((P–>R)&(Q–>R)) <–> ((PvQ)–>R) = TKDprDqrDApqr

D13: P <–> ~~P = TpNNp

D14: (~P–>P) –> P = DDNppp

D15: (P–>~P) –>~P = DDpNpNp

D16: (P<–>~P) –> P = DTpNpp

D17: (P<–>~P) –>~P = DTpNpNp

D18: ((P–>Q)&~Q) –>~P = DKDpqNqNp

D19: ((P–>)&(P–>~Q)) –>~P = DKDpqDpNqNp

D20: ((~P–>Q)&(~P–>~Q)) –> P = DKDNpqDNpNqp

D21: (P–>(Q&~Q)) –>~P = DDpKqNqNp

D22: (~P–>(Q&~Q)) –> P = DDNpKqNqp

D23: ~(P<–>~P) = NTpNp

D24: ~(P–>Q) –> P = DNDpqp

D25: ~(P–>Q) –> ~Q = DNDpqNq

D26: P–> (~Q–>~(P–>Q)) = DpDNqNDpq

A formulation of Aristotle’s Principle of Contradiction would be: “Two* contradictory sentences are not both true in the same language”* or *“Two contradictory sentences are not both true, if the words occurring in those sentences have the same meanings”. *

In Two Valued Logic, there is a Theorem known as the Law of Overfilling, or Implicational Law of Overfilling, or Dun Scotus Law, or L2 Theorem 1.

** L2 Theorem 1**: **P—> (~P—>Q)**

If an assertion implies its contradiction, then that assertion implies any and all statements.

“A deductive system…is called inconsistent, if its theses include two such which contradict one another, that is such that one is the negation of the other, e.g., (P) and (~ P) . If any inconsistent system is based on a two valued logic, then by the implicational law of overﬁlling one can obtain in it as a thesis any formula P which is meaningful in that system. It suffices…to apply the rule of modus ponens twice[ to P—> (~P—>Q)]. A system in which any meaningful formula is a thesis shall be termed overﬁlled.”

- Assume: P—> (~P—>Q)
- Modus Ponens: P
- Conclusion: ~P—>Q
- Modus Ponens: ~P
- Conclusion: Q

“[T]he problem of the logic of inconsistent systems is formulated here in the following manner: the task is to find a system of the sentential calculus which: (1) when applied to the inconsistent systems would not always entail overfilling, (2) would be rich enough to enable practical inference, (3) would have an intuitive justification. “

This means that Discussive Logic does not have the theorem of implicational law of overfilling. The theorem is not always true in Discussive Logic. Discussive Logic does not entail that a contradiction does not always entail any and all assertions. So Discussive Logic rejects the truth of the theorem P—> (~P—> Q), which is a theorem is two value logic, i.e. been proven true under conditions of two value logic.

**Kolmogorov’s System**

Here are Four axioms from Hilbert’s positive logic, and one axiom introduced by Kolmogorov.

K 1:** P—> (Q—>P)**

K 2: **(P—> (P—>Q))—> (P—>Q)**

K 3: **(P—> (Q—>R))—> ((Q—> (P—>R))**

K 4: **(Q—>R)—> ((P—>Q)—> (P—>R))**

K 5: **(P—>Q)—> ((P—>~Q)—>~P)**

Under these axioms, Two valued logic cannot be proved. Implicational Law of Overfilling not being provable in Discussive Logic implies that Two Valued logic cannot be proved in Discussive Logic. This entails that there might be overlap between Two Valued logic and Discussive Logic, but there is not a total overlap between Two Valued logic and Discussive Logic. Not all theorems of Two Valued logic will be theorems in Discussive logic (like law of overfilling), but some theorems of two valued logic are theorems in Discursive logic.

From these Axioms and the rule of inference known as Modus Ponens, there is one theorem which has some similarities to implicational law of overfilling.

K 9**: P—> (~P—>~Q)**

It is not the only Theorem that can be derived from the Axioms and Modus Ponens. Here is a list of some Theorems that can be derived by using Modus Ponens on the Axioms.

K 6 : **(P—>Q)—> ((Q—R)—> (P—>R))**

K 7: **((Q—>P)—>R)—> (P—>R)**

K 8: **P—> ((Q—>~P)—> ~Q)**

K 9: **P—> (~P—>~Q)**

Proof of how K 6 – K 9 are derived are ignored for here. All that needs to be known is that K3 and applications of Modus Ponens is equal to If K4 then K 6. K 6 and applications of Modus Ponens is equal to If K 1 then K 7. K 7 and applications of Modus Ponens is equal to If K 5 then K 8. K 6 and applications of Modus Ponens is equal to If K 8 then K 7 implies K 9.

This forms Kolmogorov’s System.

