## 20 Feb 2009

### Stoic Logic, Mates, Chapter 1, §1 The Problem

Benson Mates

Stoic Logic

Chapter I: Introduction

§1 The Problem

Logic was first formalized by Aristotle. It remained largely the same up through Kant. But within fifty years following Kant, logic developed rapidly. It transformed into a discipline "as exact and adequate as any part of mathematics." (1-2)

In fact, some of these modern developments in logic were innovated by the Stoics. However, Stoic logic was neglected. So these advances had to be re-discovered many hundreds of years later.

Lukasiewicz showed two essential ways that Stoic logic differed from Aristotelian logic.
1) Stoic logic was a logic of propositions, while Aristotelian logic was a logic of classes.

2) Stoic logic was a theory of "inference-schemas," while Aristotelian logic was a theory of "logically true matrices." (2-3)

Also note that
a) Stoics used truth-functional definitions for all the common propositional connectives.

b) Stoics clearly distinguished arguments from the corresponding conditional propositions, and, most importantly,

c) the Stoics had a kind of calculus of inference-schemas: they took five inference-schemas as valid without poof and rigorously derived other valid schemas from these. (3a)

Mates' text will build from Lukasiewicz' findings, and make four additions:
1) The Stoics make a semantical distinction. It resembles Frege's and Carnap's sense-denotation and intension-extension distinctions. And, Stoic logic deals with propositions rather than sentences.

2) Many know about the Stoic debate regarding hypothetical propositions. One type of implication form has been termed the "Diodorean implication." But it hitherto has been misunderstood. Mates will rectify the misunderstanding regarding this type of implication as well as Chrysippus' "strict implication."

3) Lukasiewicz notes a Stoic logical principle similar to an important modern theorem:
an argument is valid if and only if the conditional proposition having the conjunction of the premises as antecedent and the conclusion as consequent is logically true. (4bc)
This Stoic principle resembles the "principle of conditionalization" and the "deduction theorem."

4) The Stoics believed that their system of propositional logic was 'complete.' For them, this means that every valid argument could be reduced to a series of arguments of five basic types. Yet, we cannot now know if this is true, because we are missing some of their meta-rules. (4c)

From:
Mates, Benson. Stoic Logic. Berkeley: University of California Press, 1973. [Originally published in 1953 as Volume 26 of the University of California Publications in Philosophy.]