Propositional Logic - Syntax

As we showed above, propositional logic is consistent of symbols which take the place of atomic propositions and logical operators. There are two other building blocks in propositional logic which include its syntax and semantics.

Syntax refers to how to set up grammatically meaningful propositions in a symbolic logic system.

The following are examples of syntax:

‘The dog is brown.’ Is a sentence.

‘The dog is brown, and the dog is microscopic.’ Is a sentence with two propositions connected by a conjunction.

‘The dog and’ is not a full sentence.

In all of these cases we are looking at the grammar of the sentences, not the truth content nor interpretation. It is unlikely that the sentence in point (2) is true, but it can be said to be true or false as a proposition. It is impossible for sentence in point (3) to be converted into a proposition.

One aspect to using proper syntax in any symbolic logic language is to be sure that the propositions expressed are well formed formulas. Well Formed Formulas (wffs) are collections of symbols which properly represent a proposition according to formation rules. There is a collection of wffs that will be used to understand formation rules:

Atomic wffs: a wff that consists of a single propositional letter. Ex: P, R, S, D 

Complex wffs: a wff that contains at least one propositional letter and a truth functional operator. 

Literal wffs: a wff that consists of an atomic wff P, or a negated wff $ \neg P $ 

Subformula: a subformula Q of P, is any wff occurring as a part of P.

These concepts will be utilized in the production of truth trees. However, it may be useful to give some examples of subformulas for certain compound propositions:

Propositions

Subformulas

 

 

 

$ \neg P $

$ P $

 $ \neg P $

 

 

$ \neg(P \implies Q) $

 $ P $ $ Q $ $ P \implies Q $ $ \neg(P \implies Q) $

Table 1

Finally, it is useful to identify main operators and the scope of an operator. The scope of an operator in a wff P, is the smallest subformula of P that contains that occurrence of an operator. Meanwhile, the main operator in a wff is the operator with the greatest scope. The following are examples of these concepts:

1. What is the scope of $ \neg $ in $ \neg P $ ? It is $ \neg P $

2.  What is the scope of $ \implies $ in $ \neg ( P \implies Q ) $ ? It is $ P \implies Q $

3. What is the main operator of $ \neg ( P \land Q) $ ? It is $ \neg $

4.  The main operator of $ \neg P \implies \neg Q $ ? It is $ \implies $

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