Leibniz’s Lawa = b ↔ ∀X(Xa ↔ Xb)
The Identity of Indiscernibles∀X(Xa ↔ Xb) → a = b
The Indiscernibility of Identicalsa = b → ∀X(Xa ↔ Xb)
∃Z(Zab & Zcd)
∀Z(∀x∀y(Zxy → ~Zyx) → ∀x~Zxx)
Al is a frog.Beth is a frog.∴ Al and Beth have something in common.(Nolt 382)
Fa, Fb ⊢ ∃xHabx(Nolt 382)
Fa, Fb ⊢ ∃X(Xa & Xb)where ‘X’ is not a specific predicate, but a variable replaceable by predicates – a variable that stands for properties. Thus the conclusion asserts that there is a | property X which both Al and Beth have [...].(Nolt 382-283)
A logic which quantifies over both individuals and properties of individuals is called a second-order logic, as opposed to systems such as classical predicate logic, which quantify only over individuals and are therefore called first-order logics.(Nolt 383)
Socrates is snub-nosed.
Being snub-nosed is an undesirable property.
∴ Socrates has a property that has a property.
Logics which quantify over properties of properties in this fashion are called third-order logics. And there are fourth-order logics, fifth-order logics, and so on. Any logic of the second order or higher is called a higher-order logic. Higher-order logics use a different type of variable for each domain of quantification (individuals, properties of individuals, properties of properties of individuals, and so on). An infinite hierarchy of higher-order logics is called a theory of types.(Nolt 383)
∃x∃y(~x = y & ∀z(Mz ↔ (z = x ∨ z = y)))
∃x∃y(~x=y & ∀z(Pz ↔ (z = x ∨ z = y))(Nolt 384)
More generally, using the predicate variable ‘X’, we can say that exactly two things have property X like this:∃x∃y(~x=y & ∀z(Xz ↔ (z = x ∨ z = y))This expression is in effect a one-place predicate whose instances are properties rather than individuals.(Nolt 384)
mathematics itself is nothing more than logic. More precisely, they argued that all mathematical ideas can be defined in terms of purely logical ideas and that all mathematical truths are logical truths. This thesis, known as logicism, has, however, met with serious technical difficulties and is now in disrepute.(Nolt 384-385)
is the principle that objects are identical to one another if and only if they have exactly the same properties. In formal terms:
a = b ↔ ∀X(Xa ↔ Xb)(Nolt 385)
identity of indiscernibles:
∀X(Xa ↔ Xb) → a = b(Nolt 385)
indiscernibility of identicals:
a = b → ∀X(Xa ↔ Xb)(Nolt 385)
The first of these formulas says that objects that have exactly the same properties are identical, and the second says that identical objects have exactly the same properties. Leibniz's law is equivalent to their conjunction.(385)
Al loves Beth.∴ Al has some relation to Beth.
‘Lab ⊢ ∃ZZab’(Nolt 385)
a stands to b as c stands to dFor instance:Washington D.C. is to the USA as Moscow is to Russia,the analogy here being the relationship between a country and its capital city(Nolt 386)
∃Z(Zab & Zcd)This says that there is some respect in which a stands to b as c stands to d.(Nolt 386)
A relation R is asymmetric in the set A if, for every x and y in A, whenever xRy, then it is not the case yRx. In symbols:R asymmetric in A ↔ (x)(y)[x ∈ A & y ∈ A & xRy → –(yRx)].The relation of being a mother is asymmetric, for obvious biological reasons.(Suppes 214)
A relation R is irreflexive in the set A if, for every x in A it is not the case that xRx. In symbols:] Nolt explains that
R irreflexive in A ↔ (x)(x ∈ A → –(xRx)).
The relation of being a mother is irreflexive in the set of people, since no one is his own mother. The relation < is irreflexive in the set of real numbers, since no number is less than itself.
Nolt then makes this formalization:An asymmetric relation is one such that if it holds between x and y it does not hold between y and x. An irreflexive relation is one that does not hold between any object and itself.(Nolt 386)
‘All asymmetric relations are irreflexive’
∀Z(∀x∀y(Zxy → ~Zyx) → ∀x~Zxx)