Catalog of named tautologies Some propositional taulogies have names, for example, Modus Ponens, Modus Tollens, Contrapositon, ...
Is there a catalog of all named propositional taulogies?
In particular, does the following tautology has a name:
$(p \to q) \equiv (p \& q \equiv p)$   ?
 A: You may find here list of various tautologies.
I don't think that this tautology has some specific name.Left hand side of the equivalence is expanded by using of few logical definitions and laws,so we may reduce right hand side:
$((p\land q)\Rightarrow p)\land (p \Rightarrow (p\land q))$  -definition of equivalence
$\top \land (p \Rightarrow (p\land q))$  -simplification law
$\lnot p \lor (p \land q)$ -definition of implication
$(\lnot p \lor p) \land (\lnot p \lor q)$ -distributive law
$\top \land (\lnot p \lor q)$
$(\lnot p \lor q)$ -definition of implication
$p \Rightarrow q$
A: One problem that anyone trying to name tautologies probably runs into with naming "modus ponens", modus tollens, etc. a tautology comes as that by doing so, you haven't exactly made clear the distinction between a rule of inference and a tautology (modus ponens is a rule of inference, not a tautology... or at least you'll have to not consider modus ponens a rule of inference, and instead have a rule of detachment, or conditional elimination).  Another comes as that you can find different tautologies which one might call "modus ponens", for example (p->((p->q)->q)) and ((p^(p->q))->q).  That said in front of me I have Jan Lukasiewicz's Elements of Mathematical Logic where a number of tautologies appear with names.  In his notation he puts all logical operations (connectives) before their arguments, and uses C for the material conditional, N for logical negation, A for logical disjunction (which he calls alternation), K for conjunction, E for logical equivalence, and D for alternative denial (NAND).  He names certain tautologies as follows:
CCpqCCqrCpr-law of hypothetical syllogism
CCNppp-law of Clavius (I don't think he names this tautology this in this text, but does name it that I think his book on Aristotle's Syllogistic)
CCqrCCpqCCrsCps-sorites
Cpp-law of identity
Epp-law of identity
CqCpq-law of simplification
CpCCpqq-modus ponens
CCpCqrCqCpr-law of commutation
CApqAqp-law of commutativity of alternation
CNNpp-law of double negation
CpNNp-law of double negation
CCpqCNqNp
CCpNqCqNp
CCNpqCNqp
CCNpNqCqp -laws of transposition
CApqAqp-law of commutativity of alternation
ApNp-law of the excluded middle
EKppp-law of tautology for conjunction
CCKpqrCpCqr-law of exportation
CCpCqrCKpqr-law of importation
CKpCpqq-modus ponens
CKpqKqp-law of commutativity of conjunction
NKpNp-law of contradiction
CANpNqNKpq
CNKpqANpNq
CKNpNqNApq
CNApqKNpNq-De Morgan's laws
CDpqDqp-law of commutativity of alternative denial
Epp-law of identity in the form of an equivalence
ECpCqrCqCpr-law of commutation in the form of an equivalence
EpNNp-law of double negation in the form of an equivalence
ECpqCNqNp-law of transposition in the form of an equivalence
One might also call Cpp the weak law of identity, and Epp the strong law of identity.  Or CEEpqrEpEqr a weak law of the associativity of equivalence, and EEEpqrEpEqr the strong law of the associativity of equivalence. 
