Karnpo asked:

What is the relation between Stoic logic and symbolic logic?

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As in modern propositional logic, the patterns of reasoning described by Stoic logic are the patterns of
interconnection between propositions.

The Stoics examined a number of ways in which two propositions can be combined to give a third,
more complicated proposition.

The key idea behind the Stoics' approach to logic was that you do not (have to) know what the
constituent propositions are about, or even whether each constituent proposition is true or false. All
that you know is that any proposition must be either true or false. When the Stoics came to analyze
the combining of two propositions by one of the connectives, they did so by looking at the pattern of
truth and falsity.

For example, in the case of 'and', the pattern is straightforward:

If both p and q are true, then the proposition 'p and q' will be true; if one or both of p and q are
false, then 'p and q' will be false.

It is of interest to note that at no time did the Stoics hit upon the idea of using algebraic notation, with
letters denoting arbitrary propositions and symbols denoting connectives. They wrote everything out
in ordinary language. This often resulted in their having to write down long and complicated
sentences that are difficult to follow, and that almost certainly hampered their possible progress in
logic.

The Stoics expressed the truth pattern for 'p and q' in the following cumbersome fashion:

If the first and if the second, then the first and the second. If not the first, then not the first and
the second. If not the second, then not the first and the second.

Present-day logicians bring out the abstract pattern of connectives such as 'and' by using algebraic
notation. The letters p, q, r are generally used to denote arbitrary propositions, and a symbol such as
& is used to abbreviate the word 'and'. Thus, [p&q] denotes the proposition [p and q]. Modern
logicians generally display such a 'pattern of truth' in a tabular form, using a truth table, a nineteenth
century device not available to the Stoics. These innovations led to modern formal or symbolic logic.

The Stoics formulated five rules of inference. Expressed in modern-day algebraic notation (with the
word 'aut' denoting 'or' in the exclusive sense, the symbol -> expressing 'if...then...' and the symbol ~
expressing 'not'), they are:

From [p -> q] and p deduce q.

From [p -> q] and ~q deduce ~p.

From ~[p & q] and p deduce ~q.

From [p aut q] and p deduce ~q.

From [p aut q] and ~q deduce p.

The first of these rules is the modern-day logical inference rule of 'modus ponens'. Here is how the
Stoics themselves expressed this rule:

If the first then the second, and if the first, then the second.

Starting with their five inference rules, the Stoics were able to deduce a number of other patterns of
reasoning. For example, they showed that the following deduction is valid:

From [p -> [p -> q]] and p deduce q.

Using the Stoics' own terminology, this was expressed like this:

If the first then if the first then the second, and if the first, then the second.

Given algebraic notation and the modern technique of truth tables, much of Stoic logic reduces to
some simple algebraic manipulations together with the filling-in of truth-values in a table. However, it
took over two thousand years for Mankind to reach that stage. Not having access to such modern
tools, the Stoics had a much harder time establishing their results. But still they established them.

Together with Aristotelean logic, Stoic Logic paved the way for all subsequent work in logic, right up
to the present day, and led to much of twentieth century logic and computer science.

Simone Klein