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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-cad 1601 | Definition of the "carry" output of the full adder. It is true when at least two arguments are true, so it is equal to the "majority" function on three variables. See cador 1602 and cadan 1603 for alternate definitions. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) | ||
Theorem | cador 1602 | The adder carry in disjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) |
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ 𝜒) ∨ (𝜓 ∧ 𝜒))) | ||
Theorem | cadan 1603 | The adder carry in conjunctive normal form. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 25-Sep-2018.) |
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) | ||
Theorem | cadbi123d 1604 | Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → (𝜃 ↔ 𝜏)) & ⊢ (𝜑 → (𝜂 ↔ 𝜁)) ⇒ ⊢ (𝜑 → (cadd(𝜓, 𝜃, 𝜂) ↔ cadd(𝜒, 𝜏, 𝜁))) | ||
Theorem | cadbi123i 1605 | Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜒 ↔ 𝜃) & ⊢ (𝜏 ↔ 𝜂) ⇒ ⊢ (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂)) | ||
Theorem | cadcoma 1606 | Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜑, 𝜒)) | ||
Theorem | cadcomb 1607 | Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) |
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓)) | ||
Theorem | cadrot 1608 | Rotation law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜒, 𝜑)) | ||
Theorem | cadnot 1609 | The adder carry distributes over negation. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) |
⊢ (¬ cadd(𝜑, 𝜓, 𝜒) ↔ cadd(¬ 𝜑, ¬ 𝜓, ¬ 𝜒)) | ||
Theorem | cad11 1610 | If (at least) two inputs are true, then the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ ((𝜑 ∧ 𝜓) → cadd(𝜑, 𝜓, 𝜒)) | ||
Theorem | cad1 1611 | If one input is true, then the adder carry is true exactly when at least one of the other two inputs is true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 19-Jun-2020.) |
⊢ (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∨ 𝜓))) | ||
Theorem | cad0 1612 | If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2024.) |
⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) | ||
Theorem | cad0OLD 1613 | Obsolete version of cad0 1612 as of 21-Sep-2024. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) | ||
Theorem | cadifp 1614 | The value of the carry is, if the input carry is true, the disjunction, and if the input carry is false, the conjunction, of the other two inputs. (Contributed by BJ, 8-Oct-2019.) |
⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓))) | ||
Theorem | cadtru 1615 | The adder carry is true as soon as its first two inputs are the truth constant. (Contributed by Mario Carneiro, 4-Sep-2016.) |
⊢ cadd(⊤, ⊤, 𝜑) | ||
Minimal implicational calculus, or intuitionistic implicational calculus, or positive implicational calculus, is the implicational fragment of minimal calculus (which is also the implicational fragment of intuitionistic calculus and of positive calculus). It is sometimes called "C-pure intuitionism" since the letter C is sometimes used to denote implication, especially in prefix notation. It can be axiomatized by the inference rule of modus ponens ax-mp 5 together with the axioms { ax-1 6, ax-2 7 } (sometimes written KS), or with { imim1 83, ax-1 6, pm2.43 56 } (written B'KW), or with { imim2 58, pm2.04 90, ax-1 6, pm2.43 56 } (written BCKW), or with the single axiom minimp 1616. This section proves minimp 1616 from { ax-1 6, ax-2 7 }, and then the converse, due to Ivo Thomas. Sources for this section are the webpage https://web.ics.purdue.edu/~dulrich/C-pure-intuitionism-page.htm 7 on Ted Ulrich's website, and the articles C. A. Meredith, A single axiom of positive logic, Journal of computing systems, vol. 1 (1953), 169--170, and C. A. Meredith, A. N. Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, vol. 4 (1963), 171--187. We may use a compact notation for derivations known as the D-notation where "D" stands for "condensed Detachment". For instance, "D21" means detaching ax-1 6 from ax-2 7, that is, using modus ponens ax-mp 5 with ax-1 6 as minor premise and ax-2 7 as major premise. D-strings are accepted by the grammar Dstr := digit | "D" Dstr Dstr. (Contributed by BJ, 11-Apr-2021.) | ||
