# 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:

$(p \to q) \equiv (p \& q \equiv p)$ ?

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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

$p \Rightarrow q$

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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

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.

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