**Lewis System of Strict Implication**

Strict Implication is defined by modal operator of “it is possible that P” or <>P. So “P strictly implies Q” is equal to “It is not possible that both P and not Q”. But taking the conditional statement —> as strict implication means that the implicational law of overfilling is not a theorem.

Material Implication as a conditional is usually defined by the logical relationship of a conjunction.

Material Conditional: P—>Q = ~(P & ~Q)

“P implies Q” is equal to “Not both P and not Q”

Strict Conditional: P—» Q = ~<>(P & ~Q)

“P strictly implies Q” is equal to “It is not possible that both P and not Q”

Under Strict Implication, Law of Overfilling is not a theorem. Under Material Implication, Law of Overfilling is a theorem. And set of theorems which include only strict implication and not material implication is very limited.

**Many Valued Logics**

Based on a certain Three Value logical matrix, which shall be ignored, the Law of Overfilling is not a theorem. But there is another theorem in the Three Value logic which has some similarity to the Law of Overfilling.

L 1: **P—> (~P—> (~~P—> Q))**

Based on the theorem (stated above) of this specific three valued logic, it holds the overfilling of a system when it includes the inconsistent thesis of P, ~P, and ~~P. And the implicational theses of two valued calculus remains valued in the three valued logic. But the three valued logic also holds other theorems that are not in two valued logic, which are as follows.

L 2: **P—> ~~P**

L 3: **~~~P—> P**

L 4:** ~P—> ~(P—> P)**

So in the three valued logic, which is ignoring the logical matrix of this three valued logic, we cannot obtain the Law of Overfilling. The Law of Overfilling will thus be a theorem in two valued logic but not a theorem in this three valued logic. But the three valued logic has a theorem that is similar to the Law of Overfilling but is not equivalent to the Law of Overfilling. This three value logic also has some theorems that are not theorems in two valued logic. Besides the Law of Overfilling not being a theorem in the three valued logic, the rest of implicational theorems in two valued logic are theorems in the three valued logic.

**Calculus of Modal Sentences (M2)**

The Modal Sentences of (M2) will assume that modal assertions are either true or false, or simply that the Modal sentences are two valued. But now suppose that there are factors that do not allow for the assertion P to be determined strictly to be either true or false.

For example: Suppose that you are flipping a coin. Suppose that you make the assertion that “During the game heads will turn up more times than tails will” and this is represented by the variable of P. There will be certain sequences that turn up so that P is true, and there will be certain sequences that turn up so that P is false. So P may take on both true and false.

“It is necessary that P” = []P

Taking the example above, we can say that “P occurs for all possible events”.

Q is any formula that includes (1) operators —>, V, &, <—>, ~ and [], (2) and variables p,q,r,s..etc. R is any formula that is already a Q formula and is replacements of variables in Q by interpreting them as P(x), Q(x), R(x), S(x)…etc, and interpreting [] by universal quantifiers “for every x”. Every Q satisfies (1) and (2) and every R satisfies (1) and (2), and additionally satisfying (3).

The operators are implication, disjunction, conjunction, equivalence, and necessity. These are applied to variables or connects variables. When those conditions are met, then it is a formula of Q. The replacement of the variables and [] are formula of R. (1) and (2) can be recognized as P–>Q, or []P—>Q, or []P—>P. We can replace those variables to formulas in R: P(x) —> Q(x), or For every x, P(x) —> Q(x), or For every x, P(x) —> P(x).

“It is possible that P” = <>P

<>P can be taken as “it is not necessary that not P”.

<>P = ~[]~P

Like we could change []P into “for every x”, we may also change <>P into “for some x”.

**Definition of Discussive Implication and Discussive Equivalence**

As is known, even sets of those inscriptions which have no intuitive meaning at all can be turned into a formalized deductive system. In spite of this theoretical possibility, logical researches so far have been taking into consideration such deductive systems which are symbolic interpretations of consistent theories, so that theses in each such system are theorems in a theory formulated in a single symbolic language free from terms whose meanings are vague.

But suppose that theses which do not satisfy those conditions are included into a deductive system. It suffices, for instance, to deduce consequences from several hypotheses that are inconsistent with one another in order to change the nature of the theses, which thus shall no longer reﬂect a uniform opinion.** The same happens if the theses advanced by several participants in a discourse are combined into a single system, or if one person’s opinions are so pooled into one system although that person is not sure whether the terms occurring in his various theses are not slightly differentiated in their meanings.** Let such a system which cannot be said to include theses that express opinions in agreement with one another, be termed a *discussive system*. (*Italics *is authors and **Bold **is mine)

Each the theses in discussive logic are preceded so that each thesis has the speaker has the reservation such that each assertion means “in accordance with the opinion of one of the participants in the discussion” or “for a certain admissible meaning of the terms used”. So when you add an assertion to a discussive system, that assertion will have a different intuitive meaning. Discussive assertions have the implicit condition of the equivalence to <>P.