Theorem | minimp 1616 | A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. Among single axioms of this length, it is the one with simplest antecedents (i.e., in the corresponding ordering of binary trees which first compares left subtrees, it is the first one). (Contributed by BJ, 4-Apr-2021.) |
⊢ (𝜑 → ((𝜓 → 𝜒) → (((𝜃 → 𝜓) → (𝜒 → 𝜏)) → (𝜓 → 𝜏)))) | ||
Theorem | minimp-syllsimp 1617 | Derivation of Syll-Simp (jarr 106) from ax-mp 5 and minimp 1616. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) | ||
Theorem | minimp-ax1 1618 | Derivation of ax-1 6 from ax-mp 5 and minimp 1616. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | minimp-ax2c 1619 | Derivation of a commuted form of ax-2 7 from ax-mp 5 and minimp 1616. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒))) | ||
Theorem | minimp-ax2 1620 | Derivation of ax-2 7 from ax-mp 5 and minimp 1616. (Contributed by BJ, 4-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | minimp-pm2.43 1621 | Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1616. It uses the classical derivation from ax-1 6 and ax-2 7 written DD22D21 in D-notation (see head comment for an explanation) and shortens the proof using mp2 9 (which only requires ax-mp 5). (Contributed by BJ, 31-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | ||
Implicational calculus is the fragment of propositional logic that uses only material implication, and not negation. It can be axiomatized by inference rule modus ponens ax-mp 5 together with the axioms { ax-1 6, ax-2 7, peirce 201 } or the Tarski-Bernays axioms { ax-1 6, imim1 83, peirce 201 } or with the single axiom { impsingle 1622 }. From either one of these three axiom sets, all tautologies containing only material implication may be proved. In contrast, minimal implicational calculus, which is presented just before this section, cannot prove certain tautologies (peirce 201, for example ). The class of theorems proved by minimal implicational calculus is thus a subset of the class of theorems proved by implicational calculus. The primary source for this section is the paper by Jan Lukasiewicz, The Shortest Axiom of the Implicational Calculus of Propositions, Proceedings of the Royal Irish Academy, Section A, vol. 52 (1948-1950), 25--33. It will be cited as simply "Lukasiewicz" in the remainder of this section. This section proves that the above three distinct axiom sets for implicational calculus all produce the same class of theorems. (Contributed by Larry Lesyna and Jeffrey P. Machado, 1-Aug-2023.) | ||
Theorem | impsingle 1622 | The shortest single axiom for implicational calculus, due to Lukasiewicz. It is now known to be the unique shortest axiom. The axiom is proved here starting from { ax-1 6, ax-2 7 and peirce 201 }. (Contributed by Larry Lesyna and Jeffrey P. Machado, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜒 → 𝜑) → (𝜃 → 𝜑))) | ||
Theorem | impsingle-step4 1623 | Derivation of impsingle-step4 from ax-mp 5 and impsingle 1622. It is used as a lemma in proofs of imim1 83 and peirce 201 from impsingle 1622. It is Step 4 in Lukasiewicz, where it appears as 'CCCpqpCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → 𝜑)) | ||
Theorem | impsingle-step8 1624 | Derivation of impsingle-step8 from ax-mp 5 and impsingle 1622. It is used as a lemma in proofs of ax-1 6 imim1 83 and peirce 201 from impsingle 1622. It is Step 8 in Lukasiewicz, where it appears as 'CCCsqpCqp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)) | ||
