PL 120  Symbolic Logic I

Glossary of Terms

A proposition In categorical logic, an A proposition is a universal affirmative proposition.

Abbreviated Truth Table In constructing a reverse truth table, assume that all the premises are true and the conclusion is false, then consistently assign truth values to the components in an attempt to show that your assumption is correct. If the argument is invalid, this method allows you to construct one line of a truth table that demonstrates the invalidity. If the argument is valid, it is impossible consistently to assign truth values on the assumption that the argument is invalid.

Absorption (Abs.) In propositional logic, absorption is a rule of inference in which a conditional statement is given as a premise; you conclude a conditional statement with the same antecedent, and the consequent is a conjunction of the antecedent and consequent of the given proposition. The form of the rule is: p q     p (p · q)

Accent The fallacy of accent rests upon the ways in which you emphasize the words in a statement or by quoting passages out of context.

Accident An argument commits the fallacy of accident if it applies a general rule in a case in which that rule does not apply.

Ad hoc Hypothesis An ad hoc hypothesis is a hypothesis posed solely to explain why another hypothesis or theory fails in a particular case or kind of case.

Addition (Add.) In propositional logic, addition is a rule of inference which, given any statement as a premise, allows you to conclude that statement disjoined to any other statement. The form of the rule is:   p p v q

Affirmative If you claim that members of the first class named are members of  the second class, the proposition is affirmative in quality.

Ambiguous A word is ambiguous if it has more than one meaning.

Ampersand The symbol (&) for the conjunction operator.

Amphiboly Amphibolies are arguments based on loose sentence structure. In an argument that commits the fallacy of amphiboly, the referent of a word or phrase is left unclear and the meaning of the phrase shifts in the course of the argument.

Analogy Analogies are claims of similarity. Analogies can be used to illustrate, to explain, and to argue.

Antecedent The antecedent of a conditional statement is the phrase following 'if; in symbolic form, it is the statement to the left of the arrow or horseshoe.

Appeal to Force The fallacy of appeal to force occurs if you appeal to force or the threat of force to convince someone to accept a conclusion.

Arguing in a Circle One argues in a circle when one propose a series of arguments in which the conclusion of the last argument was accepted as a premise in an earlier argument.

Argument  form A general logical pattern or structure that many arguments may have in common.  More specifically, a set of two or more statement forms, one of which is designated as the conclusion and the other (or others) designated as a premise (or premises).  An argument form may be expressed by writing the formulas for its component statement forms and designating the conclusion and the premise or premises.

Argument An argument is a set of propositions in which one or more propositions (the premises) are said to provide reasons or evidence for the truth of another proposition (the conclusion).

Argumentum ad Baculum See Appeal to Force.

Argumentum ad Hominem See Personal Attack.

Argumentum ad Populum See Mob Appeal.

Aristotelian Interpretation of Categorical Logic The Aristotelian interpretation of categorical logic is an interpretation of categorical logic that ascribes existential import to universal propositions.

Aristotle A Greek philosopher who lived in Athens from 384 to 322 B.C.; the first person known to treat logic as a formal discipline.

Arrow (   )In the symbolic language for propositional logic, the arrow represents the relation of material implication. It is also known as the single arrow. See also Horseshoe.

Association (Assoc.) In propositional logic, association is an equivalence falling under the Rule of Replacement that allows you to reposition parentheses in a Statement in which the only connectives are either wedges or ampersands. The two forms of this equivalence are:  [p ·(q ·  r)] :: [(p  · q) ·r] and [p v (q v r)] :: [(p v q)  v r]

Average See Mean, Median, Midrange, and Mode.

Bandwagon A bandwagon argument is a form of the fallacy of mob appeal--basically, "Everyones doing it, so you should do it too."

Begging the Question You commit the fallacy of begging the question if the conclusion of your argument is nothing more than a restatement of one of the premises.

Biased Sample A survey is based on a biased sample if the population surveyed is not representative of the population the survey purports to represent.

Biconditional   A compound statement of the form  “p if, but only if, q.”  See also:  material equivalence.

Binary Relation A binary relation is a two-place relation. Also called a dyadic realtion.

Boole, George George Boole was the nineteenth-century British logician and philosopher who is credited with resolving the controversy over the existential import of universal propositions in favor of holding that universal propositions do not have existential import.

Boolean Interpretation of Categorical Logic The Boolean interpretation of categorical logic is an interpretation of categorical logic that ascribes existential import only to particular propositions.

Bound Variable A bound variable is a variable within the scope of a quantifier.  See also Quantifier

Categorical Syllogism A categorical syllogism is a special kind of syllogism in which both the premises and the conclusion are categorical propositions, and in which there are three terms, each occurring twice, with each term assigned the same meaning throughout the argument.

Ceremonial Function of Language The ceremonial function of language is the use of language in various ceremonies, including greetings.

Class A class is a collection of objects that have at least one characteristic in common, namely, the class-defining characteristic.

Classical Theory of Probability The classical theory of probability was developed by the seventeenth-century mathematicians Blaise Pascal and Pierre de Fermat. The classical theory is a priori insofar as it does not depend on empirical data. It assumes that all possibilities are taken into account and that all possibilities are equal. The mathematical basis for probability theory developed by Pascal and Fermat is also used in the relative frequency theory of probability and the subjective theory of probability.

Collective Use of a Word A word is used collectively if it applies to a class itself--as a whole.

Commissive Function of Language The commissive function of language is to use language to make vows and promises.

Commutation (Comm.) In propositional logic, commutation is a statement of logical equivalence found under the Rule of Replacement that allows you to switch the places of statements around a common ampersand or wedge. The two versions of the equivalence are: (p · q) :: (q · p) and (p v q) :: (q v p)

Complementary Class For any class C, a complementary class is a class containing all the objects not in C. The complement of the class of green things is the class of nongreen things.

Complex Question The fallacy of complex question involves implicitly asking two questions at once, one explicitly and one implicitly, and an answer to the explicit question allows you to draw a conclusion regarding the implicit question.

Composition The fallacy of composition occurs if either (1) you attribute characteristics true of a part to a whole or (2) you claim that something that is true of each member of a class of objects is true of the class as a whole.

Compound Statement A compound statement is any statement that has another statement as a component.

Conclusion In an argument, the conclusion is the statement whose truth the argument is attempting to establish.

Conclusion indicators  Words or expressions used in the context of an argument to signal the conclusion.

Conclusion Indicators Conclusion indicators are words such as 'thus, 'therefore, and 'so that are used to indicate that a statement is the conclusion of an argument.