King Solomon having to decide between two harlots claiming to be the mother of a baby. Woman A claimed to be the mother of the baby and not the mother of the dead baby, and Woman B claimed to be the mother of the baby and not the mother of the dead baby. King Solomon being the arbitrator, under Discussive assertions, would have taken each Woman’s claim as having the prefix of possibility, or “it is possible that Woman A is the mother” or “it is possible that Woman A is not the mother”.

Discussive logic is not based on ordinary two valued logic. Discussive logic would not hold Modus Ponens in all cases if it did.

Take the statement P—>Q is asserted in a discussion. It would be understood to mean “It is possible that If P, then Q”. P is asserted in the same discussion. It would be understood to mean “It is possible that P”. Q would not follow from the two assertions in the discussion. For by Q would not follow in the discussion because Q stands for “It is possible that Q”. So it is invalid to infer from “It is possible that if P, then Q” and “It is possible P” that “It is possible that Q”. But people might assume the normal two value logic in which Modus Ponens holds in all cases.

For Discussive Logic, Discussive Implication is defined as such:

Definition of Discussive Implication: P**—>**Q = <>P—>Q

There is a theorem of M2 based on Discussive Implication.

M2 Theorem 1: <>(<>P—>Q) —> (<>P—><>Q)

So Modus Ponens may be used in Discussrive Logic when we understand that from (<>P—><>Q) and <>P, we may infer that <>Q by Modus Ponens.

For Discussive Logic, Discussive Equivalence is defined as such:

Definition of Discussive Equivalence: P **<—>** Q = (<>P—>Q) & (<>Q—><>P)

There is a theorem of M2 based on Discussive Implication.

M2 Theorem 2: <> (P**<—>**Q) —> (<>P—> <>Q)

M2 Theorem 3: <> (P **<—>** Q) —> (<>Q —> <>P)

**Two valued Discussive System of Sentential Calculus: D2**

The system of D2 (i.e. Discussive Logic) of two valued discussive sentential calculus is marked by the formula T, and are marked by the following properties: (1) Sentential variables and functors of Discussive Implication, Discussive Equivalence, Disjunction, Conjunction, and Negation. (2) precedening T with the symbol of <> yields a theorem in two valued sentential calculus of modal sentences M2.

As the author says, “The system defined in this way is discussive, i.e., its theses are provided with discussive assertion which implicitly includes the functor <>/ This is an essential fact, since even such a simple law as P—>P, on replacement of —> with **—-> **(i.e. discussive implication leads) to a new theorem.”

D2 Theorem 1: P**—>**P

D2 is not a theorem in M2, specifically because M2 did not have discussive implication. But in order to make D2 theorem 1 into a theorem in M2, you have to add <> to D2 theorem 1 like this:

M2 Theorem 4: <>(P—>P)

System M2 is decidable, so the discussive sentential calculus D2, defined by an interpretation in M2 is decidable too.

Methodological Theorem 1: “Every thesis T in two valued sentential calculus which does not include constant symbols of —>, <—>, V, becomes a thesis in T(d) in discusive sentential calculus D2 when in T the implication symbols is replaced by the [discurssive implication], and the equivalence symbols are replaced by [discusrive equivalence]. “

“Proof. Consider a formula T(d) constructed so as the theorem to be proved describes. It is to be demonstrated that <>T(d) is a thesis in M2. It is claimed that <>T(d) is equivalent to some other formulae; the equivalences will be proved gradually.”

Here are a couple more M2 Theorems.

M2 Theorem 5: <>(P**—>**Q) <—> (<>P—><>Q)

M2 Theorem 6: <>(P **<—>** Q) <—> (<>P <—> <>Q)

M2 Theorem 7: <>(P v Q) <—> (<>P v <>Q)

These theorems are about the distribution of <> over the variables. For example, M2 Theorem 5 distributes <> over implication, and M2 Theorem 6 distributes <> over equivalence, and M2 Theorem 7 distributes over Disjunction. M2 Theorem 5 and M2 Theorem 6 have Discussive Implication and Discussive Implication as the antecedents, respectively.

This shows how we can replace Discurssive Implication with regular implication and how we can replace Discurssive Equivalence with regular equivalence. So from <>(P**—>**Q), which contains Discurssive Implication, can be replaced with regular implication as <>P—><>Q. The form <>(P**<—>**Q), which contains Discurssive Equivalence, can be replaced with regular implication as <>P,—><>Q. Discurssive assertion like <>(PvQ) has the equivalence in M2, or Modal Logic, as <>P v <>Q.