Theorem | impsingle-ax1 1625 | Derivation of impsingle-ax1 (ax-1 6) from ax-mp 5 and impsingle 1622. Lukasiewicz was used to construct this proof. Every formula corresponding to a detachment step was converted to its corresponding Metamath formula. mmj2 was used to unify each formula using ax-mp 5, which in turn produced this proof. With extremely high confidence, this result shows that the Lukasiewicz proof of ax-1 6 (step 27) is correct and that Metamath correctly verifies the proof. The same comments apply to the proofs that follow this one. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | impsingle-step15 1626 | Derivation of impsingle-step15 from ax-mp 5 and impsingle 1622. It is used as a lemma in proofs of imim1 83 and peirce 201 from impsingle 1622. It is Step 15 in Lukasiewicz, where it appears as 'CCCrqCspCCrpCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → (𝜒 → 𝜃)) → ((𝜑 → 𝜃) → (𝜒 → 𝜃))) | ||
Theorem | impsingle-step18 1627 | Derivation of impsingle-step18 from ax-mp 5 and impsingle 1622. It is used as a lemma in proofs of imim1 83 and peirce 201 from impsingle 1622. It is Step 18 in Lukasiewicz, where it appears as 'CCCCrpCspCCCpqrtCuCCCpqrt' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → (((𝜓 → 𝜃) → 𝜑) → 𝜏)) → (𝜂 → (((𝜓 → 𝜃) → 𝜑) → 𝜏))) | ||
Theorem | impsingle-step19 1628 | Derivation of impsingle-step19 from ax-mp 5 and impsingle 1622. It is used as a lemma in proofs of imim1 83 and peirce 201 from impsingle 1622. It is Step 19 in Lukasiewicz, where it appears as 'CCCCspqCrpCCCpqrCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜓)) → (((𝜓 → 𝜒) → 𝜃) → (𝜑 → 𝜓))) | ||
Theorem | impsingle-step20 1629 | Derivation of impsingle-step20 from ax-mp 5 and impsingle 1622. It is used as a lemma in proofs of imim1 83 and peirce 201 from impsingle 1622. It is Step 20 in Lukasiewicz, where it appears as 'CCCCrppCspCCCpqrCsp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜑 → 𝜓) → 𝜓) → (𝜒 → 𝜓)) → (((𝜓 → 𝜃) → 𝜑) → (𝜒 → 𝜓))) | ||
Theorem | impsingle-step21 1630 | Derivation of impsingle-step21 from ax-mp 5 and impsingle 1622. It is used as a lemma in the proof of imim1 83 from impsingle 1622. It is Step 21 in Lukasiewicz, where it appears as 'CCCCprqqCCqrCpr' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) → ((𝜒 → 𝜓) → (𝜑 → 𝜓))) | ||
Theorem | impsingle-step22 1631 | Derivation of impsingle-step22 from ax-mp 5 and impsingle 1622. It is used as a lemma in proofs of imim1 83 and peirce 201 from impsingle 1622. It is Step 22 in Lukasiewicz, where it appears as 'Cpp' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜑) | ||
Theorem | impsingle-step25 1632 | Derivation of impsingle-step25 from ax-mp 5 and impsingle 1622. It is used as a lemma in the proof of imim1 83 from impsingle 1622. It is Step 25 in Lukasiewicz, where it appears as 'CCpqCCCprqq' using parenthesis-free prefix notation. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → 𝜓) → 𝜓)) | ||
Theorem | impsingle-imim1 1633 | Derivation of impsingle-imim1 (imim1 83) from ax-mp 5 and impsingle 1622. It is step 29 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | impsingle-peirce 1634 | Derivation of impsingle-peirce (peirce 201) from ax-mp 5 and impsingle 1622. It is step 28 in Lukasiewicz. (Contributed by Larry Lesyna and Jeffrey P. Machado, 2-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) | ||
Theorem | tarski-bernays-ax2 1635 | Derivation of ax-2 7 from the Tarski-Bernays axiom set { ax-1 6, imim1 83, peirce 201 }. Note that no inference rule other than ax-mp 5 is used in this proof. This proof establishes a circle of equivalence: From { impsingle 1622 }, { ax-1 6, imim1 83, peirce 201 } was proved. From { ax-1 6, imim1 83, peirce 201 }, { ax-1 6, ax-2 7 and peirce 201 } was proved. From { ax-1 6, ax-2 7 and peirce 201 }, { impsingle 1622 } was proved. Therefore, the theorems that can be proved by selecting any one of these three distinct axiom sets must be equivalent. Prover9 and mmj2 were used to help constuct this proof. (Contributed by Larry Lesyna and Jeffrey P. Machado, 1-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | meredith 1636 |