Conditional  In sentential logic a compound statement of the form “If p then q.”  See also: material implication. A conditional statement is a statement expressing the relation of material implication. A conditional statement is true except when its antecedent is true and its consequent is false.

Conditional Proof In propositional and predicate logic, you construct a conditional proof by assuming a statement as an additional premise, working out the consequences of this assumption, and discharging the assumption by constructing a conditional statement in which the antecedent is the assumption for conditional proof and the consequent is the conclusion reached in the previous line of the proof.

Confirmation of a Hypothesis If an experimental procedure based on a hypothesis yields the predicted consequence, the experiment tends to confirm the hypothesis, that is, it provides some evidence that the hypothesis is true.

Conjunct A conjunct is a statement in a conjunction.

Conjunction (Conj.) In propositional logic, conjunction is a rule of inference in which two premises are given and the conclusion is a conjunction of the two premises. The form of the rule is:   p, q  p · q

Conjunction In propositional logic, conjunction is represented by the ampersand (&) or dot ( · ). A statement of the form     (p · q) is true if and only if both p and q are true.

Connotative Meaning; Connotation The connotative meaning of a term consists of all the characteristics or properties that are common to all the members of the class denoted by a term. Also known as the intension of a term.

Consequent In a conditional statement, the consequent is the statement following the word 'then; in symbolic form, it is the statement to the right of the arrow or horseshoe.

Constructive Dilemma (C.D.) In propositional logic, constructive dilemma is a rule of inference in which the first premise is a conjunction of two conditional statements, the second premise is a disjunction of the antecedents of the conditionals in the first premise, and the conclusion is a disjunction of the consequents of the conditionals in the first premise. The form of the rule is:           (p q), (r s), p v   q v s

Consistency A set if statements is consistent if an only if it is possible for all of the  members of the set to be simultaneously true.

Constant   A symbol that stands for a specifically identified thing.  In QL, individual constants such as a, b, c are symbols that abbreviate singular terms and  predicate constants are symbols such as, A, B, and C that abbreviate predicates.

Constant  A symbol that stands for one specified thing.

Constituent element  In a negation, the constituent element is the component that is negated, the statement whose truth value is “flipped” by the negation.

Contingent Proposition The truth value of a contingent proposition is determined by whether the proposition corresponds to the world. All simple propositions are contingent.

Contingent statement  (in truth-functional logic)  A statement with the following feature: the final column of its truth-table show at least one T and at least one F.

Contradiction (in truth-functional logic)  A statement with the following feature: the final column of its truth-table shows all F's.

Contradiction, Contradictories In categorical logic, two statements are contradictories if they are formally related to one another in such a way that if one is true the other must be false. In propositional logic, a contradiction or self-contradiction is a statement that is false solely in virtue of its form.

Contradictory statement form.  A statement form all of whose instances are contradictions.  All contradictory statement forms are equivalent to p · ~p.

Contrapositive, Contraposition The contrapositive of a given categorical proposition is formed by replacing the subject term with the complement of the predicate term, and replacing the predicate term with the complement of the subject term. A and 0 propositions are logically equivalent to their contrapositives; E and I propositions are not logically equivalent to their contrapositives.

Contrariety, Contraries In Aristotelian logic, contrariety is the formal relationship between two universal propositions with the same subject and predicate terms but opposite qualities such that it is possible for both to be false but not for both to be true. Two propositions so related are known as contraries.

Conventional Connotation of a Term The conventional connotation of a term consists of the properties of a thing that the members of a certain linguistic community consider common to the things a term denotes.

Converse, Conversion The converse of a given categorical proposition is formed by reversing the position of the subject and the predicate term. Only E and I propositions are logically equivalent to their converses.

Convertand A convertand is a proposition that is to be converted. Copula The copula is a form of the verb 'to be.

Criterion, Criteria A criterion is a standard for judging on a certain topic or subject matter.

Crucial Experiment A crucial experiment is an experiment that provides strong evidence that one of two opposing hypotheses is correct by showing that the predictions of one hypothesis correspond to the observational data whereas the predictions of the other do not.

DeMorgan, Augustus.  A brilliant mathematician noted for his advances in algebra and logic. He was a friend of Charles Babbage (inventor of the Analytical Engine, forerunner of the modern computer), coined the term 'mathematical induction', and, with George Boole, he was a founder of symbolic logic as it developed in England . He is perhaps best known as the creator of De Morgan's Theorems.

De Morgans Theorems (DeM). In propositional logic, a pair of equivalences falling under the Rule of Replacement specifying (1) that the denial of a conjunction of two propositions is equivalent to the disjunction of the denial of each proposition, and (2) that the denial of the disjunction of two propositions is equivalent to the conjunction of the denial of each proposition. The forms of these two equivalences are: ~(p v q) :: (~p · ~q) and ~(p ·q) :: (~p v ~q)

Decision procedure    (or algorithm")  A problem-solving method that has the following features:  (a) the method has precisely specified steps; (b) following the steps is a matter of following exact rules requiring no creativity or ingenuity; and (c) the method is guaranteed to give a definite solution in a finite number of those steps.

Deductive Argument A deductive argument is one in which the premises, if true, guarantee the trugh to the conclusion, that is, the conclusion is a logical consequence of the premises.  A sound deductive argument provides conclusive evidence for the truth of its conclusion.

Deductive Counterexample You construct a deductive counterexample to an argument of a given form by constructing an instance of the same form with true premises and a false conclusion. This shows that the argument form is invalid.

Deductively sound argument  A deductive argument that is valid and that also has true premises

Deductive-Nomological Explanation A deductive-nomological explanation tells why a phenomenon is as it is by showing that the truth of a description of the phenomenon follows deductively from a statement of a natural law and an initial state description (statement of antecedent conditions).

Definiendum The definiendum is the word defined in a definition.

Definiens The word or words that define the definiendum are known as the definiens.

Definite Description A definite description is a phrase of the form "the so and so." A definite description is a complex linguistic expression that picks out exactly one thing.

Definition by Genus and Difference A definition by genus and difference is a connotative definition in which the definiendum is treated as the name of a class; in which the definiens identifies a more general class of which it is a part (the genus) and the properties that are unique to that species; and in which the definiens differentiates that species from other members of the genus.

Definition by Subclass A definition by subclass is a denotative definition in which you name the subclasses of a class denoted by a term.

Denotative Definition A denotative definition defines a word by reference to objects in the terms denotation.

