A reminder of some definitions
A reminder of some definitions
Here are some of the definitions we discussed, roughly in the order that
we will encounter them.
- Arity: a property of predicates or functions. 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. The arity of a function
is the number of symbolic terms that must be combined with
the function in order to form a symbolic term. Thus, if
A is an arity one function, and b is a name (a term) of our
language, then Ab is a symbolic term; in this case, it acts
just like a name.
- 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 (for which we write
φ(α)) and no quantifiers, then if we put a
quantifier with that variable before the formula (either
∃αφ(α)) the variable alpha is now
- Contradictory sentence: a sentence that must
- Contingent sentence: a sentence that could be
true or 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 and not 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
- Predicate: that part of our language used to
express 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 constants/names that
they require to make a sentence.
- Sound Argument: a valid argument with true
- Symbolic Term: either a name, variable, indefinite
term, or arbitrary term. A constant or a name is used to
"pick out" or refer to exactly one thing. Indefinite names
pick out a single unknown thing. An arbitrary term picks
out an arbitrary instance. We have been using "a, b, c..."
as constants/names, and "x, y, z..." as variables, and "p, q,
r..." as indefinite names; and "x', y', z'..." as arbitrary
- Tautology: a sentence that must be true.
- Valid Argument: an argument in which, if the
premises are true, then the conclusion must be true. (Note,
some books, including ours, include a terrible definition of
"valid" -- something like, an argument made with a logical
system. This is truly terrible 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 it; or any two well-formed
formulas with a ^, v, →, or ↔ between them, are well