INTRODUCTION TO SYMBOLIC LOGIC
QUANTIFICATION RULES SIMPLIFIED VERSION
When dealing with formal proofs involving quantified expressions always remember the following;
The nineteen rules of deduction do not apply to quantified expressions directly. Every quantified proposition must be instantiated before 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:Generalization:
1. Universally quantified expressions can be universally instantiated into either i) any arbitrarily chosen variable, often called the "dummy vaiable," (for example: y), or ii) can be instantiated into a constant (e.g a, b, c...t) which is symbolizing a particular "named" individual. Whenever a universally quantified expression is instantiated, the rule which applies is Universal Instantiation (U.I.) even if the instantiation is made into a constant term.
2. Existentially Quantified expressions can be existentially instantiated only into a constant which has not previously occurred in the deduction. The rule which applies, whenever an existentially quantified expression is instantiated is Existential Instantiation (E.I.) Note two important restrictions on "E.I." (which do not apply to "U.I."): a) You can use E.I. only with constants (e.g a, b, c...t) . b) If you use E.I. more than once each subsequent use must introduce a new instantiating constant.
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 be universally generalized only when the instantiating symbol over which the generalization is made is a variable (the so-called "dummy" variable, y) not a constant. You can use U.G. only on variables.Two Restricting Conditions to remember:
4. Existential generalization may be performed on any expression instantiated into a constant.
I. Whenever a variable is instantiated always remember to replace every occurrence of that variable with the same instantiating term.
II Any expression with a negation in front of the quantifier cannot be instantiated. The negatlion symbol must first be removed by "flipping it over" the quantifier and changing universal to existential or vice versa as required by the quantifier Negation rules. Expressions in which the negation appears between quantifier 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.