Denotative Meaning; Denotation The denotative meaning of a term consists of all the things to which a term is correctly applied. Also known as the extension of a term.

Dichotomy A dichotomy is a division of a class into two mutually exclusive and exhaustive subclasses, that is, a division of a class such that every member of the original class is a member of one of the two subclasses and no member of the original class is a member of both subclasses.

Directive Function of Language The directive function of language is the use of language to request information, plead for action, and issue orders.

Disjunct A disjunct is a statement in a disjunction.

Disjunction Disjunction is represented by the wedge, or vee ( v ) or the del .   A statement of the form "p v q" is true except when both p and q are false.

Disjunctive Syllogism (D.S.) In propositional logic, disjunctive syllogism is a rule of inference in which the first premise is a disjunction, the second premise is the negation of the first disjunct in the first premise and the conclusion is the second disjunct in the first premise. The form of the rule is: p v q, ~p  q

Distribution (Dist.) In propositional logic, distribution is an equivalence falling under the Rule of Replacement that specifies the relationship between a statement conjoined to a disjunction and a conjunction of disjunctions, or a statement conjoined to a disjunction and a disjunction of conjunctions. The two forms of distribution are:

[p · (q v r)] :: [(p · q) v (p · r)] and [p v (q · r)] :: [(p v q) · (p v r)]

Distribution A term in a categorical proposition is distributed if and only if it refers to all the members of the class denoted by the term.

Distributive Use of a Word A word is used distributively if it applies to each and every member of a class taken individually.

Division The fallacy of division occurs if either (1) you attribute to a part characteristics that are true only of the corresponding whole or (2) you attribute to a member of a class a property that is true of a class of objects as a whole.

Domain of a variable   The set of things the variable can take as values.  The values of a variable are just the entities represented by the singular terms that can replace the variable.  The domain of a variable is also known as the universe of discourse for the statement containing the variable.

Domain of Discourse A domain of discourse is a set of objects that are assumed to exist for the sake of the argument.

Double Arrow In the symbolic language for propositional logic, the double arrow ( «  ) represents material equivalence.  See also Triple Bar (º).

Double Negation (D.N.) In propositional logic, double negation is an equivalence falling under the Rule of Replacement indicating that any statement is equivalent to its double negative. The form of the rule is: p :: ~~p

Drawing an Affirmative Conclusion from a Negative Premise (ACNP) In categorical logic, a syllogism commits the fallacy of drawing an affirmative conclusion from a negative premise if its conclusion is an affirmative proposition and at least one of its premises is a negative proposition.

Dyadic connective  (or "dyadic operator")  A connective that joins together two statements to form a compound statement.

Dyadic Relation A dyadic relation is a two-place relation.

E Proposition In categorical logic, an E proposition is a universal negative proposition.

Element  A member of a universe of discourse.  The elements in a universe of discourse are the individual or objects that may have or lack the properties used in a quantified formula.

Emotive Function of Language The emotive function of language is the use of language to express or evoke emotions.

Empirical Evidence, Empirical Data Empirical evidence is evidence drawn from experience; empirical data is data based on empirical experience.

Empty Denotation A term has an empty denotation if there is nothing it denotes.

Enthymeme, Enthymematic Argument An enthymeme is an argument in which one of the premises or the conclusion is not stated.

Enumerative Definition An enumerative definition is a denotative definition in which the definiendum is defined by naming objects in the denotation of the term.

Epithet An epithet is a descriptive word or phrase used to characterize a person, thing, or idea. You can commit the fallacy of begging the question by ascribing an epithet to a person, thing, or idea that assumes what you are trying to establish.

## Equivalence    Two statements are equivalent if and only if it would be impossible for the two to differ in items of truth and falsity, so that if one is true, the other is true, and if one is false, the other is false.  In short, two statements are equivalent if and only if they imply each other.  See also: material equivalence.

Equivocate If you shift from one meaning of a word to another within a piece of discourse, you equivocate.

Equivocation An argument commits the fallacy of equivocation if the meaning of a word shifts in the context of the argument and the persuasive force of the conclusion depends upon that shift.

Evidence Evidence is that which tends to prove or disprove the truth of a statement.

Exceptive Proposition An exceptive proposition is a proposition beginning with 'All except or 'All but. Exceptive propositions are complex. "All except S are P" means both "All non-S are P" and "No S is P."

Exclusive Premises (EP) In categorical logic, a syllogism commits the fallacy of exclusive premises if both of its premises are negative.

Existential Fallacy (EF) In categorical logic under the Boolean Interpretation, a syllogism commits the existential fallacy if its conclusion is a particular proposition and both of its premises are universal propositions.

Existential Generalization (EG) In predicate logic, the rule of existential generalization allows you to introduce the existential quantifier given that a statement is true of some individual.

Existential Import Existential import is the property of a proposition if its truth entails the existence of at least one object.

Existential Instantiation (EI) In predicate logic, existential instantiation is a rule that allows you to eliminate the existential quantifier by introducing a hypothetical name to represent the "at least one thing of which the statement is true. The name introduced must be new to the proof.

Existential Quantifier In predicate logic, the quantifier in a particular proposition. The symbol of the backward E ( x) is the existential quantifier.

Explanation An explanation is a discourse that answers the question why something is as it is.

Explicit contradiction A statement of the form P & ~ P.

Exportation In propositional logic, exportation is a logical equivalence falling under the Rule of Replacement asserting that a conditional statement with a conjunction as an antecedent is logically equivalent to a conditional statement in which the first conjunct is the antecedent and the consequent is a conditional statement in which second conjunct of the equivalent statement is the antecedent and the consequent of the equivalent statement is the consequent. The form the equivalence is:

[(p · q) r] :: [p ( q r)].

Extension The extension of a term consists of all the things to which a term is correctly applied. Also known as the denotation of a term.

External Consistency Experimental evidence is externally consistent with accepted scientific theories when the accepted theories help explain the evidence.

Fallacy A fallacy is an error in reasoning. It is an argument that seems to be sound but is not. Invalid deductive arguments are formally fallacious: the form of the argument does not guarantee that if the premises are true, the conclusion is also true. Informal fallacies rest on the content of the argument.

Fallacy of Four Terms (4T) In categorical logic, an argument commits the fallacy of four terms if either there are more than three terms in the argument, or a term is assigned different meanings at different points in the argument. If an argument commits the fallacy of four terms, by definition it is not a categorical syllogism.

False Cause An argument commits the fallacy of false cause if it misidentifies the cause of an event and draws a conclusion.

