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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 3or6 1201 | Analog of or4 675 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011.) |
⊢ (((φ ∨ ψ) ∨ (χ ∨ θ) ∨ (τ ∨ η)) ↔ ((φ ∨ χ ∨ τ) ∨ (ψ ∨ θ ∨ η))) | ||
Theorem | mp3an1 1202 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ φ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((ψ ∧ χ) → θ) | ||
Theorem | mp3an2 1203 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ ψ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((φ ∧ χ) → θ) | ||
Theorem | mp3an3 1204 | An inference based on modus ponens. (Contributed by NM, 21-Nov-1994.) |
⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((φ ∧ ψ) → θ) | ||
Theorem | mp3an12 1205 | An inference based on modus ponens. (Contributed by NM, 13-Jul-2005.) |
⊢ φ & ⊢ ψ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (χ → θ) | ||
Theorem | mp3an13 1206 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ φ & ⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (ψ → θ) | ||
Theorem | mp3an23 1207 | An inference based on modus ponens. (Contributed by NM, 14-Jul-2005.) |
⊢ ψ & ⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (φ → θ) | ||
Theorem | mp3an1i 1208 | An inference based on modus ponens. (Contributed by NM, 5-Jul-2005.) |
⊢ ψ & ⊢ (φ → ((ψ ∧ χ ∧ θ) → τ)) ⇒ ⊢ (φ → ((χ ∧ θ) → τ)) | ||
Theorem | mp3anl1 1209 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ φ & ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) ⇒ ⊢ (((ψ ∧ χ) ∧ θ) → τ) | ||
Theorem | mp3anl2 1210 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ ψ & ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) ⇒ ⊢ (((φ ∧ χ) ∧ θ) → τ) | ||
Theorem | mp3anl3 1211 | An inference based on modus ponens. (Contributed by NM, 24-Feb-2005.) |
⊢ χ & ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) ⇒ ⊢ (((φ ∧ ψ) ∧ θ) → τ) | ||
Theorem | mp3anr1 1212 | An inference based on modus ponens. (Contributed by NM, 4-Nov-2006.) |
⊢ ψ & ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (χ ∧ θ)) → τ) | ||
Theorem | mp3anr2 1213 | An inference based on modus ponens. (Contributed by NM, 24-Nov-2006.) |
⊢ χ & ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (ψ ∧ θ)) → τ) | ||
Theorem | mp3anr3 1214 | An inference based on modus ponens. (Contributed by NM, 19-Oct-2007.) |
⊢ θ & ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (ψ ∧ χ)) → τ) | ||
Theorem | mp3an 1215 | An inference based on modus ponens. (Contributed by NM, 14-May-1999.) |
⊢ φ & ⊢ ψ & ⊢ χ & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ θ | ||
Theorem | mpd3an3 1216 | An inference based on modus ponens. (Contributed by NM, 8-Nov-2007.) |
⊢ ((φ ∧ ψ) → χ) & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ ((φ ∧ ψ) → θ) | ||
Theorem | mpd3an23 1217 | An inference based on modus ponens. (Contributed by NM, 4-Dec-2006.) |
⊢ (φ → ψ) & ⊢ (φ → χ) & ⊢ ((φ ∧ ψ ∧ χ) → θ) ⇒ ⊢ (φ → θ) | ||
Theorem | biimp3a 1218 | Infer implication from a logical equivalence. Similar to biimpa 280. (Contributed by NM, 4-Sep-2005.) |
⊢ ((φ ∧ ψ) → (χ ↔ θ)) ⇒ ⊢ ((φ ∧ ψ ∧ χ) → θ) | ||
Theorem | biimp3ar 1219 | Infer implication from a logical equivalence. Similar to biimpar 281. (Contributed by NM, 2-Jan-2009.) |
⊢ ((φ ∧ ψ) → (χ ↔ θ)) ⇒ ⊢ ((φ ∧ ψ ∧ θ) → χ) | ||
Theorem | 3anandis 1220 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 18-Apr-2007.) |
⊢ (((φ ∧ ψ) ∧ (φ ∧ χ) ∧ (φ ∧ θ)) → τ) ⇒ ⊢ ((φ ∧ (ψ ∧ χ ∧ θ)) → τ) | ||
