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Logic

Symbols Made Simple

A quick and friendly introduction to symbolic logic by Stephen Szanto.

Most non-professional philosophers are deterred from attending lectures and reading books by academics who use symbolic logic. Some even claim it is an elitist attempt to make presentations deliberately inaccessible to the uninitiated. In any case, I believe it is worth studying and needn’t be a scary as it at first looks. I hope this ‘child’s guide’ to modern philosophical formalism will provide a bridge between these two groups of philosophers.

The two most intimidating symbols are ‘∃’, standing for ‘one’ or ‘some’ or ‘somebody’, and ‘∀’, standing for ‘all’ or ‘every’ or ‘everybody’. They were designed by the Italian mathematician and logician Giuseppe Peano (1858-1912) and they are usually combined with another letter or letters which stand for the statement of our choice. Don’t panic, here is an example:

(∀x) (∃y) (y causes x)

This means “for all x there exists a y such that y causes x,” or, to put it in more everyday terms, “everything has a cause.” Which is a true statement, if we forget about quantum mechanics which has no bearing on our logic in an everyday sense. ∃ and ∀ are called ‘quantifiers’; ∃ the existential quantifier and ∀ the universal quantifier. Let’s now change the order of the quantifiers and we get:

(∃y) (∀x) (y causes x)

This means “there is one y such that for all x, y causes x,” or, more simply put, “there is one cause for everything.” This is considered false by atheistic humanists and materialists, but is passionately declared true by monotheistic devotees, who believe in one God as the maker and cause of the universe. However, if we restrict the statement to nature alone without bringing in metaphysical concepts of theology, we can say that the second proposition is false. For this reason, the illusion that we can reverse the quantifiers in a true statement and end up with another true statement is called the Quantifier Shift Fallacy. Even before we used philosophical symbols this type of fallacy was known to philosophers and theologians. The great Aristotle himself was accused of having committed it at the beginning of his Nicomachean Ethicsas was the equally great Thomas Aquinas whose argument from causation is also held to be fallacious in this way.

“Hah”, I can hear you exclaiming, “So if all this could be done without those confusing symbols, what’s their use then?” “Well,” the formalists would say in defence of their abbreviated symbolic expressions, “isn’t it simpler, neater and more elegant to express our arguments in this concise way than scribbling pages and pages of convoluted arguments?”

The philosopher Immanuel Kant hated his contemporary psychiatrists (whose introspective experiments he regarded as philosophically untenable) so much that he wanted philosophers to attend court cases instead of the experts in the field. One can only imagine how much he would have loved to translate large volumes of psychiatric wisdom on the paranoid personality type who thinks that everybody hates him – but whose paranoia actually stems from his unconscious hatred of everybody else – into the following two elegant formulations:

(∀x) (∃y) (x hates y)

meaning “everybody hates me” and

(∃x) (∀y) (x hates y)

meaning “I hate everybody”, which is the cause of the person’s psychiatric problem. Unfortunately in Kant’s time such elegant symbolic formulations had not been invented. I hope you realised that this is another example of the Quantifier Shift Fallacy.

“Yes, yes,” you moan sadly, “but I turned to philosophy to find out what it is all about. Can symbolic logic help me with this?” Well, Frege, Russell and Wittgenstein, in his earlier ‘phase’, as well as the recently deceased Quine and several other outstanding contemporary philosophers think that the answer is ‘yes’. By introducing this strict new logical language we can learn a great deal about the world as it really is. To make this more tangible, let me quote a widely-known example. For a long time mankind was aware of the morning star and the evening star and regarded them as two different bright heavenly bodies. They were mysterious and many stories were attached to them. Then astronomers discovered that they were just one star, the planet Venus. Similarly, with the help of formal logic, some modern philosophers hope to purge our language of obscurity and confusion and show the world in its stark reality.

On the other side more and more equally brilliant thinkers claim that this enterprise has failed and is ‘dead’. Whatever the case, the enterprise is an outstanding achievement in the history of philosophy, and it is well worth the effort to form at least an intuitive insight of its essence. This brief introduction was meant to give you some confidence that it can be understood ‘without tears’ and perhaps also to whet your appetite.

Stephen Szanto is a medical doctor and also has a PhD in philosophy from London University.