Synergy User Manual and Tutorial. - THE CORE MEMORY

Synergy User Manual and Tutorial
Fido is a dog
Therefore, Fido is a mammal
The three statements:
p: All dogs are mammals
q: Fido is a dog
r: Fido is a mammal
are of the form:
p
q
∴ r
can be independently evaluated under propositional logic but cannot be evaluated to
derive the conclusion “r: Fido is a mammal” because “therefore” (‘∴’) is not a legitimate
propositional logic operator. We need to expand propositional calculus and set theory to
make use of the predicate calculus.
We use the universal quantifier ∀, which means for all or for every, to establish a
symbolic statement that includes all of the things in a set X that we are considering as
such:
∀x[Px→Qx]
The brackets define the scope of the quantifier. This example is read “For every variable
x in set X, if Px then Qx”. Applied to the example above, we could reword the statement
“All dogs are mammals” by letting Px be: “if x is a mammal” and Qx be “then x is a
mammal”. We have:
“For all x, if x is a dog, then x is a mammal”.
This is called a statement form and will become a statement when x is given a value. Let
f = Fido. A syllogism is a predicate calculus argument with two premises sharing a
common term.
∀x[Px→Qx]
Pf
∴ Qf
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