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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | merco1lem1 1601 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | retbwax4 1602 | tbw-ax4 1590 rederived from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | retbwax2 1603 | tbw-ax2 1588 rederived from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem2 1604 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem3 1605 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem4 1606 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem5 1607 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem6 1608 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem7 1609 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | retbwax3 1610 | tbw-ax3 1589 rederived from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem8 1611 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem9 1612 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem10 1613 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem11 1614 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem12 1615 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem13 1616 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem14 1617 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem15 1618 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem16 1619 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem17 1620 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem18 1621 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1600. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | retbwax1 1622 |
tbw-ax1 1587 rederived from merco1 1600.
This theorem, along with retbwax2 1603, retbwax3 1610, and retbwax4 1602, shows that merco1 1600 with ax-mp 5 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco2 1623 |
A single axiom for propositional calculus offered by Meredith.
This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1600. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem1 1624 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem2 1625 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem3 1626 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem4 1627 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem5 1628 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem6 1629 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem7 1630 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem8 1631 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw1 1632 | tbw-ax1 1587 rederived from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw2 1633 | tbw-ax2 1588 rederived from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw3 1634 | tbw-ax3 1589 rederived from merco2 1623. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw4 1635 |
tbw-ax4 1590 rederived from merco2 1623.
This theorem, along with re1tbw1 1632, re1tbw2 1633, and re1tbw3 1634, shows that merco2 1623, along with ax-mp 5, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-bijust 1636 | Justification for rb-imdf 1637. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-imdf 1637 | The definition of implication, in terms of and . (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | anmp 1638 | Modus ponens for axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-ax1 1639 | The first of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-ax2 1640 | The second of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-ax3 1641 | The third of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-ax4 1642 | The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rbsyl 1643 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rblem1 1644 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rblem2 1645 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rblem3 1646 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rblem4 1647 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rblem5 1648 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rblem6 1649 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rblem7 1650 | Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1axmp 1651 | ax-mp 5 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re2luk1 1652 | luk-1 1542 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re2luk2 1653 | luk-2 1543 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re2luk3 1654 |
luk-3 1544 derived from Russell-Bernays'.
This theorem, along with re1axmp 1651, re2luk1 1652, and re2luk2 1653 shows that rb-ax1 1639, rb-ax2 1640, rb-ax3 1641, and rb-ax4 1642, along with anmp 1638, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
The Greek Stoics developed a system of logic called Stoic logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm). In this section we show that the propositional logic system we use (which is non-modal) is at least as strong as the non-modal portion of Stoic logic. We show this by showing that our system assumes or proves all of key features of Stoic logic's non-modal portion (specifically the Stoic logic indemonstrables, themata, and principles). "In terms of contemporary logic, Stoic syllogistic is best understood as a substructural backwards-working Gentzen-style natural-deduction system that consists of five kinds of axiomatic arguments (the indemonstrables) and four inference rules, called themata. An argument is a syllogism precisely if it either is an indemonstrable or can be reduced to one by means of the themata (Diogenes Laertius (D. L. 7.78))." (Ancient Logic, Stanford Encyclopedia of Philosophy https://plato.stanford.edu/entries/logic-ancient/). There are also a few "principles" that support logical reasoning, discussed below. For more information, see "Stoic Logic" by Susanne Bobzien, especially [Bobzien] p. 110-120, especially for a discussion about the themata (including how they were reconstructed and how they were used). There are differences in the systems we can only partly represent, for example, in Stoic logic "truth and falsehood are temporal properties of assertibles... They can belong to an assertible at one time but not at another" ([Bobzien] p. 87). Stoic logic also included various kinds of modalities, which we do not include here since our basic propositional logic does not include modalities. A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 5, modus tollendo tollens (modus tollens) mto 181, modus ponendo tollens I mptnan 1655, modus ponendo tollens II mptxor 1656, and modus tollendo ponens (exclusive-or version) mtpxor 1658. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1658 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1657. After we prove the indemonstratables, we then prove all the Stoic logic themata (the inference rules of Stoic logic; "thema" is singular). This is straightforward for thema 1 (stoic1a 1659 and stoic1b 1661) and thema 3 (stoic3 1664). However, while Stoic logic was once a leading logic system, most direct information about Stoic logic has since been lost, including the exact texts of thema 2 and thema 4. There are, however, enough references and specific examples to support reconstruction. Themata 2 and 4 have been reconstructed; see statements T2 and T4 in [Bobzien] p. 110-120 and our proofs of them in stoic2a 1662, stoic2b 1663, stoic4a 1665, and stoic4b 1666. Stoic logic also had a set of principles involving assertibles. Statements in [Bobzien] p. 99 express the known principles. The following paragraphs discuss these principles and our proofs of them. "A principle of double negation, expressed by saying that a double-negation (Not: not: p) is equivalent to the assertible that is doubly negated (p) (DL VII 69)." In other words, as proven in notnot 297. "The principle that all conditionals that are formed by using the same assertible twice (like 'If p, p') are true (Cic. Acad. II 98)." In other words, as proven in id 22. "The principle that all disjunctions formed by a contradiction (like 'Either p or not: p') are true (S. E. M VIII 282)" Remember that in Stoic logic, 'or' means 'exclusive or'. In other words, as proven in xorexmid 1427. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1411 and ancom 456) and the principle of contraposition (con4 108) (pointing to DL VII 194). In short, the non-modal propositional logic system we use is at least as strong as the non-modal portion of Stoic logic. For more about Aristotle's system, see barbara 2393 and related theorems. | ||
Theorem | mptnan 1655 | Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1656) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.) |
Theorem | mptxor 1656 | Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or . See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.) |
Theorem | mtpor 1657 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1658, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if is not true, and or (or both) are true, then must be true." An alternative phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) |
Theorem | mtpxor 1658 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1657, one of the five "indemonstrables" in Stoic logic. The rule says, "if is not true, and either or (exclusively) are true, then must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1657. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1656, that is, it is exclusive-or df-xor 1409), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1656), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.) |
Theorem | stoic1a 1659 |
Stoic logic Thema 1 (part a).