The procedure yields the formula W, which is equivalent to <>T(d) and includes (1) only the symbols —>, <—>, and V, (2) variables, and (3) symbols <> in certain special positions, like each variable is directly preceded by <> and each symbol <> directly precedes a variable. Forming T(d) from the thesis T belonging to two value logic is possibly be seen that W can be obtained from T by preceding each variable by <>. For example, precede the variable P by <>P or precede the variable Q by <>Q. This procedure would yield the following theorems in M2.

(a) W is a result of the substitution in T

(b) <>T(d) is equivalent to W.

Hence T(d) is a thesis of D2

The theorems just listed above, immediately yields these theorems in Discussive Logic:

D2 Theorem 2: (P**<—>**Q) **<—>** (Q**<—>**P)

D2 Theorem 3: (P**—>**Q) **—>** ((Q**—>**P) **—>** (P<**—>**Q))

Each of the connectives in the D2 theorem just listed are Discurssive Equivalence for D2 Theorem 2 and Discurssive Implication for D2 Theorem 3.

Methodological Theorem 2: If T is a thesis in the two valued sentential calculus and includes variables and at the most the functors V, &, ~, then (1) T and (2) ~T **—>** q, are thesis in D2. The implication of (2) is Discurssive Implication.

Proof: The symbols V, &, and ~ retain respective meanings in M2 and D2, and that (3) []T is a thesis in M2. The symbols V, &, and ~ retain respective meanings in M2 and D2 and that (3) []T is a thesis in M2. Hence (1) by M2 Theorem 8 []P—><>P and (2) by M2 Theorem 9 []P—><>(<>~P—>Q).

M2 Theorem 8: []P—><>P

M2 Theorem 9: []P—> <>(<>~P—>Q)

We may apply the Methodological Theorem 2 to the Two Valued Logic Theorem of ~(Pv~P), i.e. Aristotle’s Principle of Contradiction.

L2 Theorem 3: ~(P&~P)

“Methodological Theorem 2 and Law of Contradiction in Two valued logic yieleds – in view of the law of double negation- the following theorem of Discussive logic.”

D2 Theorem 4: ~(P&~P) [Law of Contradiction]

D2 Theorem 5: (P&~P) —> Q [Conjunctional Law of Overfilling]

What these two theorems are basically stating is this: Suppose that we have an individual in a discussion, and this individual holds to the Discussive assertion of (P&~P), this individual would hold inconsistent opinions. And in Discussive logic, when an individual holds to inconsistent opinion, the persons opinion implies any and all discussive assertions. This basically forbids an individual from holding to discussive assertions that are contradictory to one another by D2 Theorem 4, and if we do hold to contradictory discussive assertions then any discussive assertion follows from the conjunction of contradictory discussive assertions. This is similar to Law of Overfilling in two value logic but not exactly the same.

We also have the following theorems in Discussive Logic.

D2 Theorem 6: (P&Q) **—>** P

D2 Theorem 7: P**—>** (P&P)

D2 Theorem 8: (P&Q) **<—>** (Q&P)

D2 Theorem 9: (P& (Q&R)) **<—>** ((P&Q) & R)

D2 Theorem 10: (P**—>** (Q**—>**R)) **—>** ((P&Q) **—>**R) [Law of Importation]

D2 Theorem 11:((P**–>**Q) & (P**—>**R)) **<—>** (P**—>**(Q & R))

D2 Theorem 12: ((P**—>**R) & (Q**—>**R)) **<—>** ((PvQ) **—>** R)

D2 Theorem 13:P **<—>** ~~P

D2 Theorem 14:(~P**—>**P) **—>**P

D2 Theorem 15:(P**—>**~P)** —**>~P

D2 Theorem 16:(P**<—>**~P) **—>**P

D2 Theorem 17: (P**<—>**~P) **—>**~P

D2 Theorem 18: ((P**—>**Q) & ~Q) **—>** ~P

There are laws of inference by reductio ad absurdum that remain valid in Discurssive Logic.

D2 Theorem 19:((P**—>**Q) & (P**—>**~Q)) **—>** ~P

D2 Theorem 20:((~P**—>**Q) & (~P**—>**~Q)) **—**> P

D2 Theorem 21:(P**—**> (Q&~Q)) **—>** ~P

D2 Theorem 22: (~P **—>** (Q&~Q)) **—>**P

Here are some other theorems of Discurssive Logic.

D2 Theorem 23:~(P **<—>** ~P)

D2 Theorem 24:~(P**—>**Q) **—>**P

D2 Theorem 25: ~(P**—>**Q) **—>**~Q

D2 Theorem 26: P** —>** (~Q**—>**~(P**—>**Q))