Theorem | 3anandirs 1221 | Inference that undistributes a triple conjunction in the antecedent. (Contributed by NM, 25-Jul-2006.) (Revised by NM, 18-Apr-2007.) |
⊢ (((φ ∧ θ) ∧ (ψ ∧ θ) ∧ (χ ∧ θ)) → τ) ⇒ ⊢ (((φ ∧ ψ ∧ χ) ∧ θ) → τ) | ||
Theorem | ecased 1222 | Deduction form of disjunctive syllogism. (Contributed by Jim Kingdon, 9-Dec-2017.) |
⊢ (φ → ¬ χ) & ⊢ (φ → (ψ ∨ χ)) ⇒ ⊢ (φ → ψ) | ||
Theorem | ecase23d 1223 | Variation of ecased 1222 with three disjuncts instead of two. (Contributed by NM, 22-Apr-1994.) (Revised by Jim Kingdon, 9-Dec-2017.) |
⊢ (φ → ¬ χ) & ⊢ (φ → ¬ θ) & ⊢ (φ → (ψ ∨ χ ∨ θ)) ⇒ ⊢ (φ → ψ) | ||
Even though it isn't ordinarily part of propositional calculus, the universal quantifier ∀ is introduced here so that the soundness of definition df-tru 1229 can be checked by the same algorithm that is used for predicate calculus. Its first real use is in axiom ax-5 1312 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1234 may be adopted and this subsection moved down to the start of the subsection with wex 1358 below. However, the use of dftru2 1234 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | wal 1224 | Extend wff definition to include the universal quantifier ('for all'). ∀xφ is read "φ (phi) is true for all x." Typically, in its final application φ would be replaced with a wff containing a (free) occurrence of the variable x, for example x = y. In a universe with a finite number of objects, "for all" is equivalent to a big conjunction (AND) with one wff for each possible case of x. When the universe is infinite (as with set theory), such a propositional-calculus equivalent is not possible because an infinitely long formula has no meaning, but conceptually the idea is the same. |
wff ∀xφ | ||
Even though it isn't ordinarily part of propositional calculus, the equality predicate = is introduced here so that the soundness of definition df-tru 1229 can be checked by the same algorithm as is used for predicate calculus. Its first real use is in axiom ax-8 1372 in the predicate calculus section below. For those who want propositional calculus to be self-contained i.e. to use wff variables only, the alternate definition dftru2 1234 may be adopted and this subsection moved down to just above weq 1369 below. However, the use of dftru2 1234 as a definition requires a more elaborate definition checking algorithm that we prefer to avoid. | ||
Syntax | cv 1225 |
This syntax construction states that a variable x, which has been
declared to be a setvar variable by $f statement vx, is also a class
expression. This can be justified informally as follows. We know that
the class builder {y ∣ y
∈ x} is a class by cab 2004.
Since (when
y is
distinct from x) we
have x = {y ∣ y
∈ x} by
cvjust 2013, we can argue that the syntax "class x " can be viewed as
an abbreviation for "class
{y ∣ y ∈ x}". See the discussion
under the definition of class in [Jech] p.
4 showing that "Every set can
be considered to be a class."
While it is tempting and perhaps occasionally useful to view cv 1225 as a "type conversion" from a setvar variable to a class variable, keep in mind that cv 1225 is intrinsically no different from any other class-building syntax such as cab 2004, cun 2888, or c0 3197. For a general discussion of the theory of classes and the role of cv 1225, see http://us.metamath.org/mpeuni/mmset.html#class. (The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1369 of predicate calculus from the wceq 1226 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.) |
class x | ||
Syntax | wceq 1226 |
Extend wff definition to include class equality.