False Dichotomy An argument commits the fallacy of false dichotomy if it presents two alternatives as the only alternatives with respect to an issue when in fact there are other options, and rejects one of the alternatives and concludes that you must accept the other. What the argument claims is a dichotomy is not.

False, Falsehood A proposition is false if and only if it does not correspond with the world. Falsehood is a characteristic of a statement or proposition.

Falsifying Interpretation  An assignment of  values to the elements in a universe of discourse that makes a particular formula false in that universe under that interpretation.

Fermat, Pierre de Pierre de Fermat was the seventeenth-century French mathematician who, with Blaise Pascal, developed the classical theory of probability.

Figure of a Categorical Syllogism The figure of a categorical syllogism specifies the position of the middle in the syllogism in standard form.

Form of an Argument The form of an argument is the structural pattern of an argument. It is like the design of a house insofar as there may be many arguments of the same form.

Formal Fallacy A formal fallacy is an error in reasoning based solely on the form of the argument, not on its content.

Free Variable In predicate logic, a free variable is any variable not bound by a quantifier.

General Conjunction Rule In the probability calculus, the general disjunction rule allows you to calculate the probability of the second event on the assumption that the first event occurred. Where A and B are two events, the formula for the general conjunction rule is:

General Disjunction Rule In the probability calculus, the general disjunction rule allows you to calculate the probability of either of two independent events whether or not they are mutually exclusive. Where A and B are two events, the formula for the general disjunction rule is: P(A or B) = P(A or B) = P(A) + P(B) - P(A and B).

Guide Columns In a truth table, guide columns specify the truth values of each of the different simple statements in the argument.

Hasty Generalization An argument commits the fallacy of hasty generalization if a general conclusion--a conclusion pertaining to all or most things of a kind--is based on an atypical case or cases.

Horseshoe ( ⊃ ) In the symbolic language for propositional logic, the arrow represents the relation of material implication. It is also known as the single arrow. See also Arrow.

Hypothesis A hypothesis is an educated guess or hunch regarding a necessary or sufficient condition for a particular phenomenon or kind of phenomenon.

Hypothetical Syllogism In propositional logic, the rule of hypothetical syllogism states that given two premises that are conditional statements in which the consequent of the first is the antecedent of the second, the conclusion is a conditional statement in which the antecedent is the antecedent of the first premise and the consequent is the consequent of the  second premise. The form of the rule is: p q, q   p  r.  Sometimes called Chain Argument.

I Proposition In categorical logic, an I proposition is a particular affirmative proposition.

Identity           To say that is identical with y is to say that x and y are one and the same entity.

Ignoratio Elenchi See Irrelevant Conclusion.

Illicit Process of the Major Term (IMa) In categorical logic, a syllogism commits the fallacy of illicit process of the major term (illicit major) if the major term is distributed in the conclusion but not in the major premise.

Illicit Process of the Minor Term (IMi) In categorical logic, a syllogism commits the fallacy of illicit process of the minor term (illicit minor) if the minor term is distributed in the conclusion but not in the minor premise.

Immediate Inference An immediate inference is an inference you can correctly draw regarding the truth value of a proposition given nothing more than the truth value of one other proposition.

Implication A logical relation that obtains between two statements whenever it is impossible for the first of the two to be true while the second is false.

Incomplete Truth Table An incomplete truth table is a truth table in which you construct the guide columns and the column for the conclusion, and, proceeding from the upper right corner to the lower left, you fill in only those rows in which the conclusion is false. Further, you assign truth values to the statements in a row only until you find a false premise, and you continue assigning truth values on the table only until you find a row in which all the premises are true and the conclusion is false.  See also Abbreviated Truth Table and Truth Value Analysis

Inconsistency A set of statements is inconsistent if and only if it is  impossible that all of the statements in the set are true together.

Inconsistent; Inconsistency Two propositions are inconsistent with one another if one asserts what the other denies.

Independent Events In probability theory, two or more events are independent if and only if the occurrence of any one of them has no influence on the occurrence of any of the others.

Indirect Proof In propositional and predicate logic, an indirect proof is constructed by assuming the denial of the proposition you want to prove as an additional premise and showing that this enlarged set of premises yields a pair of contradictory statements. This procedure shows that the proposition you wanted to prove follows from the original premises.

Inductive Arguments Inductive arguments provide some, but not conclusive, evidence for the truth of their conclusions..

Inference  A person "draws an inference" when he or she asserts a conclusion on the basis of one or more premises.  Inferring is a psychological act that a person undertakes.

Informal Fallacy An argument commits an informal fallacy if it is psychologically persuasive but not logically persuasive, and its logical error rests on the material presented in the argument.

Informative Function of Language The informative function of language is the use of language to convey information.

Intension The intension of a term consists of all the characteristics or properties that are common to all the members of the class denoted by a term. Also known as the connotation of a term.

Internal Consistency A theory is internally consistent if the statements in the theory do not entail self-contradictory statements.

Interpretation of a Universe of Discourse A specification of the properties of and relations between the elements of a universe of discourse.

Invalid, Invalidity Invalidity is a characteristic of an argument form. An argument form is invalid if and only if the truth of the premises do not guarantee the truth of the conclusion.

Irrelevant Conclusion You commit the fallacy of irrelevant conclusion if your premises seem to lead you to one conclusion and you draw an entirely different conclusion.

Justification    (of an inference)  In a proof, a citation of the rule used in drawing the inference and the lines to which the rule was applied.

Law of Noncontradiction   The principles that contradictions are impossible.

Legal Realism (American)  American Legal Realism is a theory of law which holds that the behavior of judges is far more important to understanding what law is than are the abstract verbal formulations of written rules.  The most extreme version of the theory holds that "The law is what the judge says it is."  Less extreme versions treat written rules as predictions of what judges will do in concrete cases.  The view is associated with Jerome Frank, Karl Llewellyn, Wesley Newcomb Hohfeld and Oliver Wendell Holmes.  Also called Rule Skepticism.

Lexical Definition A lexical definition states the convention governing the use of a word.

Logical Equivalence Two statement forms are logically equivalent if they express the same proposition and if they are true under exactly the same conditions.

Logical truth   A statement that can be known on the basis of logical theory alone, without investigating the physical world.

Major Premise In a categorical syllogism, the major premise is the premise containing the major term.

Major Term In a categorical syllogism, the major term is the predicate term of the conclusion.