The first thema of the four Stoic logic themata, in its basic form, was: "When from two (assertibles) a third follows, then from either of them together with the contradictory of the conclusion the contradictory of the other follows." (Apuleius Int. 209.9-14), see [Bobzien] p. 117 and https://plato.stanford.edu/entries/logic-ancient/ We will represent thema 1 as two very similar rules stoic1a 1659 and stoic1b 1661 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.) |
Theorem | stoic1aOLD 1660 | Obsolete proof of stoic1a 1659 as of 20-May-2020. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | stoic1b 1661 | Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1659. (Contributed by David A. Wheeler, 16-Feb-2019.) |
Theorem | stoic2a 1662 | Stoic logic Thema 2 version a. Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two." Bobzien uses constructs such as ; in Metamath we will represent that construct as . This version a is without the phrase "or both"; see stoic2b 1663 for the version with the phrase "or both". We already have this rule as syldan 477, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic2b 1663 | Stoic logic Thema 2 version b. See stoic2a 1662. Version b is with the phrase "or both". We already have this rule as mpd3an3 1369, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic3 1664 | Stoic logic Thema 3. Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic Thema 3. "When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external assumption, another follows, then other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4a 1665 |
Stoic logic Thema 4 version a. Statement T4 of [Bobzien] p. 117 shows a
reconstructed version of Stoic logic Thema 4: "When from two
assertibles a third follows, and from the third and one (or both) of the
two and one (or more) external assertible(s) another follows, then this
other follows from the first two and the external(s)."
We use to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1666 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4b 1666 | Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1665 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
Here we extend the language of wffs with predicate calculus, which allows us to talk about individual objects in a domain of discussion (which for us will be the universe of all sets, so we call them "setvar variables") and make true/false statements about predicates, which are relationships between objects, such as whether or not two objects are equal. In addition, we introduce universal quantification ("for all", e.g. ax-4 1686) in order to make statements about whether a wff holds for every object in the domain of discussion. Later we introduce existential quantification ("there exists", df-ex 1668) which is defined in terms of universal quantification. Our axioms are really axiom schemes, and our wff and setvar variables are metavariables ranging over expressions in an underlying "object language." This is explained here: mmset.html#axiomnote. Our axiom system starts with the predicate calculus axiom schemes system S2 of Tarski defined in his 1965 paper, "A Simplified Formalization of Predicate Logic with Identity" [Tarski]. System S2 is defined in the last paragraph on p. 77, and repeated on p. 81 of [KalishMontague]. We do not include scheme B5 (our sp 1942) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1880 below). Theorem spw 1880 can be used to prove any instance of sp 1942 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 1942 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 1942 as part of what we call "Tarski's system" because we want it to be the smallest set of axioms that is logically complete with no redundancies. We later prove sp 1942 as theorem axc5 32467 using the auxiliary axiom schemes that make our system metalogically complete. Our version of Tarski's system S2 consists of propositional calculus (ax-mp 5, ax-1 6, ax-2 7, ax-3 8) plus ax-gen 1673, ax-4 1686, ax-5 1762, ax-6 1809, ax-7 1855, ax-8 1893, and ax-9 1900. The last three are equality axioms that represent three sub-schemes of Tarski's scheme B8. Due to its side-condition ("where is an atomic formula and is obtained by replacing an occurrence of the variable by the variable "), we cannot represent his B8 directly without greatly complicating our scheme language, but the simpler schemes ax-7 1855, ax-8 1893, and ax-9 1900 are sufficient for set theory and much easier to work with. Tarski's system is exactly equivalent to the traditional axiom system in most logic textbooks but has the advantage of being easy to manipulate with a computer program, and its simpler metalogic (with no built-in notions of "free variable" and "proper substitution") is arguably easier for a non-logician human to follow step by step in a proof (where "follow" means being able to identify the substitutions that were made, without necessarily a higher-level understanding). In particular, it is logically complete in that it can derive all possible object-language theorems of predicate calculus with equality, i.e. the same theorems as the traditional system can derive. However, for efficiency (and indeed a key feature that makes Metamath successful), our system is designed to derive reusable theorem schemes (rather than object-language theorems) from other schemes. From this "metalogical" point of view, Tarski's S2 is not complete. For example, we cannot derive scheme sp 1942, even though (using spw 1880) we can derive all instances of it that don't involve wff metavariables or bundled setvar variables. (Two setvar variables are "bundled" if they can be substituted with the same setvar variable i.e. do not have a $d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-10 1919, ax-11 1924, ax-12 1937, and ax-13 2092 that are metatheorems of Tarski's system (i.e. are logically redundant) but which give our system the property of "metalogical completeness," allowing us to prove directly (instead of, say, by induction on formula length) all possible schemes that can be expressed in our language. | ||
