Formal Logic
The last article introduced the concept of deductive validity, which I mentioned was a central concept to formal logic. This article will be more of an interest piece and overview than anything else, to give you an idea of what formal logic is, and also some (hopefully) interesting history behind logic.
Aristotle developed one of the earliest logical systems, based on the observation that certain arguments are valid based on their form alone, irrespective of their content.
For instance,
All Greeks are men
All men are mortal
Therefore,
All Greeks are mortal
This argument is clearly valid, but what makes it valid has nothing at all to do with Greeks or men or mortality; the validity comes from its form;
All A are B
All B are C
Therefore,
All A are C
We could substitute all sorts of things for A, B and C and the validity would remain. Hence we have formal logic.
Aristotle created a system which showed that arguments of certain forms were valid, and argued (incorrectly) that all valid arguments could be reduced or broken down into these arguments. These were labelled syllogisms.
Syllogisms had a few features. They were three lines; there were two premises and a conclusion. The form of the premises and the conclusion was restricted to a few different types of sentences, such as;
All A are B
No A are B
Some A are B
Some A are not B
You have probably heard the famous argument attributed to Aristotle that goes;
Socrates is a man
All men are mortal
Therefore,
Socrates is a mortal
This argument was never made by Aristotle, as it does not fit into his system at all! This very simple example is one of the types of limitations that Aristotelian logic comes up against. Even philosophers as recent as Kant claimed that logic was a complete science, when there had been only minor additions to Aristotle’s work at the time. By that time, syllogistic logic has been expanded to include ways of dealing with disjunctions and conditionals, as well as singular entities like ‘Socrates’.
This really is a massive simplification of Aristotle’s work on logic, and I do not mean to undermine its importance, sophistication or impact. I did want to briefly outline some of its limitations though, to help explain why modern logical techniques were invented. Logic is not a done deal, there is still debate going on and progress being made about various issues in logic.
Until the late 19th century, syllogistic logic remained as the standard, albeit with additions and variations to Aristotle’s original work. This makes it the last of Aristotle’s works to be superseded, having lasted a remarkable two millennia. Frege is credited with founding predicate logic (or predicate calculus), which is a different type of logic that allows us to deal with a much wider range of sentences, although many logicians have contributed to it. Propositions are expressed in symbolic form, and it also unites logic with mathematics; allowing logical proofs for mathematical concepts. There are some modern philosophers that still use syllogistic logic, but they are rare.
This invention of predicate logic had a profound effect on western thought as well as technology in the 20th century, so its impact and importance cannot be understated. I am not going to go into much detail here because it is rather more complex than Aristotelian logic and it uses symbols rather than regular language. For those interested, I may produce articles on formal logic after this critical thinking series.
I should make a few important points about syllogisms to round this discussion off. In modern usage, a ‘syllogism’ is not the same think as in Aristotelian logic, it is less limited. Be careful when using the term, as its modern meaning is different from the arguments put forward by Aristotle. Check out Wikipedia’s article on syllogisms for a nice overview at http://en.wikipedia.org/wiki/Syllogism.
There are also some commonly made logical inferences that are syllogisms, and we use the term as a kind of shorthand even in modern logical systems, so do not be confused if someone is talking about a disjunctive syllogism while they are dealing with predicate logic.