Carew Meredith's sole axiom for propositional calculus. This amazing
formula is thought to be the shortest possible single axiom for
propositional calculus with inference rule ax-mp 5,
where negation and
implication are primitive. Here we prove Meredith's axiom from ax-1 6,
ax-2 7, and ax-3 8. Then from it we derive the Lukasiewicz
axioms
luk-1 1650, luk-2 1651, and luk-3 1652. Using these we finally rederive our
axioms as ax1 1661, ax2 1662, and ax3 1663,
thus proving the equivalence of all
three systems. C. A. Meredith, "Single Axioms for the Systems (C,N),
(C,O) and (A,N) of the Two-Valued Propositional Calculus", The
Journal of
Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be
close to a proof that this axiom is the shortest possible, but the proof
was apparently never completed.
An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues." (Contributed by NM, 14-Dec-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Wolf Lammen, 28-May-2013.) |
⊢ (((((𝜑 → 𝜓) → (¬ 𝜒 → ¬ 𝜃)) → 𝜒) → 𝜏) → ((𝜏 → 𝜑) → (𝜃 → 𝜑))) | ||
Theorem | merlem1 1637 | Step 3 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (The step numbers refer to Meredith's original paper.) (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜒 → (¬ 𝜑 → 𝜓)) → 𝜏) → (𝜑 → 𝜏)) | ||
Theorem | merlem2 1638 | Step 4 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜑) → 𝜒) → (𝜃 → 𝜒)) | ||
Theorem | merlem3 1639 | Step 7 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜓 → 𝜒) → 𝜑) → (𝜒 → 𝜑)) | ||
Theorem | merlem4 1640 | Step 8 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜏 → ((𝜏 → 𝜑) → (𝜃 → 𝜑))) | ||
Theorem | merlem5 1641 | Step 11 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (¬ ¬ 𝜑 → 𝜓)) | ||
Theorem | merlem6 1642 | Step 12 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜒 → (((𝜓 → 𝜒) → 𝜑) → (𝜃 → 𝜑))) | ||
Theorem | merlem7 1643 | Between steps 14 and 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃))) | ||
Theorem | merlem8 1644 | Step 15 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜓 → 𝜒) → 𝜃) → (((𝜒 → 𝜏) → (¬ 𝜃 → ¬ 𝜓)) → 𝜃)) | ||
Theorem | merlem9 1645 | Step 18 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜃 → (𝜓 → 𝜏)))) → (𝜂 → (𝜒 → (𝜃 → (𝜓 → 𝜏))))) | ||
Theorem | merlem10 1646 | Step 19 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜃 → (𝜑 → 𝜓))) | ||
Theorem | merlem11 1647 | Step 20 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | merlem12 1648 | Step 28 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → 𝜑) → 𝜑) | ||
Theorem | merlem13 1649 | Step 35 of Meredith's proof of Lukasiewicz axioms from his sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → (((𝜃 → (¬ ¬ 𝜒 → 𝜒)) → ¬ ¬ 𝜑) → 𝜓)) | ||
Theorem | luk-1 1650 | 1 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | luk-2 1651 | 2 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||
Theorem | luk-3 1652 | 3 of 3 axioms for propositional calculus due to Lukasiewicz, derived from Meredith's sole axiom. (Contributed by NM, 14-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||
Theorem | luklem1 1653 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 23-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | luklem2 1654 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → ¬ 𝜓) → (((𝜑 → 𝜒) → 𝜃) → (𝜓 → 𝜃))) | ||
Theorem | luklem3 1655 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜃 → 𝜒))) | ||
Theorem | luklem4 1656 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((¬ 𝜑 → 𝜑) → 𝜑) → 𝜓) → 𝜓) | ||
Theorem | luklem5 1657 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | luklem6 1658 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | ||
Theorem | luklem7 1659 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) | ||
Theorem | luklem8 1660 | Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) | ||
