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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nottru 1301 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
Theorem | notfal 1302 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubitru 1303 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | trubifal 1304 | A identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
Theorem | falbitru 1305 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | falbifal 1306 | A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | truxortru 1307 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | truxorfal 1308 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxortru 1309 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | falxorfal 1310 | A identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
The Greek Stoics developed a system of 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). For more about Aristotle's system, see barbara and related theorems. 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 7, modus tollendo tollens (modus tollens) mto 587, modus ponendo tollens I mpto1 1311, modus ponendo tollens II mpto2 1312, and modus tollendo ponens (exclusive-or version) mtp-xor 1313. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtp-xor 1313 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtp-or 1314. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mpto1 1311 | 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 mpto2 1312) 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, 2-Mar-2018.) |
Theorem | mpto2 1312 | 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, 2-Mar-2018.) |
Theorem | mtp-xor 1313 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, 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 mtp-or 1314. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1312, that is, it is exclusive-or df-xor 1266), 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 mpto2 1312), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 2-Mar-2018.) |
Theorem | mtp-or 1314 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1313, 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 | stoic2a 1315 |
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 1316 for the version with the phrase "or both". We already have this rule as syldan 266, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic2b 1316 |
Stoic logic Thema 2 version b. See stoic2a 1315.
Version b is with the phrase "or both". We already have this rule as mpd3an3 1232, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic3 1317 |
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 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 1318 |
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 1319 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | stoic4b 1319 |
Stoic logic Thema 4 version b.
This is version b, which is with the phrase "or both". See stoic4a 1318 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.) |
Theorem | syl6an 1320 | A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
Theorem | syl10 1321 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
Theorem | exbir 1322 | Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexp 1323 | impexp 250 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicom 1324 | 3impexp 1323 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | 3impexpbicomi 1325 | Deduction form of 3impexpbicom 1324. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | ancomsimp 1326 | Closed form of ancoms 255. (Contributed by Alan Sare, 31-Dec-2011.) |
Theorem | expcomd 1327 | Deduction form of expcom 109. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | expdcom 1328 | Commuted form of expd 245. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | simplbi2comg 1329 | Implication form of simplbi2com 1330. (Contributed by Alan Sare, 22-Jul-2012.) |
Theorem | simplbi2com 1330 | A deduction eliminating a conjunct, similar to simplbi2 367. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.) |
Theorem | syl6ci 1331 | A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.) |
Theorem | mpisyl 1332 | A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.) |
The universal quantifier was introduced above in wal 1240 for use by df-tru 1245. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Axiom | ax-5 1333 | Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-7 1334 | Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-gen 1335 | 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 spi 1426 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, 5-Aug-1993.) |
Theorem | gen2 1336 | Generalization applied twice. (Contributed by NM, 30-Apr-1998.) |
Theorem | mpg 1337 | Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.) |
Theorem | mpgbi 1338 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | mpgbir 1339 | Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Theorem | a7s 1340 | Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alimi 1341 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | 2alimi 1342 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
Theorem | alim 1343 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
Theorem | al2imi 1344 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
Theorem | alanimi 1345 | Variant of al2imi 1344 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Syntax | wnf 1346 | Extend wff definition to include the not-free predicate. |
Definition | df-nf 1347 |
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 1657). An example of where this is used is
stdpc5 1473. See nf2 1555 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 , 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). (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfi 1348 | Deduce that is not free in from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | hbth 1349 |
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, 5-Aug-1993.) |
Theorem | nfth 1350 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfnth 1351 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
Theorem | nftru 1352 | The true constant has no free variables. (This can also be proven in one step with nfv 1418, but this proof does not use ax-17 1416.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
Theorem | alimdh 1353 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
Theorem | albi 1354 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimih 1355 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | albii 1356 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
Theorem | 2albii 1357 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
Theorem | hbxfrbi 1358 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Theorem | nfbii 1359 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfr 1360 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfxfrd 1361 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alcoms 1362 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
Theorem | hbal 1363 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | alcom 1364 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrimdh 1365 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | albidh 1366 | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.26 1367 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
Theorem | 19.26-2 1368 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | 19.26-3an 1369 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
Theorem | 19.33 1370 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | alrot3 1371 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | alrot4 1372 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
Theorem | albiim 1373 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
Theorem | 2albiim 1374 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
Theorem | hband 1375 | Deduction form of bound-variable hypothesis builder hban 1436. (Contributed by NM, 2-Jan-2002.) |
Theorem | hb3and 1376 | Deduction form of bound-variable hypothesis builder hb3an 1439. (Contributed by NM, 17-Feb-2013.) |
Theorem | hbald 1377 | Deduction form of bound-variable hypothesis builder hbal 1363. (Contributed by NM, 2-Jan-2002.) |
Syntax | wex 1378 | Extend wff definition to include the existential quantifier ("there exists"). |
Axiom | ax-ie1 1379 | is bound in . Axiom 9 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-ie2 1380 | Define existential quantification. means "there exists at least one set such that is true." Axiom 10 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Theorem | hbe1 1381 | is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | nfe1 1382 | is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | 19.23ht 1383 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Theorem | 19.23h 1384 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Theorem | alnex 1385 | Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if can be refuted for all , then it is not possible to find an for which holds" (and likewise for the converse). Comparing this with dfexdc 1387 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.) |
Theorem | nex 1386 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
Theorem | dfexdc 1387 | Defining given decidability. It is common in classical logic to define as but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1388. (Contributed by Jim Kingdon, 15-Mar-2018.) |
DECID | ||
Theorem | exalim 1388 | One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1387. (Contributed by Jim Kingdon, 29-Jul-2018.) |
The equality predicate was introduced above in wceq 1242 for use by df-tru 1245. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Theorem | weq 1389 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1389 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1242. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1389 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1242. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |
Syntax | wcel 1390 |
Extend wff definition to include the membership connective between
classes.
