She argues that such arguments were neither part of Aristotle's dialectic, nor simply the result of an adoption of elements of Stoic logic, but the outcome of a long, gradual development that begins with Aristotle's logic as preserved in his Topics and Prior Analytics; and that, as a result, we have a Peripatetic logic of hypothetical inferences which is a far cry both from Stoic logic and from classical propositional logic
, but which sports a number of interesting characteristics, some of which bear a cunning resemblance to some twentieth-century theories.
He begins by picking up discrete calculus, including proof by induction, and moves to selected area computation (pi, anyone?), limit's and Taylor's theorem, including series representations and Taylor polynomials, infinite series, including both the positive and the general, beginning logic, including propositional logic
, predicates and quantifiers, and proofs, real numbers, functions such as derivatives and a substantial pair of chapters on integrals.
Nevertheless, the restriction to propositional logic
has its downside.
A Semantical Analysis of Modal Logic I: Normal Modal Propositional Logic
. Zeitschrift fur Mathematische Logik under Grundlagen der Mathematik 9:67-97.
Using a quantified propositional logic
involving the operators "it is known that" and "it is possible to know that," the author lists formalizations of various interesting philosophical claims involved in the realism debate.
Thus, he articulates a new Stoic ethics employing modern scientific categories, and uses modern propositional logic
to support it, as Chapter 4 and its accompanying appendix demonstrate.
For example, one might use propositional logic
to describe the effects of actions, which would make it quite awkward to describe actions with universally quantified effects, such as a machine shop spray paint action that coats all objects in the hopper.
Work is not limited to first-order logic and propositional logic
He contends that Edwards himself developed a viable alternative to the classical-modern philosophical outlook by drawing explicitly upon the pre-modernist Renaissance propositional logic
of Peter Ramus.
In sections on informal logic, formal logic, and inductive logic, they consider such aspects as language: meaning and definition, categorical syllogisms, natural deduction in propositional logic
, causality and Mill's methods, and science and superstition.
In late 1950s, logician Hao Wang at IBM wrote a code called Program II (Wang 1960) to discover deep theorems in propositional logic
. For Wang, a theorem is deep if it is short but requires a long proof.
Some members of classical logic which are still being used up until now are Propositional Logic
[22,23,24], First-Order Logic [25,26,27,28], and Second-Order Logics [29,30,31] among others.