## LOGICAL SYNTAX

Logic, like ordinary language, has a syntax, a set of rules determining which strings of symbols are meaningful within a particular language.  When a string of symbols satisfies the syntax rules, that string is said to be well-formed.  In logic well-formed formulas are called WFF's (pronounced "woofs").  It is important to be able to separate out WFF's from strings of symbols that are syntactically incorrect.  Below are several definitions and the rules of logical syntax, the rules of WFF formation.

Definitions:

 'Statement Letter' Any capital letter representing a simple statement 'Connective' a tilde(~), dot(●), wedge(▼), arrow (→), or double arrow (↔) 'Grouper' a parenthesis, bracket, brace, etc. 'unicode' a statement letter, connective, or grouper 'Formula' any horizontal string of symbols 'Left-hand grouper' a '(', '[', or '{' 'Matching right-hand grouper' the mirror image of a left-hand grouper 'Binary Connective' any connective other than a tilde

With these definitions we can now define a WFF

 A WFF is either (a) a statement letter (b) a tilde followed by a WFF, or (c) a left-hand grouper followed by a WFF followed by a   binary connective followed by a WFF followed by matching right-hand grouper*

NO other formula is a WFF!

*We will, for the sake of simplicity, allow a formula which would be a WFF under (c) if the formula were encased in a set of parentheses to count as a WFF.  So, while 'A B' is not, technically speaking, a WFF we will treat it as one since (A B) is a WFF.  We shall act as if the outermost set of parentheses were present.

Each  compound WFF is a substitution instance of a basic statement form.

Remember, our 5 basic statement forms are:

 Negation ~p Conjunction p ● q Disjunction p ▼ q Conditional p → q Biconditional p ↔ q

Remember, in a statement form the lower case letters are sentential variables, they represent sentences but are not themselves sentences.  The logical operators in our statement  forms are constants, they do not change.

A compound WFF  Φ is a substitution instance of the statement form Ψ if, but only if,  Φ can be obtained by replacing each sentential variable in  Ψ with a WFF, using the same WFF for the same sentential variable throughout.

Here's how it works:

The compound WFF (P Q) (if you doubt that this is a WFF, check it using the formal definition above) is a substitution instance of p q because if we replace the p in p q with P  and the q with Q we get  (P Q) in which the constant (in red) remains the same, but each sentential variable (in green and blue, respectively) are replaced by WFF's of the appropriate color.

To take a more complex example, the compound WFF  [(P   Q)  (~R S)] is a substitution instance of p q because if we replace the p in p q with (P   Q) and the q with (~R S) we get  [(P   Q)  (~R S)].

Finally, ~((P (Q ▼ ~R)) ((Q ● P) P)) is a substitution instance of ~p.  In this example the negated WFF, ((P (Q ▼ ~R)) ((Q ● P) P)) is a substitution instance of p q.  The left hand side of the biconditional, (P (Q ▼ ~R)) is a substitution instance of p q and the right hand side of the biconditional, ((Q ● P) P), is also a substitution instance of p q.  The consequent of the first conditional, (Q ~R), is a substitution instance of p q, with ~R being a substitution instance of ~p.  Similarly, the antecedent of the second conditional, (Q P), is a substitution instance of p q

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