Margin of Error In a survey, the margin of error is the percentage by which past experience suggests actual behavior might deviate from the results of the survey within a certain "level of confidence," which is typically 95 percent. So if the results of a survey show that a certain presidential candidate can expect 48 percent of the votes and the survey has a 2 percent margin of error, this indicates that there is a 95 percent chance that the candidate will receive between 46 and 50 percent of the votes.

Material Equivalence (Equiv.) In propositional logic, material equivalence is a statement of logical equivalence falling under the Rule of Replacement that allows you to introduce or replace a statement containing a double arrow or triple bar. The two forms of the equivalence are: (p   q) :: [(pq) ·(qp)] and (p   q) ::[(p·q) v  (~p ·~q)].

Material Equivalence In propositional logic, material equivalence is represented by the double-arrow ( 𔹜   ) or the triple bar ( ). A statement of material equivalence is true whenever the truth values of p and q are the same; otherwise it is false.

Material Implication (Impl.) In propositional logic, material implication is a statement of logical equivalence falling under the Rule of Replacement that allows you to replace a conditional statement with a disjunction consisting of the denial of the antecedent of the conditional and the consequent or to replace such a disjunction with a conditional statement. The two forms of the equivalence are: (p q) ::  (~p v q) and (~p q) :: (p v q).

Material Implication In propositional logic, material implication is represented by the single arrow ( ) or horseshoe ( ). A statement of the form "p ®  q" is true except when p is true and q is false.   Sometimes called “material conditional” or merely ‘the conditional.”

Matrix  In QL the body of a formula following the quantifier(s).

Mean The mean is the arithmetic average. It is calculated by dividing the sum of the individual values by the total number of individuals in the reference class.

Mechanical Jurisprudence  A theory of legal reasoning which holds that a valid legal decision must be the conclusion of a logically sound deductive argument.  The task of the judge is to apply the law, as set down by the legislature, to the facts, as found by the jury, in a logically rigorous, mechanical way.  This theory eliminates discretion on the part of the judge.

Median The median of a set of numerical data is the middle value when arranged in ascending order: there are as many values above as below. 'Median is one of the meanings of 'average.

Mention A word is mentioned in a statement if a statement refers to the word itself. By convention, you mention a word by placing it in single quotation marks.

Metaphor A metaphor is an analogy in which an implicit comparison is made between two things. The statement "Language is a picture of the world" is a metaphorical statement.

Middle Term In a categorical syllogism, the middle term is the term that is in both premises but not in the conclusion.

Midrange The midrange is the point in the arithmetic middle of a range, and it is determined by adding the highest number in the range to the lowest number and dividing by two. 'Midrange is one of the meanings of 'average.

Minor Premise In a categorical syllogism, the minor premise is the premise containing the minor term.

Minor Term In a categorical syllogism, the minor term is the subject term of the conclusion.

Mob Appeal You commit the fallacy of mob appeal by appealing to the emotions of the crowd--the desire to be loved, accepted, respected, etc.--rather than to the relevant facts.

Mode In a set of numerical data, the mode is the value that occurs most frequently. 'Mode is one of the meanings of 'average.

Modus Ponens (M.P.) In propositional logic, modus ponens is a rule of inference in which the first premise is a conditional statement, the second premise is the antecedent of that conditional, and the conclusion is the consequent of the conditional. The form of the argument is: pq, p q.

Modus Tollens (M.T.) In propositional logic, modus tollens is a rule of inference in which the first premise is a conditional statement, the second premise is the denial of the consequent of that conditional, and the conclusion is the denial of the antecedent of that conditional. The form of the argument is:   p q, ~q  ⊢~p.

Monadic connective   (or "monadic operator")  A connective that is joined to just one statement to form a compound statement.

Monadic predicate  A predicate phrase that contains one blank and is used attribute a property to just one thing.

Mood of a Syllogism In categorical logic, the mood of a syllogism consists of the kinds of propositions of which the syllogism is composed. In stating the mood of a syllogism, you state the letter representing the major premise first, then the letter representing the minor premise, and finally the letter representing the conclusion..

Mutually Exclusive Events In probability theory, two or more events are mutually exclusive if and only if they are distinct and the occurrence of one of the events precludes the occurrence of any of the others.

Natural deduction method     A method in which we deduce a conclusion from a set of premises through a series of valid inferences corresponding to more or less natural patterns of reasoning.

Natural deduction system      A system consisting of (a) a formal language that can be used to symbolize information; and (b) a set of natural deduction rules that can be used to deduce conclusions from premises.

Necessary and Sufficient Condition A necessary and sufficient condition is a condition in whose absence a given phenomenon will not occur and in whose presence the phenomenon occurs. A necessary and sufficient condition is expressed by a biconditional.

Necessary Condition A necessary condition is a condition is whose absence a given phenomenon will not occur. A necessary condition is expressed by the consequent of a conditional.

Negation In propositional logic, negation is represented by the tilde (~). Placing a tilde in front of a statement changes a true statement into a false statement and a false statement into a true statement.

Negation Rule In the probability calculus, the negation rule tells you that the probability that some event A occurs is equal to one minus the probability that A does not occur. The form of the rule is:

Negative If you deny that members of the first class named are members of the second, the proposition is negative in quality.

Non Causa Pro Causa ("not the cause for the cause") This is a variety of the false cause fallacy. If you incorrectly take something to be the cause of something else without any reference or allusion to the temporal order of events, you commit the fallacy of non causa pro causa.

Normal Probability Distribution A normal probability distribution is the mark of a random survey. You have a normal probability distribution if the mean, the mode, the median, and the midrange are the same.

O Proposition In categorical logic, an 0 proposition is a particular negative proposition.

Objective Connotation The objective connotation of a term consists of all the characteristics common to all the things a term denotes.

Obverse, Obversion In categorical logic, the obverse of a categorical proposition is obtained by changing the quality of a given categorical proposition from affirmative to negative and replacing the predicate term with its complement. Every categorical proposition is logically equivalent to its obverse.

Obvertend The obvertend is a proposition to be obverted.

Open sentence  An incomplete sentence that is lacking a subject expression and that has a space where a subject expression may be placed.

Operational Definition An operational definition is a connotative definition in which the definiens specifies an experimental procedure or operation that provides a criterion for the application of a term.

Oppositions In categorical logic, oppositions are immediate inferences among categorical propositions.

Ostensive Definition An ostensive definition is a denotative definition in which the definiendum is defined by pointing to objects denoted by the word.

Overlapping quantifiers  In a sentence with overlapping quantifiers, one quantifier appears within the scope of another quantifier.

