Opposition to Same-Sex Unions is Mathematically/Logically Wrong

Gary Hill

Using the basic principles of mathematical set theory, the conservative Christian activist Andrew Schafley has attempted to demonstrate the moral superiority of opposite-sex to same sex unions on the basis that opposite sex unions create a larger set:

“Traditional marriage provides a greater set than otherwise: the union of A = {a, b, c, d} and B = {a, b, c, e} is merely {a, b, c, d, e}, while the union of M (man) = {a, b, c, d} and W (woman) = {e, f, g, h} is {a, b, c, d, e, f, g, h}, which is a broader and more diverse set.”

The glaring problem with this formulation is obvious: on what grounds does Schafley maintain that a larger mathematical set is representative of a higher moral state of affairs than a smaller mathematical set? It's simply an assertion, the logical deduction is not obvious.

In response, two simple mathematically-based arguments are presented demonstrating that Schafley's moral opposition to same-sex unions is not logically possible unless the assumption is made that one biological sex possesses a greater degree of moral and personal autonomous value than the other. Consider the following statements and their binary truth values:

S1: All human beings, regardless of their biological sex, hold equal value (true/false)

S2: Same-sex unions are inherently and always immoral (true/false)

I demonstrate mathematically/logically that one cannot hold to both S1 and S2 as being equally true moral statements; specifically, if S1 is maintained to be true, then S2 must be false.

Let:

M = male; F = female

X = a morally permissible union; Y = an immoral union

⇒ (M + F = X)

⇒ (M + M = Y)

⇒ (F + F = Y)

If (M + F = X) ⇒ (X = 1M + 1F)

If (M + M = Y) ⇒ (Y = 2M)

If (F + F = Y) ⇒ (Y = 0M)

(each case: Y is determined solely by M)

∴ (M > F; i.e., M possesses greater value in apportioning moral status to a union than F)

∴ If S1 is claimed to be true, S2 must be false

Solution:

(M + F = X) ⇔ (M + M = X) & (M + F = X) ⇔ (F + F = X)

S1 is true & S2 is false

Let:

M = male; F = female

P = personhood, i.e., autonomous personal being able to understand and consent to union

1 = numerical value assigned to P (∴ 0 = ¬ P)

∴ MP = 1; FP = 1

⇒ MP + FP = (1P + 1P)

⇒ MP + MP = (1P + 1P)

⇒ FP + FP = (1P + 1P)

∵ from (i): (M + F) ≠ (M + M)

∵ from (i): (M + F) ≠ (F + F)

∴ (MP > FP; i.e., MP possesses greater value in apportioning moral status to a union than FP; MP possesses a higher degree of personhood than FP).

S1 is false

Solution:

From (i): (M + F + X) ⇔ (M + M = X) & (M + F = X) ⇔ (F + F = X)

∴ (MP + FP) = (MP + MP) & (MP + FP) = (FP + FP)

S1 is true, S2 is false

Of course, both formulae can be redefined with biological sex transposed and the opposite result obtained (i.e., F > M or FP > FM). This does not, of course, refute the argument in any way; although the formulae would no longer reflect the attitude of those most actively objecting to same-sex unions, the mathematical processes and conclusion would be unaffected; if S1 is considered to be objectively true, then S2 is false.

“Traditional marriage provides a greater set than otherwise: the union of A = {a, b, c, d} and B = {a, b, c, e} is merely {a, b, c, d, e}, while the union of M (man) = {a, b, c, d} and W (woman) = {e, f, g, h} is {a, b, c, d, e, f, g, h}, which is a broader and more diverse set.”

The glaring problem with this formulation is obvious: on what grounds does Schafley maintain that a larger mathematical set is representative of a higher moral state of affairs than a smaller mathematical set? It's simply an assertion, the logical deduction is not obvious.

In response, two simple mathematically-based arguments are presented demonstrating that Schafley's moral opposition to same-sex unions is not logically possible unless the assumption is made that one biological sex possesses a greater degree of moral and personal autonomous value than the other. Consider the following statements and their binary truth values:

S1: All human beings, regardless of their biological sex, hold equal value (true/false)

S2: Same-sex unions are inherently and always immoral (true/false)

I demonstrate mathematically/logically that one cannot hold to both S1 and S2 as being equally true moral statements; specifically, if S1 is maintained to be true, then S2 must be false.

**(i) Values placed on the moral standing of possible unions between two biological sexes**Let:

M = male; F = female

X = a morally permissible union; Y = an immoral union

⇒ (M + F = X)

⇒ (M + M = Y)

⇒ (F + F = Y)

If (M + F = X) ⇒ (X = 1M + 1F)

If (M + M = Y) ⇒ (Y = 2M)

If (F + F = Y) ⇒ (Y = 0M)

(each case: Y is determined solely by M)

∴ (M > F; i.e., M possesses greater value in apportioning moral status to a union than F)

∴ If S1 is claimed to be true, S2 must be false

Solution:

(M + F = X) ⇔ (M + M = X) & (M + F = X) ⇔ (F + F = X)

S1 is true & S2 is false

**(ii) Values placed on the autonomous person, regardless of biological sex**Let:

M = male; F = female

P = personhood, i.e., autonomous personal being able to understand and consent to union

1 = numerical value assigned to P (∴ 0 = ¬ P)

∴ MP = 1; FP = 1

⇒ MP + FP = (1P + 1P)

⇒ MP + MP = (1P + 1P)

⇒ FP + FP = (1P + 1P)

∵ from (i): (M + F) ≠ (M + M)

**⇒**(1P + 1P) ≠ (1P + 1P) ⇒ (MP ≠FP) & (MP > FP)∵ from (i): (M + F) ≠ (F + F)

**⇒**(1P + 1P) ≠ (1P + 1P) ⇒ (MP ≠ FP) & (MP > FP)∴ (MP > FP; i.e., MP possesses greater value in apportioning moral status to a union than FP; MP possesses a higher degree of personhood than FP).

S1 is false

Solution:

From (i): (M + F + X) ⇔ (M + M = X) & (M + F = X) ⇔ (F + F = X)

∴ (MP + FP) = (MP + MP) & (MP + FP) = (FP + FP)

S1 is true, S2 is false

Of course, both formulae can be redefined with biological sex transposed and the opposite result obtained (i.e., F > M or FP > FM). This does not, of course, refute the argument in any way; although the formulae would no longer reflect the attitude of those most actively objecting to same-sex unions, the mathematical processes and conclusion would be unaffected; if S1 is considered to be objectively true, then S2 is false.