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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | merco1lem7 1601 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | retbwax3 1602 | tbw-ax3 1581 rederived from merco1 1592. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem8 1603 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem9 1604 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem10 1605 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem11 1606 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem12 1607 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem13 1608 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem14 1609 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem15 1610 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem16 1611 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem17 1612 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | merco1lem18 1613 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1592. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | retbwax1 1614 |
tbw-ax1 1579 rederived from merco1 1592.
This theorem, along with retbwax2 1595, retbwax3 1602, and retbwax4 1594, shows that merco1 1592 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 1615 |
A single axiom for propositional calculus offered by Meredith.
This axiom has 19 symbols, sans auxiliaries. See notes in merco1 1592. (Contributed by Anthony Hart, 7-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem1 1616 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem2 1617 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem3 1618 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem4 1619 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem5 1620 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem6 1621 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem7 1622 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | mercolem8 1623 | Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw1 1624 | tbw-ax1 1579 rederived from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw2 1625 | tbw-ax2 1580 rederived from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw3 1626 | tbw-ax3 1581 rederived from merco2 1615. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re1tbw4 1627 |
tbw-ax4 1582 rederived from merco2 1615.
This theorem, along with re1tbw1 1624, re1tbw2 1625, and re1tbw3 1626, shows that merco2 1615, 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 1628 | Justification for rb-imdf 1629. (Contributed by Anthony Hart, 17-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-imdf 1629 | 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 1630 | Modus ponens for axiom systems. (Contributed by Anthony Hart, 12-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | rb-ax1 1631 | 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 1632 | 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 1633 | 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 1634 | 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 1635 | 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 1636 | 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 1637 | 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 1638 | 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 1639 | 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 1640 | 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 1641 | 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 1642 | 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 1643 | ax-mp 5 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re2luk1 1644 | luk-1 1534 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re2luk2 1645 | luk-2 1535 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | re2luk3 1646 |
luk-3 1536 derived from Russell-Bernays'.
This theorem, along with re1axmp 1643, re2luk1 1644, and re2luk2 1645 shows that rb-ax1 1631, rb-ax2 1632, rb-ax3 1633, and rb-ax4 1634, along with anmp 1630, 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 179, modus ponendo tollens I mptnan 1647, modus ponendo tollens II mptxor 1648, and modus tollendo ponens (exclusive-or version) mtpxor 1650. The first is an axiom, the second is already proved; in this section we prove the other three. Note that modus tollendo ponens mtpxor 1650 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 1649. 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 1651 and stoic1b 1653) and thema 3 (stoic3 1656). 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 1654, stoic2b 1655, stoic4a 1657, and stoic4b 1658. 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 292. "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 23. "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 1419. [Bobzien] p. 99 also suggests that Stoic logic may have dealt with commutativity (see xorcom 1403 and ancom 451) and the principle of contraposition (ax-3 8) (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 2368 and related theorems. | ||
Theorem | mptnan 1647 | 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 1648) 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 1648 | 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 1649 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1650, 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 1650 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1649, 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 1649. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1648, that is, it is exclusive-or df-xor 1401), 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 1648), 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 1651 |
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 1651 and stoic1b 1653 to represent each side. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof shortened by Wolf Lammen, 21-May-2020.) |
Theorem | stoic1aOLD 1652 | Obsolete proof of stoic1a 1651 as of 20-May-2020. (Contributed by David A. Wheeler, 16-Feb-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Theorem | stoic1b 1653 | Stoic logic Thema 1 (part b). The other part of thema 1 of Stoic logic; see stoic1a 1651. (Contributed by David A. Wheeler, 16-Feb-2019.) |
