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) or ∃αφ(α)) the variable alpha is now bound.
• Contradictory sentence: a sentence that must be false.
• 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 free.
• 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 premises.
• 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 terms.
• 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 formed formulas).