Symbolization into Propositional and Predicate Logic [Gentzen syntax]

Logical System


The symbolization of English statements or propositions can be done using instruments of varying degrees of logical sophistication. The presentation here is restricted to propositional and predicate logic. Researchers in linguistics, philosophical logic, advanced computer science, or artificial intelligence would likely use some more advanced form of logic. However, predicate logic, encompassing propositional logic, is not introductory and it is entirely adequate for many purposes. Being familiar with it is worthwhile intellectually (and it is non-trivial to learn).

An unfortunate consideration (which might mean more to you later)

Not all logicians, and logical texts, use the same symbols for the so-called 'logical connectives'. [In fact, pretty well every logical text is slightly different from every other one, usually in small details of the symbols, the syntax, or the rules. This is inconvenient, unnecessary (and pathetic). Imagine what would happen in mathematics if each text used a different symbol for the addition '+' sign, or for the integration sign!]

However, it is what it is and we have to face it. Here are typical possibilities

'not' : ∼ (the 'tilde'), ¬ (looks like the top right corner of a box)

'and': ∧, & (the ampersand), . (just a period)

'or': ∨ (usually just this, vel)

'implication': ⊃ , →

'equivalence': ≡, ↔

'existential quantifier': ∃, ∑

'universal quantifier':∀, ∏

So, in a logic book, you might see (A&B)→C and that is just the same as (A∧B)⊃C.

And you might see (∀x)(Fx ⊃ Gxy) and that might be just the same as ∀x(F(x)→G(x,y)).

The software running here can easily manage or render any of these. But the written English text of the Notes and explanations has a fixed form. So we have to go with one choice. You should find it easy though to translate to other forms if the choices of the present text are not to your liking.

We will start with propositional logic, then move on to the more advanced predicate logic.

Starting on propositional logic ...