Theorem | stoic2a 1654 | 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 1655 for the version with the phrase "or both". We already have this rule as syldan 472, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic2b 1655 | Stoic logic Thema 2 version b. See stoic2a 1654. Version b is with the phrase "or both". We already have this rule as mpd3an3 1361, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic3 1656 | 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 1657 |
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 1658 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4b 1658 | Stoic logic Thema 4 version b. This is version b, which is with the phrase "or both". See stoic4a 1657 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 1678) 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 1660) 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 1912) of system S2 since [KalishMontague] shows it to be logically redundant (Lemma 9, p. 87, which we prove as theorem spw 1859 below). Theorem spw 1859 can be used to prove any instance of sp 1912 having mutually distinct setvar variables and no wff metavariables. However, it seems that sp 1912 in its general form cannot be derived from only Tarski's schemes. We do not include B5 i.e. sp 1912 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 1912 as theorem axc5 32164 using the auxiliary axioms 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 1665, ax-4 1678, ax-5 1751, ax-6 1797, ax-7 1841, ax-8 1872, and ax-9 1874. 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 1841, ax-8 1872, and ax-9 1874 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 1912, even though (using spw 1859) we can derive all instances of it that don't involve wff metavariables or bundled set metavariables. (Two set metavariables are "bundled" if they can be substituted with the same set metavariable i.e. do not have a $d distinct variable proviso.) Later we will introduce auxiliary axiom schemes ax-10 1889, ax-11 1894, ax-12 1907, and ax-13 2055 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 1435 for use by df-tru 1440. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Syntax | wex 1659 | Extend wff definition to include the existential quantifier ("there exists"). |
Definition | df-ex 1660 | 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 1661 | Theorem 19.7 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.) |
Theorem | eximal 1662 | 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 1699. (Contributed by BJ, 12-May-2019.) |
Syntax | wnf 1663 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1664 |
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 2175). An example of where this is used is
stdpc5 1965. See nf2 2018 for an alternative 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 1844), 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 2579 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Axiom | ax-gen 1665 | 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 30837 shows the special case . Theorem spi 1917 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 1666 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
Theorem | mpg 1667 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
Theorem | mpgbi 1668 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | mpgbir 1669 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | nfi 1670 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1671 |
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 1672 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nftru 1673 | The true constant has no free variables. (This can also be proven in one step with nfv 1754, but this proof does not use ax-5 1751.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | nex 1674 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
Theorem | nfnth 1675 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nffal 1676 | The false constant has no free variables (see nftru 1673). (Contributed by BJ, 6-May-2019.) |
Theorem | sptruw 1677 | Version of sp 1912 when is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) |
Axiom | ax-4 1678 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1679 for labeling consistency. It should be used only by alim 1679. (Contributed by NM, 21-May-2008.) (New usage is discouraged.) |
Theorem | alim 1679 | Restatement of Axiom ax-4 1678, for labeling consistency. It should be the only theorem using ax-4 1678. (Contributed by NM, 10-Jan-1993.) |
Theorem | alimi 1680 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.) |
Theorem | 2alimi 1681 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
Theorem | al2im 1682 | Closed form of al2imi 1683. Version of ax-4 1678 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | al2imi 1683 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.) |
Theorem | alanimi 1684 | Variant of al2imi 1683 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Theorem | alimdh 1685 | Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1679. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1686 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.) |
Theorem | albii 1687 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1688 | Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Theorem | alrimih 1689 | Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 1962 and 19.21h 1964. (Contributed by NM, 9-Jan-1993.) |
Theorem | hbxfrbi 1690 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2551 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | nfbii 1691 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfr 1692 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfrd 1693 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alex 1694 | Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.) |
Theorem | exnal 1695 | Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Theorem | 2nalexn 1696 | Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | 2exnaln 1697 | Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | 2nexaln 1698 | Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | alimex 1699 | 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 eximal 1662. (Contributed by BJ, 12-May-2019.) |
Theorem | aleximi 1700 | A variant of al2imi 1683: instead of applying quantifiers to the final implication, replace them with . A shorter proof is possible using nfa1 1954, sps 1918 and eximd 1935, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.) |
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