Glossary of Terms
Ambiguity: Having more than one meaning. Natural languages, their words, phrases and
sentences are inherently ambiguous (according to Bertrand Russell). The language tools of Logic
can help to disambiguate or eliminate ambiguity by showing the logical possibilities of a given
statement and presenting them in a way that is careful, clear, precise, and explicit. (See Logical
Languages)
Analogy: A comparison. In reasoning, analogies are used to compare arguments with respect to
both their form and their content. Analogies of content are the basis of one common form of
Inductive argument (the argument from analogy). Analogies of form are the basis of providing a
counter argument as a demonstration of invalidity in deductive logic, and for recognizing validity
and invalidity in different arguments that share a common form. (See Counter Example; and
Rules of Inference)
Antecedent: In a conditional statement, the statement on the left hand side of the operator; the
“if” part of an “if-then” statement. The antecedent states the conditions under which the
consequent is claimed to be true. (see Consequent)
Argument: 1) A set of statements in which the assumed truth of one (or more) of those
statements (called premise(s)) is intended to provide reasons, evidence, or grounds for the truth
of another (called a conclusion). 2) An expression of our reasoning. 3) A premise or set of
premises offered to support the truth of a conclusion..
Arguments are a type of discourse. We use this type of discourse to persuade,
demonstrate or otherwise validate the truth of our beliefs and opinions and to dispute the validity
of beliefs and opinions we do not agree with. Arguments are classified as one of two types,
inductive or deductive. Both types of arguments express our reasoning. Both are equally useful,
in specific contexts. In both inductive and deductive categories, there are specific criteria for
evaluating or determining if a given argument reflects good or bad reasoning.
Argument analysis: the separation and identification of the individual statements in an argument
as premise(s) and conclusion. In this class the process continues through “showing the argument
form.” The final step of argument analysis is the presentation of its form using a logical
language.
Argument evaluation: assessing the quality, value or worth of an argument. We can assess
arguments according to their content, their structure and their persuasive force. Inductive and
deductive arguments have different standards of evaluation within these categories.
Artificial language: A symbolic language developed for the specific purpose of revealing the
logical structure of a statement or argument. In this course we will use two such languages:
Sentential and Predicate. (See Logical Language).
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Assumption: In a deductive proof, a statement that is temporarily introduced for the specific
purpose of furthering the proof. An assumption may be entered at any time in a proof and
functions as a temporary premise, but unlike a premise, an assumption must be closed in
accordance with a specific rule. In this course, there are three rules of assumption: Conditional
Introduction, Negation Introduction/Elimination, and Disjunction Elimination.
Bi-conditional: A logical operator that abbreviates the expression, (p e q) C (q e p). It is
symbolized using the three-bar equivalence (/), as p / q. A double arrow ( ] ) is an alternative
symbol. A bi-conditional is true when both sides of the equivalence have the same truth value. It
is false when each side has a different truth value. Also called a logical equivalence. (See logical
equivalence)
Binary logic: A logic based on two values. This logic class teaches a binary logic. It works
with two values. These values are True (T) and False (F).
Categorical Statement: One of several statement forms that makes assertions about the
relationship between two things (classes or individuals). Categorical statements either present
generalized claims (using quantifiers “all” or “some,”). The basic forms of categorical
statement are:
All S is P, Some S is P, No S is P and Some S is not P,
where “S” stands for the grammatical subject term, and “P” stands for grammatical predicate
term, respectively.
There are also two forms of statements where the grammatical subject term is a named or
otherwise specified individual (an individual constant) and is linked to a specific class or
quality named by the grammatical predicate term.
Categorical Syllogism: A specific type of deductive argument that consists entirely of
categorical statements. A categorical syllogism consists of exactly two premises and one
conclusion, and relates exactly three identifiable terms. Each premise of the argument relates
two terms. One of these terms (the middle term) appears in both premises but not in the
conclusion. The two other terms appear once in each of the premises and are combined in the
conclusion.
Compound Statement: A truth functional statement that combines at least one truth
functional simple statement and a logical operator; a statement that contains another statement
as a component. The basic forms of truth functional compound statements are: negation,
conjunction, disjunction, conditional and bi-conditional. (See Simple Statement)
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Conclusion: A statement that is the logical endpoint in a process of reasoning. In an
argument, a conclusion is said to be true based on the assumed truth of its premise statements.
