Logic — inference, quantification, modus ponens and modus tollens

Wittgenstein argues in LW 6.1224 why logic was called the science of forms and reasoning and that the logical laws must not themselves be subject to logical laws LW 6.123.

Based on https://www.taoke.de/ke/Logic/index.html.

John Sowa writes in [Sowa2017a]: “From 1880 to 1885, Peirce developed his general algebra of relations, which Giuseppe Peano (1889) adopted as the basis for the modern notation of predicate calculus. Gottlob Frege (1879) has developed an equivalent notation for first-order logic, which he called Begriffsschrift (concept writing), but no one else ever used it.”

OWL existential and universal quantification is described in section OWL Quantification. For DL semantics see [BaMc2003]; for OWL-oriented modelling [AlHe2020].

The modus ponens rule represented by the OntoGraph in Figure modus-ponens may be written in sequent notation:
p → q ∧ p ∴ q.

In [Sowa2018a] the proof of the modus ponens implication is given by John Sowa using the existential graphs of Peirce. Wittgenstein argues in LW 6.1264 that the modus ponens cannot be expressed by a proposition, but that every meaningful proposition has the form of a proof and every proposition of logic is a modus ponens represented in signs. In addition, he argues in LW 6.1271 that the number of logical laws is arbitrary.

The modus tollens rule represented by the OntoGraph in Figure modus-tollens may be written in sequent notation:
p → q ∧ ¬q ∴ ¬p.