(The purpose of introducing here is to allow us to express i.e. "prove" the wel 1391 of predicate calculus in terms of the wceq 1242 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables and are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.) |
Theorem | wel 1391 |
Extend wff definition to include atomic formulas with the epsilon
(membership) predicate. This is read " is an element of
," " is a member of ," " belongs to ,"
or " contains
." Note: The
phrase " includes
" means
" is a subset of
;" to use it also
for
, as some authors occasionally do, is poor form
and causes
confusion, according to George Boolos (1992 lecture at MIT).
This syntactical construction introduces a binary non-logical predicate symbol (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments. (Instead of introducing wel 1391 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1390. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1391 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1390. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.) |
Axiom | ax-8 1392 |
Axiom of Equality. One of the equality and substitution axioms of
predicate calculus with equality. This is similar to, but not quite, a
transitive law for equality (proved later as equtr 1592). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1392 through ax-16 1692 are the axioms having to do with equality, substitution, and logical properties of our binary predicate (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1692 and ax-17 1416 are still valid even when , , and are replaced with the same variable because they do not have any distinct variable (Metamath's $d) restrictions. Distinct variable restrictions are required for ax-16 1692 and ax-17 1416 only. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-10 1393 |
Axiom of Quantifier Substitution. One of the equality and substitution
axioms of predicate calculus with equality. Appears as Lemma L12 in
[Megill] p. 445 (p. 12 of the preprint).
The original version of this axiom was ax-10o 1601 ("o" for "old") and was replaced with this shorter ax-10 1393 in May 2008. The old axiom is proved from this one as theorem ax10o 1600. Conversely, this axiom is proved from ax-10o 1601 as theorem ax10 1602. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-11 1394 |
Axiom of Variable Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent is a way of
expressing "
substituted for in
wff " (cf.
sb6 1763). It
is based on Lemma 16 of [Tarski] p. 70 and
Axiom C8 of [Monk2] p. 105,
from which it can be proved by cases.
Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1705, ax11v2 1698 and ax-11o 1701. (Contributed by NM, 5-Aug-1993.) |
Axiom | ax-i12 1395 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever is
distinct from and
, and
is true,
then quantified with is also true. In other words,
is irrelevant to the truth of
. Axiom scheme C9' in
[Megill]
p. 448 (p. 16 of the preprint). It apparently does not otherwise appear
in the literature but is easily proved from textbook predicate calculus by
cases.
This axiom has been modified from the original ax-12 1399 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
Axiom | ax-bnd 1396 |
Axiom of bundling. The general idea of this axiom is that two variables
are either distinct or non-distinct. That idea could be expressed as
. However, we instead choose an axiom
which has many of the same consequences, but which is different with
respect to a universe which contains only one object.
holds
if and are the same variable,
likewise for and ,
and holds if
is distinct from
the others (and the universe has at least two objects).
As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability). This axiom implies ax-i12 1395 as can be seen at axi12 1404. Whether ax-bnd can be proved from the remaining axioms including ax-i12 1395 is not known. The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.) |
Axiom | ax-4 1397 |
Axiom of Specialization. A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all , it is true for any
specific (that
would typically occur as a free variable in the wff
substituted for ). (A free variable is one that does not occur in
the scope of a quantifier: and are
both free in ,
but only is free
in .) Axiom
scheme C5' in [Megill]
p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski]
p. 67 (under his system S2, defined in the last paragraph on p. 77).
Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1335. Conditional forms of the converse are given by ax-12 1399, ax-16 1692, and ax-17 1416. Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1655. (Contributed by NM, 5-Aug-1993.) |
Theorem | sp 1398 | Specialization. Another name for ax-4 1397. (Contributed by NM, 21-May-2008.) |
Theorem | ax-12 1399 | Rederive the original version of the axiom from ax-i12 1395. (Contributed by Mario Carneiro, 3-Feb-2015.) |
Theorem | ax12or 1400 | Another name for ax-i12 1395. (Contributed by NM, 3-Feb-2015.) |
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