Theorem | ax1 1661 | Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | ax2 1662 | Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | ||
Theorem | ax3 1663 | Standard propositional axiom derived from Lukasiewicz axioms. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) | ||
Prove Nicod's axiom and implication and negation definitions. | ||
Theorem | nic-dfim 1664 | This theorem "defines" implication in terms of 'nand'. Analogous to nanim 1492. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓)))) | ||
Theorem | nic-dfneg 1665 | This theorem "defines" negation in terms of 'nand'. Analogous to nannot 1493. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to a definition ($a statement). (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 ⊼ 𝜑) ⊼ ¬ 𝜑) ⊼ (((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑)) ⊼ (¬ 𝜑 ⊼ ¬ 𝜑))) | ||
Theorem | nic-mp 1666 | Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply 𝜒, this form is necessary for useful derivations from nic-ax 1668. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⇒ ⊢ 𝜓 | ||
Theorem | nic-mpALT 1667 | A direct proof of nic-mp 1666. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⇒ ⊢ 𝜓 | ||
Theorem | nic-ax 1668 | Nicod's axiom derived from the standard ones. See Introduction to Mathematical Philosophy by B. Russell, p. 152. Like meredith 1636, the usual axioms can be derived from this and vice versa. Unlike meredith 1636, Nicod uses a different connective ('nand'), so another form of modus ponens must be used in proofs, e.g., { nic-ax 1668, nic-mp 1666 } is equivalent to { luk-1 1650, luk-2 1651, luk-3 1652, ax-mp 5 }. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | nic-axALT 1669 | A direct proof of nic-ax 1668. (Contributed by NM, 11-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | nic-imp 1670 | Inference for nic-mp 1666 using nic-ax 1668 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⇒ ⊢ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) | ||
Theorem | nic-idlem1 1671 | Lemma for nic-id 1673. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜃 ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))) ⊼ (((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃))) | ||
Theorem | nic-idlem2 1672 | Lemma for nic-id 1673. Inference used by nic-id 1673. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜂 ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ 𝜃)) ⇒ ⊢ ((𝜃 ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))) ⊼ 𝜂) | ||
Theorem | nic-id 1673 | Theorem id 22 expressed with ⊼. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏)) | ||
Theorem | nic-swap 1674 | The connector ⊼ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜃 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))) | ||
Theorem | nic-isw1 1675 | Inference version of nic-swap 1674. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜃 ⊼ 𝜑) ⇒ ⊢ (𝜑 ⊼ 𝜃) | ||
Theorem | nic-isw2 1676 | Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜓 ⊼ (𝜃 ⊼ 𝜑)) ⇒ ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜃)) | ||
Theorem | nic-iimp1 1677 | Inference version of nic-imp 1670 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓)) & ⊢ (𝜃 ⊼ 𝜒) ⇒ ⊢ (𝜃 ⊼ 𝜑) | ||
Theorem | nic-iimp2 1678 | Inference version of nic-imp 1670 using left-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ⊼ 𝜓) ⊼ (𝜒 ⊼ 𝜒)) & ⊢ (𝜃 ⊼ 𝜑) ⇒ ⊢ (𝜃 ⊼ (𝜒 ⊼ 𝜒)) | ||
Theorem | nic-idel 1679 | Inference to remove the trailing term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⇒ ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜒)) | ||
Theorem | nic-ich 1680 | Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓)) & ⊢ (𝜓 ⊼ (𝜒 ⊼ 𝜒)) ⇒ ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜒)) | ||
Theorem | nic-idbl 1681 | Double the terms. Since doubling is the same as negation, this can be viewed as a contraposition inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⇒ ⊢ ((𝜓 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜑 ⊼ 𝜑))) | ||
