**INTRODUCTION TO SYMBOLIC LOGIC**
**QUANTIFICATION RULES SIMPLIFIED VERSION**

When dealing with formal proofs involving quantified expressions always remember the following;

Instantiation:

The nineteen rules of deduction do not apply to quantified expressionsdirectly. Every quantified propositionmustbeinstantiatedbefore it can enter into the deduction through one of the two instantiation rules. In instantiating an expression the quantifier is dropped and every occurrence of the quantified symbol is replaced by the instantiating symbol. In other words, we reason from generalized (quantified) premisses by considering an "instance" to which that generalized statement applies. Thus we introduce two "instantiation rules" to apply each to universally quantified and existentially quantified expressions:

1. Universally quantifiedexpressions can beuniversally instantiatedintoeither i)any arbitrarily chosen variable, often called the "dummy vaiable," (for example: y),orii)can be instantiated into aconstant(e.ga, b, c...t) which is symbolizing a particular "named" individual.Whenevera universally quantified expression is instantiated, the rule which applies isUniversal Instantiation (U.I.)even if the instantiation is made into aconstantterm.2.

Existentially Quantifiedexpressions can beexistentially instantiatedonly into a constantwhich hasnot previously occurred in the deduction. The rule which applies, whenever an existentially quantified expression is instantiated isExistential Instantiation (E.I.) Note two important restrictions on "E.I."(which do not apply to "U.I."): a)You can use E.I.onlywith constants (e.ga, b, c...t) .b)If you use E.I.more than onceeach subsequent use must introduce anew instantiating constant.

Generalization:

Any argument with a general (i.e.quantified) statement as its conclusion must terminate with a move from instantiated statements to a general statement a3. Expressions may beuniversally generalizedonlywhen the instantiating symbol over which the generalization is made is avariable(the so-called "dummy" variable,y)nota constant.You can use U.G. only on variables.4.

Existential generalizationmay be performed on any expression instantiated into aconstant.

Two Restricting Conditions to remember:

I. Whenever a variable is instantiated always remember to replaceeveryoccurrence of that variable with the same instantiating term.II Any expression with a

negationin front of the quantifier cannot be instantiated. The negatlion symbol must first be removed by "flipping it over" the quantifier andchanging universal to existentialorvice versaas required by thequantifier Negationrules. Expressions in which the negation appearsbetweenquantifier and the expression quantified (i.e.the propositional function) may be instantiated, but the negation sign is part of the expression and must remain in its instantiated form.