Chapter Two – Glossary of Terms

Philosophy 202

ARGUMENT: A series of statements in which one (the conclusion) purportedly follows from the others (the premises).

ARITY: The number of argument places that a predicate or function has; that is, the number of individual constants that must be assigned to a predicate in order to form an atomic sentence, or the number of terms that a function combines with to yield another term. Arity derives from terminology such as unary and binary, as applied to predicates or functions that have one argument place and two arguments places, respectively.

ATOMIC SENTENCE: A sentence formed by assigning the right number of individual constants to a predicate (i.e., a number equal to that predicate's arity).

FITCH-STYLE SYSTEM: A deductive system, named after logician Frederic Fitch, in which one can present formal proofs. This is the deductive system we use.

FUNCTION: A symbol that combines with a term or terms (e.g., an individual constant, a name, etc.) to form another term. For example, left-foot(Steve), mother(Constance).

INDISCERNABILITY OF IDENTICALS: If a=b, then anything that can be proved about a holds also for b.

INDIVIDUAL CONSTANT: A symbol used to refer to some fixed individual or other, but never to more than one.

INFIX NOTATION: Notation that involves placing the predicate symbol between its arguments (e.g., MichaelTALLERJoshua).

LOGICAL CONSEQUENCE: A relation between one claim and other claims when the former follows logically from the latter. Some claim is a logical consequence of claims if whenever are true, is also true.

LOGICAL SUBJECTS: Terms, such as individual constants, that fill the argument places of predicates.

PREDICATE: A symbol used to denote some property of objects or relation between objects.

PREFIX NOTATION: Notation that involves placing the predicate symbol in front of its arguments (e.g., TALLER(Michael, Joshua) ).

PROOF: A step-by-step demonstration that a given conclusion is a logical consequence of some premises; that is, it establishes with certainty that whenever the premises are true, the conclusion is also true. Proofs can be informal, typically stated in natural language and lacking mention of the more obvious steps, or formal, presented in stylized fashion according to certain fixed rules.

PROPERTY: A quality, such as red or heavy, that can belong to particular objects.

PROPOSITION: (Also, claim) What is expressed by a sentence and has a truth value.

REFLEXIVITY: A property exhibited by a relation R if R(a,a), for arbitrary a. For example, identity is reflexive, since a=a, for arbitrary a.

REFLEXIVITY OF IDENTITY: (REFL =) A rule of inference according to which any sentence of the form n=n can be validly inferred from any premises whatsoever, or from no premises at all.

RELATION: A quality, such as taller-than or father-of, that obtains between two or more objects ordered in a particular way.

REITERATION: (Reit) A rule of inference according to which any sentence that you have validly inferred can be validly reasserted; that is, this is a rule that allows you to repeat an earlier step in a proof.

SYMMETRY: A property exhibited by a relation R if whenever R(a,b), then R(b,a). For example, identity is symmetric, since if a=b, then b=a.

TERM: A noun phrase that intuitively refers to individuals and behaves like an individual constant (e.g., Jack, father(Jack), father(father(Jack))).

TRANSITIVITY: A property exhibited by a relation R if whenever R(a,b) and R(b,c), then R(a,c). For example, identity is transitive, since if a=b and b=c, then a=c.

TRUTH VALUE: A value assigned to a claim made by a sentence depending on whether or not the item (or items) in the world named by the individual constants have the property (or stand in the relation) expressed by the predicate. The values we recognize are TRUE and FALSE.

VALIDITY: (Also, logical validity) A property of arguments whose conclusions must be true on the assumption that their premises are true. An argument is logically valid if its conclusion is a logical consequence of its premises.