For a general discussion of the theory of classes, see http://us.metamath.org/mpeuni/mmset.html#class. (The purpose of introducing wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1369 of predicate calculus in terms of the wceq 1226 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. For example, some parsers - although not the Metamath program - stumble on the fact that the = in x = y could be the = of either weq 1369 or wceq 1226, although mathematically it makes no difference. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-cleq 2011 for more information on the set theory usage of wceq 1226.) |
wff A = B | ||
Syntax | wtru 1227 | ⊤ is a wff. |
wff ⊤ | ||
Theorem | trujust 1228 | Soundness justification theorem for df-tru 1229. (Contributed by Mario Carneiro, 17-Nov-2013.) (Revised by NM, 11-Jul-2019.) |
⊢ ((∀x x = x → ∀x x = x) ↔ (∀y y = y → ∀y y = y)) | ||
Definition | df-tru 1229 | Definition of the truth value "true", or "verum", denoted by ⊤. This is a tautology, as proved by tru 1230. In this definition, an instance of id 19 is used as the definiens, although any tautology, such as an axiom, can be used in its place. This particular id 19 instance was chosen so this definition can be checked by the same algorithm that is used for predicate calculus. This definition should be referenced directly only by tru 1230, and other proofs should depend on tru 1230 (directly or indirectly) instead of this definition, since there are many alternative ways to define ⊤. (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by NM, 11-Jul-2019.) (New usage is discouraged.) |
⊢ ( ⊤ ↔ (∀x x = x → ∀x x = x)) | ||
Theorem | tru 1230 | The truth value ⊤ is provable. (Contributed by Anthony Hart, 13-Oct-2010.) |
⊢ ⊤ | ||
Syntax | wfal 1231 | ⊥ is a wff. |
wff ⊥ | ||
Definition | df-fal 1232 | Definition of the truth value "false", or "falsum", denoted by ⊥. See also df-tru 1229. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ ( ⊥ ↔ ¬ ⊤ ) | ||
Theorem | fal 1233 | The truth value ⊥ is refutable. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Mel L. O'Cat, 11-Mar-2012.) |
⊢ ¬ ⊥ | ||
Theorem | dftru2 1234 | An alternate definition of "true". (Contributed by Anthony Hart, 13-Oct-2010.) (Revised by BJ, 12-Jul-2019.) (New usage is discouraged.) |
⊢ ( ⊤ ↔ (φ → φ)) | ||
Theorem | trud 1235 | Eliminate ⊤ as an antecedent. A proposition implied by ⊤ is true. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ ( ⊤ → φ) ⇒ ⊢ φ | ||
Theorem | tbtru 1236 | A proposition is equivalent to itself being equivalent to ⊤. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (φ ↔ (φ ↔ ⊤ )) | ||
Theorem | nbfal 1237 | The negation of a proposition is equivalent to itself being equivalent to ⊥. (Contributed by Anthony Hart, 14-Aug-2011.) |
⊢ (¬ φ ↔ (φ ↔ ⊥ )) | ||
Theorem | bitru 1238 | A theorem is equivalent to truth. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ φ ⇒ ⊢ (φ ↔ ⊤ ) | ||
Theorem | bifal 1239 | A contradiction is equivalent to falsehood. (Contributed by Mario Carneiro, 9-May-2015.) |
⊢ ¬ φ ⇒ ⊢ (φ ↔ ⊥ ) | ||
Theorem | falim 1240 | The truth value ⊥ implies anything. Also called the principle of explosion, or "ex falso quodlibet". (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ ( ⊥ → φ) | ||
Theorem | falimd 1241 | The truth value ⊥ implies anything. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((φ ∧ ⊥ ) → ψ) | ||
Theorem | a1tru 1242 | Anything implies ⊤. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Anthony Hart, 1-Aug-2011.) |
⊢ (φ → ⊤ ) | ||
Theorem | truan 1243 | True can be removed from a conjunction. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ (( ⊤ ∧ φ) ↔ φ) | ||
Theorem | truanOLD 1244 | Obsolete proof of truan 1243 as of 21-Jul-2019. (Contributed by FL, 20-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (( ⊤ ∧ φ) ↔ φ) | ||
Theorem | dfnot 1245 | Given falsum, we can define the negation of a wff φ as the statement that a contradiction follows from assuming φ. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 21-Jul-2019.) |
⊢ (¬ φ ↔ (φ → ⊥ )) | ||
Theorem | inegd 1246 | Negation introduction rule from natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ ((φ ∧ ψ) → ⊥ ) ⇒ ⊢ (φ → ¬ ψ) | ||
Theorem | pm2.21fal 1247 | If a wff and its negation are provable, then falsum is provable. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (φ → ψ) & ⊢ (φ → ¬ ψ) ⇒ ⊢ (φ → ⊥ ) | ||
Theorem | pclem6 1248 | Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.) |
⊢ ((φ ↔ (ψ ∧ ¬ φ)) → ¬ ψ) | ||