Theorem | nic-bijust 1682 | Biconditional justification from Nicod's axiom. For nic-* definitions, the biconditional connective is not used. Instead, definitions are made based on this form. nic-bi1 1683 and nic-bi2 1684 are used to convert the definitions into usable theorems about one side of the implication. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜏 ⊼ 𝜏) ⊼ ((𝜏 ⊼ 𝜏) ⊼ (𝜏 ⊼ 𝜏))) | ||
Theorem | nic-bi1 1683 | Inference to extract one side of an implication from a definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) ⇒ ⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓)) | ||
Theorem | nic-bi2 1684 | Inference to extract the other side of an implication from a 'biconditional' definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓))) ⇒ ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜑)) | ||
Theorem | nic-stdmp 1685 | Derive the standard modus ponens from nic-mp 1666. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 | ||
Theorem | nic-luk1 1686 | Proof of luk-1 1650 from nic-ax 1668 and nic-mp 1666 (and Definitions nic-dfim 1664 and nic-dfneg 1665). Note that the standard axioms ax-1 6, ax-2 7, and ax-3 8 are proved from the Lukasiewicz axioms by Theorems ax1 1661, ax2 1662, and ax3 1663. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | nic-luk2 1687 | Proof of luk-2 1651 from nic-ax 1668 and nic-mp 1666. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((¬ 𝜑 → 𝜑) → 𝜑) | ||
Theorem | nic-luk3 1688 | Proof of luk-3 1652 from nic-ax 1668 and nic-mp 1666. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | ||
Theorem | lukshef-ax1 1689 |
This alternative axiom for propositional calculus using the Sheffer Stroke
was discovered by Lukasiewicz in his Selected Works. It improves on
Nicod's axiom by reducing its number of variables by one.
This axiom also uses nic-mp 1666 for its constructions. Here, the axiom is proved as a substitution instance of nic-ax 1668. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜃 ⊼ (𝜃 ⊼ 𝜃)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | lukshefth1 1690 | Lemma for renicax 1692. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((((𝜏 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜏) ⊼ (𝜑 ⊼ 𝜏))) ⊼ (𝜃 ⊼ (𝜃 ⊼ 𝜃))) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜒))) | ||
Theorem | lukshefth2 1691 | Lemma for renicax 1692. (Contributed by NM, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜏 ⊼ 𝜃) ⊼ ((𝜃 ⊼ 𝜏) ⊼ (𝜃 ⊼ 𝜏))) | ||
Theorem | renicax 1692 | A rederivation of nic-ax 1668 from lukshef-ax1 1689, proving that lukshef-ax1 1689 with nic-mp 1666 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 31-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃))))) | ||
Theorem | tbw-bijust 1693 | Justification for tbw-negdf 1694. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) ↔ (((𝜑 → 𝜓) → ((𝜓 → 𝜑) → ⊥)) → ⊥)) | ||
Theorem | tbw-negdf 1694 | The definition of negation, in terms of → and ⊥. (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥) | ||
Theorem | tbw-ax1 1695 | The first of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | ||
Theorem | tbw-ax2 1696 | The second of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜑)) | ||
Theorem | tbw-ax3 1697 | The third of four axioms in the Tarski-Bernays-Wajsberg system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑) | ||
Theorem | tbw-ax4 1698 |
The fourth of four axioms in the Tarski-Bernays-Wajsberg system.
This axiom was added to the Tarski-Bernays axiom system (see tb-ax1 35790, tb-ax2 35791, and tb-ax3 35792) by Wajsberg for completeness. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (⊥ → 𝜑) | ||
Theorem | tbwsyl 1699 | Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | tbwlem1 1700 | Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
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