In deductive reasoning, the conclusion’s truth is claimed to follow with absolute certainty
based on the assumed truth of the premises. (See Inference, Derivation)
Conditional: A logical operation that is only false when its antecedent is true and its
consequent is false. It reduces to the expression ~ (p C ~ q), and is most commonly expressed
in English using an “if-then” phrasing. It is symbolized by the e or “horseshoe” symbol. A
single arrow is an alternative symbol (Y)
Conjunction: A logical operation that can only be true if its component conjuncts are all true.
It is symbolized by the “dot” C symbol. Alternative symbols are the ampersand (&) and v
symbol.
Consequent: In a conditional statement, the right hand side, or the “then” statement in an “ifthen” statement. (See antecedent)
Consistent: This term refers to a set of statements. A set of statements is consistent when it is
possible that all of them are true at the same time. In a truth table this will be represented by
any row in which the set of statements are all true. A valid argument whose statement set is
consistent has a possibility of being sound. (See Inconsistent)
Contingent: A statement that has the possibility of being true or false. This is shown in a truth
table by any combination of T and F values in the column for the statement.
Contradiction: A statement that is formally false, or false under any and all circumstances as
shown by a column in a truth table. A common form of contradiction is p C ~p.
Counter Example: An argument that 1) shares the form of another argument and 2)
demonstrates the invalidity or weakness of that argument by showing its form has the
possibility of having true premises and a false conclusion. (See valid, invalid)
Deductive reasoning/ argument: Reasoning seeks certainty. Deductive arguments intend that
the truth of the premises provide absolute, certain support for the truth of the conclusion. The
conclusion of such arguments is said “to follow” from the premises with necessity. Deductive
arguments are tightly structured and will have the terms or their conclusion statement(s)
contained somewhere within the premises. (See inductive reasoning)
Deductive proof: A specific type of demonstration that shows in clear, linear, transparent
and explicit terms exactly how a given conclusion can be drawn out of a given set of premises.
Each step of a proof is either 1) an initial premise, 2) an assumption or 3) a derivation. Proofs
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are the gold standard, or the highest quality of demonstration in logic. (See Two-column
Proof)
Derivation: a conclusion drawn forward from preceding statements and justified as valid by
reference to a rule of inference. Derivations are listed after the premises in the right-hand
column of a two-column proof. (See Conclusion, Inference)
Derivation column: The left-hand column of a two-column deductive proof. It begins with
the premises of the argument and then develops a series of derivations or conclusion, and
assumptions to show exactly and precisely how the given conclusion can be pulled out or
“derived” from the initial premise set.
Disambiguate: To establish a single interpretation for a word, term, or statement that has more than
one possible meaning, or to show the multiple interpretations possible for a single statement.
Disjunction: A logical operation that expresses alternatives. An exclusive disjunction allows only
the possibility that one disjunct or alternative is true. An inclusive disjunction allows the possibility
that both disjuncts are true and is only false when both disjuncts are false. In English a disjunction is
most often expressed using an “(either)-or” phrasing. It is symbolized with the “vee” (v) symbol.
*Enthymeme: A syllogistic argument in which one statement, either a premise or the conclusion, is
implied but not explicitly stated. It is thought that the structure is sufficiently strong that the implied
statement can be easily supplied.
Existential Quantifier: A specific symbol, (›x), in the language of Predicate, that indicates “there
exists” or “there are some,” where “some” means “at least one.” Existentially quantified statements
carry the assumption that members of the class designated by the terms do exist.
Fallacy/ fallacious: A logical error. A mistake in reasoning, as in a fallacious argument. Fallacies can
be formal (a mistake in the structure of the argument) or informal (a mistake in the content of the
argument). “Affirming the consequent” and “denying the antecedent” are two common formal
fallacies.
Formal Argument: An argument that is symbolized using a formal, artificial, logical language such as
Sentential or Predicate.
Immediate Inference: An inference that can be drawn directly from a single premise. All of the
quantifier rules, and the rules of Conjunction Elimination, and Disjunction Introduction are immediate
inferences.
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Inconsistent: Any set of statements that does not have the logical possibility of being true. (Meaning at
least one statement is false.) In a truth table, no row will contain all Ts, for the statements considered.
Indicator Language: Words and phrases in any natural language that indicate how given statements
function (as premises or conclusions) in the context of an argument. There is no requirement that an
ordinary language argument make use of such language, and many indicator words have multiple
meanings. Words and phrases that indicate conclusions include thus, therefore, so, for this reason, in
conclusion, we can conclude, etc. Words and phrases that indicate premises include because, since, the
reason(s) is/are, for, etc.