The universal quantifier was introduced above in wal 1446 for use by df-tru 1451. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Syntax | wex 1667 | Extend wff definition to include the existential quantifier ("there exists"). |
Definition | df-ex 1668 | Define existential quantification. means "there exists at least one set such that is true." Definition of [Margaris] p. 49. (Contributed by NM, 10-Jan-1993.) |
Theorem | alnex 1669 | Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.) |
Theorem | eximal 1670 | A utility theorem. An interesting case is when the same formula is substituted for both and , since then both implications express a type of non-freeness. See also alimex 1707. (Contributed by BJ, 12-May-2019.) |
Syntax | wnf 1671 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1672 |
Define the not-free predicate for wffs. This is read " is not free
in ".
Not-free means that the value of cannot affect the
value of ,
e.g., any occurrence of in
is effectively
bound by a "for all" or something that expands to one (such as
"there
exists"). In particular, substitution for a variable not free in a
wff
does not affect its value (sbf 2210). An example of where this is used is
stdpc5 1995. See nf2 2046 for an alternate definition which
does not involve
nested quantifiers on the same variable.
Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, is effectively not free in the bare expression (see nfequid 1861), even though would be considered free in the usual textbook definition, because the value of in the expression cannot affect the truth of the expression (and thus substitution will not change the result). This predicate only applies to wffs. See df-nfc 2582 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Axiom | ax-gen 1673 | Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved , we can conclude or even . Theorem allt 31067 shows the special case . Theorem spi 1947 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 3-Jan-1993.) |
Theorem | gen2 1674 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
Theorem | mpg 1675 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
Theorem | mpgbi 1676 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | mpgbir 1677 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | nfi 1678 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1679 |
No variable is (effectively) free in a theorem.
This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form from smaller formulas of this form. These are useful for constructing hypotheses that state " is (effectively) not free in ." (Contributed by NM, 11-May-1993.) |
Theorem | nfth 1680 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nftru 1681 | The true constant has no free variables. (This can also be proven in one step with nfv 1765, but this proof does not use ax-5 1762.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | nex 1682 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
Theorem | nfnth 1683 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nffal 1684 | The false constant has no free variables (see nftru 1681). (Contributed by BJ, 6-May-2019.) |
Theorem | sptruw 1685 | Version of sp 1942 when is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) |
Axiom | ax-4 1686 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1687 for labeling consistency. It should be used only by alim 1687. (Contributed by NM, 21-May-2008.) Use alim 1687 instead. (New usage is discouraged.) |
Theorem | alim 1687 | Restatement of Axiom ax-4 1686, for labeling consistency. It should be the only theorem using ax-4 1686. (Contributed by NM, 10-Jan-1993.) |
Theorem | alimi 1688 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.) |
Theorem | 2alimi 1689 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
Theorem | al2im 1690 | Closed form of al2imi 1691. Version of ax-4 1686 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | al2imi 1691 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.) |
Theorem | alanimi 1692 | Variant of al2imi 1691 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Theorem | alimdh 1693 | Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1687. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1694 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.) |
Theorem | albii 1695 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1696 | Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Theorem | alrimih 1697 | Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 1992 and 19.21h 1994. (Contributed by NM, 9-Jan-1993.) |
Theorem | hbxfrbi 1698 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2559 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | nfbii 1699 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfr 1700 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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