A reminder of some definitions
A reminder of some definitions
- Arity: a property of predicates; the arity of a
predicate is the number of symbolic terms required to form
a well-formed formula with that predicate; or the number of
names required to form a sentence with that predicate. If
F is an arity one predicate, Fx is a well-formed formula,
and Fa is both a well-formed formula and also a sentence.
If G is an arity two predicate, Gxy, Gxa, and Gby are all
well-formed formulas; and Gab would be both a well-formed
formula and a sentence.
- Argument: a list of sentences, one of which we
call the conclusion, and the others of which we call premises.
(This is a strange definition for "argument," but it helps in
our logic if we define "argument" in a way that makes it a
very precisely described object that we can then study. Note
that only arguments can be valid or invalid; and note that only
sentences can be true or false. It is nonsense in our system
to talk about true arguments or valid sentences.)
- Atomic Sentence: a sentence that has no sentence
as a proper part (that is, the smallest possible kind of
sentence). Also: any of the the smallest parts of our
language that can be true or false.
- Bound variable: if a well-formed formula φ
contains a variable α and no quantifiers, then if we
put a quantifier with that variable before the formula
(either ∀α or ∃α, to make the
formulas ∀αφ or ∃αφ), the
variable α in φ is now bound.
- Contradictory sentence: a sentence that must
- Contingent sentence: a sentence that could be
true and could be false.
- Determinate: a property of predicates; a predicate
is determinate if, when it is used to form a sentence, that
sentence must be either true or false, not both or neither.
An arity n predicate followed by n names from the domain of
discourse for that language will form a sentence that is either
true or false.
- Enthymeme: an argument with missing premises.
- Fallacy: an invalid inference that is so common
we identify it with a name.
- Free variable: a variable that is not bound is
- Logically equivalent. (1) semantic definition:
sentences φ and ψ are logically equivalent if φ
and ψ have the same truth value in all situations; (2)
syntactic defintion: sentences φ and ψ are logically
equivalent if (φ ↔ ψ) is a theorem.
- Predicate: that part of our language used to refer
to properties. Predicates have an arity, which is the number of
symbolic terms that they require to make a well-formed formula;
or the number of constant that they require to make a sentence.
- Sound Argument: a valid argument with true
- Symbolic Term: either a variable, a indefinite
term, an arbitrary term, or a constant (proper name). A
constant is used to "pick out" or refer to exactly one thing.
- Tautology: a sentence that must be true.
- Theorem: a sentence that we can prove without premises.
- Valid Argument: an argument in which, necessarily
if the premises are true, then the conclusion is true.
(Note, some books and dictionaries include a bad definition
of "valid" -- something like, an argument made with a logical
system. This is bad because we used the notion of validity
to create our logical system. We could easily make a logical
system that allowed invalid proofs -- we don't do that
because we start with the notion of validity and use it to
guide our construction of our logic.)
- Well-Formed Formula: intuitively, anything that
has the correct shape to be a sentence but one or more of the
constants could be replaced by variables. A proper
definition must be recursive (it must describe the smallest
case -- a predicate of arity n followed by n symbolic terms
is a well formed formula) and show how we can build well
formed formulas (any well-formed formula with an ¬,
∀α, ∃α before it, or any two
well-formed formulas with a ^, v, →, or ↔ between
them, are well formed formulas). See pages in your book.