Parameter A parameter is a word or phrase that is added to a statement to specify its domain of discourse.

Particular Affirmative Proposition In categorical logic, a particular affirmative proposition asserts that there is at least one member of the subject class and it is also a member of the predicate class. A particular affirmative proposition can be stated in standard form as "Some S is(are) P."

Particular Negative Proposition In categorical logic, a particular negative proposition asserts that there is at least one member of the subject class and it is not a member of the predicate class. A particular negative proposition can be stated in standard form as "Some S is(are) not P."

Particular Proposition A particular proposition is a proposition that is true of at least one individual.

Pascal, Blaise Blaise Pascal was the seventeenth-century French philosopher and mathematician who, with Pierre de Fermat, developed the classical theory of probability.

Personal Attack You commit the fallacy of personal attack if you attempt to refute the conclusion of another persons argument by attacking the person who presented the argument rather than the argument itself.

Polyadic predicate A relational predicate that involves at least two variables.

Post Hoc Ergo Propter Hoc ("before that, therefore because of that") Post hoc ergo propter hoc is a variety of the false cause fallacy. If you assume that one event is the cause of another simply because it occurs first, you commit the fallacy of post hoc ergo propter hoc.

Precising Definition A precising definition is offered to set limits to the definiendum and thereby reduce vagueness.

Predicate Term In a categorical proposition, the predicate term is the term in the predicate place; it is the second term that names a class.

Premise In an argument, a premise is a statement used to provide evidence for the truth of another statement, namely, the conclusion.

Premise Indicators Premise indicators are words such as 'since, 'because, and 'given that, which are used to indicate that a statement is a premise of an argument.

Prenex Normal Form In predicate logic, a statement is in prenex normal form if all the quantifiers are placed to the left of the entire propositional function over which they range. If the quantifier is in the antecedent of a conditional, moving the quantifier outside the farthest left parentheses requires changing the quantity of the quantifier: universal to particular or particular to universal.

Principal (or Main) Connective In symbolic logic, the principal or main connective is the connective that holds an entire complex statement together. For example, in the statement "p ®  (q v r)," the arrow is the principal connective. In "(p®  q) v (r®  s)," the wedge is the principal connective.    The principal connective is sometimes called the dominant operator in a formula.

Principle of Indifference In the probability calculus, the principle of indifference is the assumption that all possible events in the class under consideration are equally probable.

Principle of Self-Identity  Each thing is identical with itself.

Principle of the Indiscernibility of Identicals  If x is identical with y, then whatever is true of x is true of y and whatever is true of y is true of x.

Probability Calculus The probability calculus consists of those mathematical formulae used to calculate the probability of an event. The probability calculus is common to the classical, relative frequency, and subjective theories of probability.

Proof  A proof is a finite series of formulas, beginning with the premises of an argument and ending with the conclusion, in which each line following the premises is justified according to accepted rules of inference and equivalence.  A proof is a formal demonstration that the conclusion of the argument follows from the premises.  Any argument for which a proof can be constructed is valid.

Proof procedure  One half of a decision procedure, the half which provides affirmative answers to the question “Is this argument valid?”  A proof procedure is any accepted method for constructing proofs.

Proposition A proposition is the information expressed by a declarative sentence or statement.

Propositional Function In predicate logic, a propositional function is a predicate with a variable as its subject.

Propositional Logic Propositional logic is a system of logic in which simple propositions are the fundamental elements of a logical schema.

Quality In categorical logic, the quality of a proposition is either affirmative or negative.

Quantifier Negation In predicate logic, quantifier negation is a rule that allows you to move the tilde across a quantifier. Moving the tilde across a quantifier changes an existential quantifier to a universal quantifier and changes a universal quantifier to an existential quantifier.

Quantifier The quantifier of a categorical statement tells you how many things the statement refers to. In categorical logic, the universal quantifiers are 'All and 'No, and the particular quantifier is 'Some. In predicate logic, the universal quantifier--all--is as upside down 'A' followed by variable in parentheses ("x), and the existential quantifier is the backward 'E' followed by a variable ∃x.

Quantity In categorical logic, the quantity of a proposition tells you how many members of the first class named are or are not members of the second.

Quaternary Relation A quaternary relation is a four-place relation.

Quine, Willard van Orman.  One of the most influential American philosophers and logicians of the 20th century. Sometimes referred to as the "philosopher's philosopher", Quine is the quintessential model of an analytic philosopher. His major writings include "Two Dogmas of Empiricism", which influentially attacked the logical positivists' conception of analytic and synthetic propositions, and Word and Object.

Random Sample The sample on which a survey is taken is random if every person or object in the population surveyed has an equal chance of being chosen for the survey.

Red Herring The  fallacy of  shifting away from the issue under consideration to something different and then drawing a conclusion.

Reductio ad Absurdum A reductio ad absurdum is literally a reduction to absurdity. It is a proof technique in which an assumption is added to a set of premises, and it is shown that adding the assumption yields a pair of contradictory statements. This shows that the original assumption was false. See also Indirect Proof.

Reflexive relation  A relation that possesses the following feature: if one thing (A) has the relation to something (B) or B has the relation to A, A has that relation to itself.

Reflexive Relation A relation is reflexive if and only if an object can stand in that relation to itself. The relation of "being the same age as" is an example of a reflexive relation.

Refutation of a Hypothesis If an experimental procedure based on a hypothesis fails to yield the predicted consequence, the experiment refutes the hypothesis, that is, it provides conclusive evidence that the hypothesis is false.

Relation A relation is a predicate of two or more places. For example, the predicates "to the left of" and "between" are relational predicates.

Relative Frequency Theory of Probability The relative frequency theory of probability is based on empirical data. This theory of probability is used in calculating such things as insurance rates.

Restricted Conjunction Rule In the probability calculus, the restricted conjunction rule is used to calculate the probability of two independent events. For the events A and B, the probability of both A and B is: P(A and B) = P(A) × P(B).

Restricted Disjunction Rule In the probability calculus, the restricted disjunction rule is used to calculate the probability that either of two or more mutually exclusive events occurs. For the events A and B, the probability of either A and B is: P(A or B) = P(A) + P(B).

Rule of Law Model  A theory of law derived from natural law traditions according to which the law is there for all, including government officials, to obey.  This theory holds that no one is above the law, that the law checks the power of government.  Frequently associated with the claim that "Ours is a government of laws, not of men."  According to this model, the law provides a neutral playing field within which citizens are free to pursue their own ends.

Rule of Replacement In propositional logic, the rule of replacement is the rule that logically equivalent expressions may replace each other wherever they occur in a proof..