Individual constant/name: In the language of Predicate, a symbol (lower-case letters a - w) that
represents a specific, named or otherwise designated individual. Individual constants are indicated in
ordinary language using a specific name (a proper noun), a descriptive phrase, or a personal or
demonstrative pronoun.
Inductive Argument/Reasoning: Reasoning / argument that intends that a set of premises supports
the truth of a conclusion with some degree of probability or likelihood. Inductive arguments can
have form and structure but the relationship between the premises, while it may provide a strong
case for the conclusion, can never guarantee the conclusion with absolute certainty. Inductive
arguments are evaluated on a scale of very weak to very strong.
Inference: A conclusion; a type of thinking that involves moving from a belief, assumption, or
truth claim, (or set of beliefs, assumptions or truth claims) to the assertion of some other claim. As a
conclusion, an inference is the end point of that process. (See Conclusion, Derivation)
Informal Argument: An argument presented (oral or in writing) in an ordinary, natural language.
(See formal argument)
Invalid(ity): The quality of a deductive argument in which it is (logically) possible for the set of
premise statements to be true and conclusion false. Invalidity is shown by a row(s) in a truth table.
(See validity)
Justification column: In a two-column proof, the right-hand column. The justification column
presents the explicit justification for each derivation made in the left-hand column by citing a
specific rule of inference and prior statement(s) that serve as premise(s) relative to that derivation.
In effect, the information in the justification column presents the premises or reasons for the validity
of the corresponding derivation. (See derivation column)
Logic: The study of methods and principles used to distinguish correct (good) from incorrect (bad)
reasoning. As a discipline, Logic offers a range of techniques, strategies and methods of analyzing
and evaluating arguments.
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Logical Equivalence: A logical relationship in which two or more statements are determined to
have the same truth value under the same circumstances. This can be demonstrated using a truth
table. If the statements are logically equivalent, the columns for these statements will show the same
pattern of values. (See bi-conditional)
Logical language: A symbolic language that has been developed for the purpose of revealing the
logical structure of a statement or argument. (See artificial language). There are two logical
languages. (See Predicate and Sentential)
Logical Operator: One of five operations in deductive logic. (See Negation, Conjunction,
Disjunction, Conditional, Bi-conditional)
Natural language: Any of the languages that we speak, such as English, Hawaiian, Russian,
Spanish, Chinese, etc. (See Ordinary language)
Negation: A logical operation that changes the truth value of a given statement. The negation is
symbolized using a tilde (~).
Nesting: In a deductive proof, the technique of entering one assumption inside of an open
assumption. To be valid, nested assumptions must be closed in reverse order. (see Assumption)
Operator: (logical operator) Any one of five basic symbols used to symbolize or represent a logical
operation. The five logical operators used in this course will be: negation (~) , conjunction (C),
(inclusive) disjunction (v), (simple) conditional (e), and bi-conditional (/).
Ordinary language: Any of the languages that we speak, such as English, Hawaiian, Russian,
Chinese, etc. (See Natural language) Arguments presented in ordinary or natural language are
informal arguments.
Predicate: A formal, artificial logical language that represents the logical relationships within
statements, using quantifiers, variables, individual constants and predicate terms.
predicate term: 1) A symbol (conventionally upper-case letters A-Z) in the language of Predicate
that represent a class, category, quality or relational term; 2) Grammatically, the predicate term that
is said of a subject term, as in All students are happy– being happy is something “said of” or said
about the subject term, “all students.” Predicate terms can be one, two, three or more placed.
predicate function: A combination of a predicate term (symbolized using a capital letter) and a
variable (symbolized by lower case letters, x, y or z). A predicate function has meaning, but no truth
value. It is, therefore, not a statement.
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Premise: The statements in an argument that provide the logical starting point of the reasoning.
Premises are assumed to be true (although they may in fact be false) and are offered as evidence,
grounds or reasons for the truth of the conclusion. All arguments have at least one premise. (See
conclusion)
Proof: see Deductive Proof A specific type of demonstration that shows in clear, linear,
transparent and explicit terms exactly how a given conclusion can be drawn out of a given set of
premises. Each step of a proof is either 1) an initial premise, 2) an assumption or 3) a derivation.
Proofs are the gold standard, or the highest quality of demonstration in logic. (See Two-column
Proof)
Quantifier: A logical symbol in Predicate that indicates quantity. There are two quantifiers in
Predicate logic, (œx) and (›x), expressing either “all” or “some,” respectively. (See Existential
quantifier and Universal quantifier)
Quantifier Rules: basic patterns of deduction that are used in deductive proofs. These rules allow
you to either eliminate, introduce or change quantifiers under specified conditions.
