Posted: May 18, 2012 7:20 am
amkerman wrote:Actually Thommo I will attempt to explain why you cannot talk about non-existent "x's" in classical logic before I end my discourse on the subject.

If x doesn't exist, there is no true statement that can be possible made about x. If x doesnt exist, x is meaningless. X->,V,=,etc y cannot possibly be true. -x is not true either, It is vacuous.

You can attempt to.

If x is some element of the domain of discourse, then x->y is not a wff in the first place, nor is ¬x, let alone them being statements (which are wffs with no free variables). Its status as existent or non existent doesn't come into the matter in the slightest. These things aren't wffs/statements in free logic either by the way. To get from a term to a wff you need a relation symbol. E.g > , = , < . For example, if x , y are terms then x=y is a formula. We can apply connectives to formulas, but not to terms, typically we represent formulas with upper case letters A, B, C... and retain lower case for variables and constants, with c, k as typical symbols for constants and x, y, z for typical variables.

Example
let x, y be terms.
let A denote the formula "x=y"
let B denote the formula "x=x"

Then A->B is a tautology. Whereas x->y is garbage.

What you're actually driving at here (because there is some truth to the loose statement "you cannot talk about non-existent "x's" in classical logic") is that in classical logic existence is not a predicate.

What this means is that whilst it is possible to discuss things which are logically contradictory (e.g. "x is a circle and x is a square") there can be no element of the domain that denotes "a thing that is a circle and is a square". So to represent Shrunk's statement about square circles, we would take a standard geometric definition of a square and represent it by a formula with a free variable that represents a set of points, and the formula S(x) would (avoiding symbolic logic for the moment) say "the set of points x forms a square" - note here that the domain is sets of points, so x as a variable symbol ranging over this is fine as long as we restrict ourselves to non-trivial (non-empty) geometries*. Similarly we would take a formula C(x) saying "the set of points x forms a circle". Thus we have square circles defined by the formula (S(x)∧C(x)).

So to emphasise, the term in this formula is "x", which represents an arbitrary set of points in plane geometry, the formula is (S(x)∧C(x)) "x is a square and x is a circle" and is unsatisfiable - i.e. there is no set of points in plane geometry which constitutes a square and which constitutes a circle. Or if you prefer ¬∃x(S(x)∧C(x)) is a valid statement in the theory of plane geometry.

*We could use free logic and allow talk of the trivial geometry too, but why bother - it's trivial.