Rules of Inference The rules of inference are rules that allow valid inferences from statements assumed as premises.

Rule of Passage  A rule governing expanding or contracting the scope of a quantifier or governing moving a negation from one side of a quantifier to the other.

Rule Skepticism  See Legal Realism

Sample A sample is a portion of a certain population of objects or people on which a poll or survey is based.

Satisfying Interpretation  An assignment of  values to the elements in a universe of discourse that makes a particular formula true in that universe under that interpretation.

Scope of a Quantifier The scope of a quantifier is the propositional function whose variables are so grouped that they are bound by the quantifier.

Scope of an operator  The operator itself along with the part of the statement that the operator applies to or links together.

Semantics  The theory of meaning for a language.  The semantics supplies the meanings for the various expressions of the language.

Sentential Variable  A lower case letter used in a statement form to represent a statement (either simple or compound).

Simile A simile is an analogy in which the word 'like makes the comparison between two things explicit.

Simple statement  A statement that does not contain within itself one or more shorter statements.  A simple statement expresses exactly one fact about the world and contains no logical operations.

Simplification (Simp.) In propositional logic, simplification is the rule of inference in which the premise is a conjunction and the conclusion is the first conjunct. The form of the rule is: p·q p

Singular Proposition or Singular Statement A singular proposition (statement) makes a claim about an individual, exactly one thing.

Singular sentence   A sentence that is made up of two parts (1) a singular subject expression in which one specific thing is singled out for discussion by a  singular term; (2) a predicate expression that contains a general term attributing a property to the specific thing designated by the subject expression

Singular term  An expression that refers to or describes one specifically identified thing.  In general, a singular term is used to single out a specific thing so that we  may say something about it.  This text addressed two types of singular term.

Slippery Slope Argument A slippery slope argument has the following structure. There is a slope--a chain of causes. It is slippery. Therefore, if you take even one step on the slope, you will slide all the way to the bottom. But the bottom is a bad place to be. So, you should not take the first step.

Slippery Slope Fallacy An argument commits the slippery slope fallacy when (and only when) at least one of the causal relations constituting the slope in a slippery slope argument does not hold.

Some As used in logic, the word 'some means at least one.

Sorites A sorites is a chain of enthymematic syllogisms in which the unstated conclusion of one syllogism is a premise for the next syllogism.

Sound Argument Sound arguments are valid deductive arguments with premises that are all true.  The conclusion of a sound argument must be true.

Square of Opposition A square of opposition is a diagram showing the immediate inferences that can be drawn given the truth or falsehood of a categorical proposition.

Standard-Form Categorical Statement In categorical logic, a statement is a standard-form categorical statement if and only if: (a) it expresses a categorical proposition; (b) its quantifier is either 'All, 'No, or 'Some; (c) it has a subject and a predicate term; (d) its subject and predicate terms are joined by a copula, a form of the verb 'to be; and (e) the order of the elements in the statement is: quantifier, subject term, copula, predicate term.

Standard-Form Categorical Syllogism A syllogism is a standard-form categorical syllogism if and only if it fulfills each of the following criteria: (a) it is a categorical syllogism; (b) the premises and conclusion are standard-form categorical statements; (c) the syllogism contains three different terms; (d) each of the terms appears twice in the argument; (e) each term is used with the same meaning throughout the argument; (f) the predicate term of the conclusion appears in the first premise; (g) the subject term of the conclusion appears in the second premise.

Statement A statement is a sentence that is true or false in virtue of the proposition it expresses.

Statement Abbreviation In propositional logic, the uppercase letters (A, B, C, . . .) are statement abbreviations and represent statements in ordinary language.

Statement connective  (or "statement operator")  A word or phrase that can be connected to one or more statements to produce a compound statement.

Statement forms   A general logical pattern or structure that many statements may have in common.  A statement form is expressed using a combination of variables and operators.

Statement letter  A capital letter used to represent a simple statement.  It is important to remember that a statement letter stands for a completer statement, a full idea, and not for a single word.

Statement Variables In propositional logic, the lowercase letters beginning with p (p, q, r, . . .) are statement variables. Statement variables can be replaced by statements of any degree of complexity. Also called sentential variables.

Stipulative Definition A stipulative definition is used to assign a meaning to a new word, to assign a new meaning to a word already in use, or to specify the meaning of a word in a particular context.

Straw Person You commit the straw person fallacy if, in replying to an argument, you either distort the original argument by suggesting that the first arguer accepted a premise that was not explicitly stated and argue that the premise is implausible, or distort the conclusion, argue against the conclusion as you have restated it, and hold your criticisms to apply to the original argument.

Subaltern In Aristotelian logic, a subaltern is a particular proposition with the same subject and predicate terms and the same quality as a given universal proposition.

Subalternation In Aristotelian logic, subalternation is an immediate inference between a universal categorical proposition and the corresponding particular proposition of the same quality that allows you to infer the truth of the particular given the truth of the universal or the falsehood of the universal given the falsehood of the particular.

Subcontrariety, Subcontraries In Aristotelian logic, subcontrariety is a formal relationship between two particular propositions with the same subject and predicate terms but opposite qualities such that it is possible for both to be true but it is not possible for both to be false. Two propositions so related are known as subcontraries.

Subject Term In a categorical proposition, the subject term is the term in the subject place; it is the first term that names a class.

Subjective Connotation The person-to-person differences in the connotations assigned to a term are known as subjective connotations.

Subjective Theory of Probability The subjective theory of probability is based on individual beliefs. This theory of probability is used in calculating such things as the outcome of sporting events.  Also called Bayesian probability.  See also, Bayes, Thomas.

Substitution instance  (of an argument form)  An argument is a substitution instance of an argument form if and only if the argument can be generated from the form by replacing only the variables in the form with TL statements and, in addition, making any necessary parenthetical adjustments.

Sufficient Condition A condition is a sufficient condition for some phenomenon if whenever that condition holds, the phenomenon in question occurs. A sufficient condition is expressed by the antecedent of a conditional.

Suppressed Evidence An argument that commits the fallacy of suppressed evidence is enthymematic. It states a premise that is true but presupposes an additional premise, a false premise, the truth of which must be assumed as grounds for accepting the conclusion.

Syllogism A syllogism is a deductive argument consisting of two premises and a conclusion.

