## Quantifier Rules

Quantifiers are symbols in logic such as ' ' or ' ' . These symbols means respectively 'for all' and 'there exists'.

The statement means that all all objects for all objects with some property, then every individual object has the property.

More precisely, suppose that X is a set of declarative sentences including the sentence and suppose D is a designator which occurs in a sentence in X. Suppose that is the sentence obtained by putting D in place of every free occurrence of 'x' in %phi then X therefore is a valid argument.

Suppose X is the set consisting of the two statements (1)

Uranium is an element (2)

We may take to be the statement and D to be 'Uranium' then is the statement (3)

The rule then states (1) and (2) imply (3).

The statement means that you can never create an inconsistency by naming a thing provided the name is not already used for something else.

More precisely suppose that X is a set of declarative sentences among which is a sentence of the form and suppose D is a proper name which occurs somewhere in X. Suppose that that is the sentence got by putting D in place of every free occurrence of 'x' in Then can be added to X without creating an inconsistency.

Suppose X is the set consisting of the three statements:   The Rule states that if X is consistent then so is the set Y consisting of X together with the sentence In fact X is inconsistent. 