# Propositional Logic

Propositional Logic (PL) allows us to by-pass many of the problems generated by the quirkiness of natural language if, instead of using the messy vernacular to express arguments, we use some much more austere and regimented formal language which is designed to be logically well-behaved and quite free of ambiguities and shifting meanings.

We start by

1. rendering the vernacular argument into a well-behaved artificial language
2. investigating the validity or otherwise of the reformulated argument.

Natural language arguments can be expressed in this artificial language and they can then be analysed for validity. How is PL designed?

1. There are three connectives:
1. Connectives connect connective free sentences. These sentences are called base class if items that express propositions. The symbols used for them are:

That is the basic alphabet of PL. To this an inference marker, ∴ and absurdity marker, ∗ , along with the comma, ,, , make up the full alphabet.

Syntax

Now the syntax. What a the rule for constructing or recognising well-formed-formulae wff? The set of wff are defined formally as:

(A1) ' P ','Q', 'R' and 'S' are atomic wffs.

(A2) Any atomic wff followed immediately by a prime ‘ ′ ’ is an atomic wff.

(A3) Nothing is an atomic wff other than what can be shown to be so by the rules (A1) and (A2).

Having fixed the class of atomic wffs, we next need to give the rules for building up more complex wffs out of simpler ones to arrive at the full class of wffs. For the record, we should state explicitly that the atomic wffs do count as belonging to the full class of wffs, atomic or molecular: so,

(W1) Any atomic wff is a wff.

(W2) If A is a wff, so is ¬A .

(W3) If A and B are wffs, so is (A∧B) .

(W4) If A and B are wffs, so is (A∨B) .

(W5) Nothing else is a wff.

Main connectives

The main connective of a non-atomic wff is the connective that is introduced at the final stage of its construction tree.

Subformulae and scope

Here are two more useful syntactic notions which again are neatly introduced using the idea of a construction tree: A wff S is a subformula of a wff A if S appears anywhere on the construction tree for A.

The scope of a connective in A is the wff on A’s construction tree where the connective is introduced.

Semantics

In the Syntax connectives were defined in an unambiguous way. This allows certain ordinary language complex statements to be translated into formal statements. This allows the analysis of arguments. However the analysis need to be interpreted back in to something close to ordinary language to communicate it effectively. However while not reintroducing the ambiguities that PL was created to eliminate.

1. No structural scope ambiguities
1. Bare conjunction, no more and no less: i.e. (A∧B) is always to be true just when A and B are both true.
2. ∨ which is stipulated invariably to signify inclusive disjunction, no more and no less. So (A∨B) is always to be true just when A is true or B is true or both. '∨' is often read ”vel”, which is Latin for inclusive ‘or’.
3. The negation prefix ‘¬’ always works as the English ‘It is not the case that’ usually works; i.e. ¬A invariably expresses the strict negation of AA, and is true just when AA is false.
4. Every occurrence of ‘∧’ and ‘∨’ is to come with a pair of brackets to indicate the scope of the connective, i.e. to demarcate exactly the limits of what they are connecting;
5. But since the negation sign ‘¬’ only combines with a single sentence, we don’t need to use brackets with that.
6. In a sentence of the form A∨(¬B∧C)), just B is being negated;
7. In (A∨¬(B∧C)), all the bracketed conjunction (B∧C) is negated;
8. In ¬(A∨(B∧C)), , the whole of (A∨(B∧C)) is negated.
• P,Q,R,S, serving as the atoms of the language. This set can be augmented using the prime symbol, e.g. P'.