Syntax | wxo 1249 | Extend wff definition to include exclusive disjunction ('xor'). |
wff (φ ⊻ ψ) | ||
Definition | df-xor 1250 | Define exclusive disjunction (logical 'xor'). Return true if either the left or right, but not both, are true. Contrast with ∧ (wa 97), ∨ (wo 616), and → (wi 4) . (Contributed by FL, 22-Nov-2010.) (Modified by Jim Kingdon, 1-Mar-2018.) |
⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))) | ||
Theorem | xoranor 1251 | One way of defining exclusive or. Equivalent to df-xor 1250. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.) |
⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ (¬ φ ∨ ¬ ψ))) | ||
Theorem | excxor 1252 | This tautology shows that xor is really exclusive. (Contributed by FL, 22-Nov-2010.) (Proof rewritten by Jim Kingdon, 5-May-2018.) |
⊢ ((φ ⊻ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (¬ φ ∧ ψ))) | ||
Theorem | xorbi2d 1253 | Deduction joining an equivalence and a left operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ((θ ⊻ ψ) ↔ (θ ⊻ χ))) | ||
Theorem | xorbi1d 1254 | Deduction joining an equivalence and a right operand to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ θ))) | ||
Theorem | xorbi12d 1255 | Deduction joining two equivalences to form equivalence of exclusive-or. (Contributed by Jim Kingdon, 7-Oct-2018.) |
⊢ (φ → (ψ ↔ χ)) & ⊢ (φ → (θ ↔ τ)) ⇒ ⊢ (φ → ((ψ ⊻ θ) ↔ (χ ⊻ τ))) | ||
Theorem | xorbin 1256 | A consequence of exclusive or. In classical logic the converse also holds. (Contributed by Jim Kingdon, 8-Mar-2018.) |
⊢ ((φ ⊻ ψ) → (φ ↔ ¬ ψ)) | ||
Theorem | pm5.18im 1257 | One direction of pm5.18dc 765, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((φ ↔ ψ) → ¬ (φ ↔ ¬ ψ)) | ||
Theorem | xornbi 1258 | A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1263. (Contributed by Jim Kingdon, 10-Mar-2018.) |
⊢ ((φ ⊻ ψ) → ¬ (φ ↔ ψ)) | ||
Theorem | xor3dc 1259 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 12-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ)))) | ||
Theorem | xorcom 1260 | ⊻ is commutative. (Contributed by David A. Wheeler, 6-Oct-2018.) |
⊢ ((φ ⊻ ψ) ↔ (ψ ⊻ φ)) | ||
Theorem | pm5.15dc 1261 | A decidable proposition is equivalent to a decidable proposition or its negation. Based on theorem *5.15 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ↔ ψ) ∨ (φ ↔ ¬ ψ)))) | ||
Theorem | xor2dc 1262 | Two ways to express "exclusive or" between decidable propositions. (Contributed by Jim Kingdon, 17-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))))) | ||
Theorem | xornbidc 1263 | Exclusive or is equivalent to negated biconditional for decidable propositions. (Contributed by Jim Kingdon, 27-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ)))) | ||
Theorem | xordc 1264 | Two ways to express "exclusive or" between decidable propositions. Theorem *5.22 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))) | ||
Theorem | xordc1 1265 | Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
⊢ ((φ ⊻ ψ) → DECID φ) | ||
Theorem | nbbndc 1266 | Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.) |
⊢ (DECID φ → (DECID ψ → ((¬ φ ↔ ψ) ↔ ¬ (φ ↔ ψ)))) | ||
Theorem | biassdc 1267 |
Associative law for the biconditional, for decidable propositions.
The classical version (without the decidability conditions) is an axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.) |
⊢ (DECID φ → (DECID ψ → (DECID χ → (((φ ↔ ψ) ↔ χ) ↔ (φ ↔ (ψ ↔ χ)))))) | ||
Theorem | bilukdc 1268 | Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (((DECID φ ∧ DECID ψ) ∧ DECID χ) → ((φ ↔ ψ) ↔ ((χ ↔ ψ) ↔ (φ ↔ χ)))) | ||
Theorem | dfbi3dc 1269 | An alternate definition of the biconditional for decidable propositions. Theorem *5.23 of [WhiteheadRussell] p. 124, but with decidability conditions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID φ → (DECID ψ → ((φ ↔ ψ) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ))))) | ||
Theorem | pm5.24dc 1270 | Theorem *5.24 of [WhiteheadRussell] p. 124, but for decidable propositions. (Contributed by Jim Kingdon, 5-May-2018.) |
⊢ (DECID φ → (DECID ψ → (¬ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))) | ||
Theorem | xordidc 1271 | Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.) |
⊢ (DECID φ → (DECID ψ → (DECID χ → ((φ ∧ (ψ ⊻ χ)) ↔ ((φ ∧ ψ) ⊻ (φ ∧ χ)))))) | ||
Theorem | anxordi 1272 | Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.) |
⊢ ((φ ∧ (ψ ⊻ χ)) ↔ ((φ ∧ ψ) ⊻ (φ ∧ χ))) | ||