Reasoning: A thinking process in which inferences are made. We express our reasoning using
arguments. (See arguments, inference)
Reductio ad Absurdum: A form of deductive reasoning that proves a statement to be true by
showing that its opposite can draw forward a contradiction (see contradiction). This form of.
Indirect Proof, and a Proof from Contradiction.
Rules of Inference: basic patterns of validity that are used in a deductive proof to draw inferences,
or conclusions forward. These patterns can be shown to be valid using a truth table method (for
Operator rules) or reflect intuitively valid patterns of reasoning (for Quantifier rules). Rules of
inference are based on formal analogies. See Operator Rules; Quantifier Rules
Sentence: A grammatical unit of meaning. There are four basic types of sentences: Declarative,
Interrogative (Question), Imperative (Command) and Exclamation, each of which has a specific
function. In conventional grammar, each of these sentence forms begins with a capital letter and
ends with a punctuation mark. Only declarative sentences are considered statements or truth claims
for logical puposes. (See Statement)
Sentential: A formal artificial logical language that represents the logical relationships between
statements. Sentential is used in symbolizing truth-funtional arguments. In Sentential, simple
statements are represented by capital letters (A-Z).
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Simple Statement: A statement that asserts a truth claim and cannot be broken down into any other
simpler, meaningful unit that asserts a truth claim. (See compound statement)
*Soroties: An extended categorical argument, having more than two premises.
Sound: A quality of deductive arguments that are both valid (well-structured) and contain only true
premises.
Standard Form Statement: In Predicate logic, a quantified statement in which the first symbol is a
quantifier.
Statement: A truth claim. To be distinguished from a sentence, which is group of words organized
into a grammatical unit. The same statement can be presented using different sentences. Several
sentences – and even an entire argument – can be contained within a single declarative sentence.
synonyms: proposition, assertion, truth claim. (See sentence, compound statement, simple
statement)
Substitution instance: any statement that substitutes for a given statement form. The substitution
instance shares the same main logical operator as the form of which it is a substitute. For example,
given the form p/q, A/B, ~A/B, and (A v B) / ~ (T C U) are all substitution instances.
Syllogism: A strongly structured type of deductive argument that consists of exactly two premises
and a single conclusion. There are several types of syllogism: categorical, hypothetical, disjunctive
are the most common.
Syntax: A set of rules for stringing together symbols to produce meaningful statements. All
languages have a syntax. In ordinary language this set of rules is often referred to as rules of
grammar.
Tautology: A statement that is formally true, or true under any and all circumstances as shown in a
truth table. Common forms of tautology are p v ~ p, and p / p. Tautologies are not simply
(contingently) true statements. They are formally true statements, meaning it is impossible for them
to be false.
Truth-Function: a function that determines the truth value of compound or complex statements
based on the truth value of its component simple statements.
Truth-Functional Logical Operator: There are five truth-functional operators: negation,
conjunction, disjunction, conditional (or material implication) and bi-conditional. Each of these
truth functional operators is defined uniquely by a specific truth table that shows the conditions
under which it is True or False.
Truth Table: A basic row-column table that presents a complete array of truth value possibilities
for a given statement or set of statements.
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Truth Value: A possible value for a statement. In a binary logic there are only two values: true (T)
and false (F).
Two-Column Proof: A model of proof in which a left hand column presents a sequence of
statements that are either given (as premises), validly derived and justified in a right-hand column
by a valid rule of inference, or introduced temporarily as an assumption, in order to show how a
given conclusion can be validly drawn from a given set of premises.
Universal Quantifier: A specific symbol (œx) in the language of Predicate that indicates “for every
x.” This symbol is commonly translated in English as “all” “any” “every” and like terms. In a
universally quantified statement there is no assumption that objects within the classes named by the
terms exist.
Unsound: In deductive reasoning, an argument that is either invalid, or valid but containing at least
one false premise. (See sound)
Valid(ity): The quality of a deductive argument in which it is impossible for the set of premise
statements to be true and conclusion false. (See invalidity)
Variable: A symbol (conventionally lower case letters x, y and z) in the language of Predicate that
serves as a place holder for an individual constant. Variables can be replaced with individual
constants, which represent named or otherwise specified individuals in accordance with quantifier
elimination rules. A variable is an unnamed “thing” as in (œx) “all things” or (›x) “some things” or
“something.” (See individual constant)
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