For classical logic, truth tables can be used to define propositional logic operations, by showing the results of those operations for all possible combinations of true (⊤) and false (⊥). Although the intuitionistic logic connectives are not as simply defined, ⊤ and ⊥ do play similar roles as in classical logic and most theorems from classical logic continue to hold. Here we show that our definitions and axioms produce equivalent results for ⊤ and ⊥ as we would get from truth tables for ∧ (conjunction aka logical 'and') wa 97, ∨ (disjunction aka logical inclusive 'or') wo 616, → (implies) wi 4, ¬ (not) wn 3, ↔ (logical equivalence) df-bi 110, and ⊻ (exclusive or) df-xor 1250. | ||
Theorem | truantru 1273 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ ∧ ⊤ ) ↔ ⊤ ) | ||
Theorem | truanfal 1274 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ ∧ ⊥ ) ↔ ⊥ ) | ||
Theorem | falantru 1275 | A ∧ identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
⊢ (( ⊥ ∧ ⊤ ) ↔ ⊥ ) | ||
Theorem | falanfal 1276 | A ∧ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ ∧ ⊥ ) ↔ ⊥ ) | ||
Theorem | truortru 1277 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊤ ∨ ⊤ ) ↔ ⊤ ) | ||
Theorem | truorfal 1278 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ ∨ ⊥ ) ↔ ⊤ ) | ||
Theorem | falortru 1279 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ ∨ ⊤ ) ↔ ⊤ ) | ||
Theorem | falorfal 1280 | A ∨ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊥ ∨ ⊥ ) ↔ ⊥ ) | ||
Theorem | truimtru 1281 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊤ → ⊤ ) ↔ ⊤ ) | ||
Theorem | truimfal 1282 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊤ → ⊥ ) ↔ ⊥ ) | ||
Theorem | falimtru 1283 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ → ⊤ ) ↔ ⊤ ) | ||
Theorem | falimfal 1284 | A → identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (( ⊥ → ⊥ ) ↔ ⊤ ) | ||
Theorem | nottru 1285 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) |
⊢ (¬ ⊤ ↔ ⊥ ) | ||
Theorem | notfal 1286 | A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (¬ ⊥ ↔ ⊤ ) | ||
Theorem | trubitru 1287 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊤ ↔ ⊤ ) ↔ ⊤ ) | ||
Theorem | trubifal 1288 | A ↔ identity. (Contributed by David A. Wheeler, 23-Feb-2018.) |
⊢ (( ⊤ ↔ ⊥ ) ↔ ⊥ ) | ||
Theorem | falbitru 1289 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊥ ↔ ⊤ ) ↔ ⊥ ) | ||
Theorem | falbifal 1290 | A ↔ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (( ⊥ ↔ ⊥ ) ↔ ⊤ ) | ||
Theorem | truxortru 1291 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ (( ⊤ ⊻ ⊤ ) ↔ ⊥ ) | ||
Theorem | truxorfal 1292 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ (( ⊤ ⊻ ⊥ ) ↔ ⊤ ) | ||
Theorem | falxortru 1293 | A ⊻ identity. (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ (( ⊥ ⊻ ⊤ ) ↔ ⊤ ) | ||
Theorem | falxorfal 1294 | 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 575, modus ponendo tollens I mpto1 1295, modus ponendo tollens II mpto2 1296, and modus tollendo ponens (exclusive-or version) mtp-xor 1297. 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 1297 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 1298. This set of indemonstrables is not the entire system of Stoic logic. | ||
Theorem | mpto1 1295 | 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 1296) 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.) |
⊢ φ & ⊢ ¬ (φ ∧ ψ) ⇒ ⊢ ¬ ψ | ||
Theorem | mpto2 1296 | 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.) |
⊢ φ & ⊢ (φ ⊻ ψ) ⇒ ⊢ ¬ ψ | ||
Theorem | mtp-xor 1297 | Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, one of the five "indemonstrables" in Stoic logic. The rule says, "if φ is not true, and either φ or ψ (exclusively) are true, then ψ must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtp-or 1298. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1296, that is, it is exclusive-or df-xor 1250), 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 1296), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 2-Mar-2018.) |
⊢ ¬ φ & ⊢ (φ ⊻ ψ) ⇒ ⊢ ψ | ||
Theorem | mtp-or 1298 | Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtp-xor 1297, 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 φ is not true, and φ or ψ (or both) are true, then ψ 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.) |
⊢ ¬ φ & ⊢ (φ ∨ ψ) ⇒ ⊢ ψ | ||
Theorem | syl6an 1299 | A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.) |
⊢ (φ → ψ) & ⊢ (φ → (χ → θ)) & ⊢ ((ψ ∧ θ) → τ) ⇒ ⊢ (φ → (χ → τ)) | ||
Theorem | syl10 1300 | A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.) |
⊢ (φ → (ψ → χ)) & ⊢ (φ → (ψ → (θ → τ))) & ⊢ (χ → (τ → η)) ⇒ ⊢ (φ → (ψ → (θ → η))) |
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