Symmetrical Relation A relation R is symmetrical if and only if when it is true that a stands in relation R to b, it is also true that b stands in relation R to a. The relation of "being a sibling" is a symmetrical relation: if John is a sibling of Mary, then Mary is a sibling of John. The relation of "being older than" is not a symmetrical relation (it is an asymmetrical relation): if John is older than Mary, then it is false that Mary is older than John.

Synonymous Definition A synonymous definition is a connotative definition in which the definiens is a single word that has the same connotation as the definiendum..

Syntax   The vocabulary and rules of grammar of a language.  Essentially, the vocabulary gives us the elements to construct expressions of the language, and the rules of grammar tell us how to construct properly formed expressions of the language.

Tautology (Taut.) In propositional logic, a tautology is a logical equivalence falling under the rule of replacement that indicates (1) that any statement is logically equivalent to a disjunction of that statement with itself and (2) that any statement is logically equivalent to a conjunction of that statement with itself. The two forms of the equivalence are: a (a ·a ) and a (a v a).  Sometimes called the property of idempotence.

Tautology In propositional logic, a tautology is a statement that is true solely in virtue of its form.

Term A term is a word or phrase that can be the subject of a sente nce.

Ternary Relation A ternary relation is a three-place relation.

Tetradic Relation A tetradic relation is a four-place relation.

Theorem  A formula that can be proven true without the use of premises.  A theorem of TD is a formula that can be proven true using a premise-free proof in TD.

Theory A scientific theory consists of a number of general, well-confirmed hypotheses that will explain why specific phenomena are as they are.

Tilde (~) In the symbolic language for propositional logic, the tilde represents negation. The tilde is the only one-place connective in our system.

Transitive Relation A relation R is transitive if and only if given that a is in relation R to b, and b is in relation R to c, it follows that a is in relation R to c.  A transitive relation possesses the following feature: if one thing bears the relation to a second thing bears the relation to a third thing, the first must bear the relation to the third.

Transposition (Trans.) In propositional logic, transposition is an equivalence falling under the rule of replacement that a conditional is logically equivalent to another statement in which the denial of the consequent of the first is the antecedent and the denial of the antecedent of the first is the consequent. The two forms of the equivalence are:        (p q) :: (~q ~p) and (~p q) :: (~q p). Sometimes called contraposition.

Triadic Relation A triadic relation is a three-place relation.

True, Truth Truth is a characteristic of a statement or proposition. A proposition is true if and only if it corresponds with the world.

Truth function  A logical rule that relates one set of truth-values to another set of truth-values.  A truth function takes a set on truth values as inputs and returns a unique truth value as the output.

Truth Functional Compound  A truth functional compound  is a compound statement in which the truth of the entire statement is determined wholly by the truth values of its component statements.

Truth Functional Expansion    In QL a method for determining the truth value of a quantified proposition by assigning properties to the elements of the universe of discourse and then comparing that assignment to a known interpretation of the universe.

Truth Functional Operator  An operator that forms a truth-functional compound statement when attached to one or more component statements.

Truth Tree The truth-tree technique is a purely mechanical method of determining whether an argument in propositional logic is valid and demonstrating that an argument in predicate logic is valid. The truth-tree technique operates by the method of reductio ad absurdum

Truth Value Truth value is a property of a proposition. The truth value of a proposition is its truth or falsehood.

Undistributed Middle In categorical logic, a syllogism commits the fallacy of undistributed middle if the middle term is undistributed.

Universal Affirmative Proposition In categorical logic, a universal affirmative proposition claims that all the members of the subject class are included in the predicate class. A universal affirmative proposition can be stated in standard form as "All S is(are) P."

Universal domain  Everything in the universe.  When the domain is everything in the universe, the domain is unrestricted or universal.

Universal Generalization (U.G.) In predicate logic, the rule of universal generalization allows you to conclude the truth of a universal proposition on the basis of propositions instantiated in terms of variables.

Universal Instantiation (U.I.) In predicate logic, the rule of universal instantiation allows you to eliminate the universal quantifier by replacing each variable in the scope of the quantifier by a constant or a variable.

Universal Negative Proposition In categorical logic, a universal negative proposition holds that no members of the subject class are included in the predicate class. A universal negative proposition can be stated in standard form as "No S is P.

Universal Quantifier In predicate logic, the universal quantifier is symbolized by the upside down A ( " ), or by a variable placed in parentheses.

Universal Statement A universal statement makes a claim about every member of its subject class.

Universe of discourse  The collection of things that the statement or statements are making a claim about.  Essentially, the collection of things we are talking about on an occasion is our universe of discourse on that occasion

Unsound argument An unsound argument is one which is either non-valid or one which, if valid, has false premises.

Use A word is used if the truth or falsehood of a statement depends upon the meaning of that word.

Vague A word is vague if its meaning is unclear or imprecise.

Valid, Validity Validity is a characteristic of an argument form. An argument form is valid if and only if the truth of the premises guarantees the truth of the conclusion.

Variable  A symbol that stands for anything from a specified group of things.  In QL, a symbol such as x, y, or z that serves a placeholder for individual constants.  A variable in QL performs the function that is performed by the word "thing" within English

Venn Diagram The Venn diagram is a pictorial means of representing categorical propositions understood according to the Boolean interpretation.

Venn, John John Venn was a nineteenth-century logician who developed a pictorial means of representing categorical propositions understood according to the Boolean interpretation.

Verbal Disputes Verbal disputes are disputes that rest on alternative meanings of terms rather than a genuine disagreement about the facts.

Wedge In the symbolic language for propositional logic, the wedge (or vee) represents disjunction.

Well-Formed Formula (WFF) I. In propositional logic, a formula is a well-formed formula in our language under the following conditions:

arrow

(1) any statement letter or simple proposition is a WFF;

(2) a tilde followed by a WFF is a WFF;

(3) if p and q are WFFs, then so are the following:

(p · q), (p v q), (p q), (p  º q) and formulas using alternative symbols for the binary connectives;

II. In predicate logic, a formula is a well-formed formula in our language under the following conditions

(1) atomic expressions: a. individual variables: x, y, z; b. individual constants: a, b, c, . . ., w; c. predicate letters: A, B, C, . . . Z; d. Connectives: i. one-place: ~; ii. two-place: , , , ; e. grouping indicators: ( ), [ ], { }; f. quantifiers: ("x), ∃x;

(2) well-formed formulae (WFFs): a. atomic WFFs: i. where R is a predicate letter and x is either an individual constant or an individual variable, then Rx is an atomic WFF; ii. where R is a predicate letter and x and y are either individual constants or individual variables, then Rxy and Ryx. are atomic WFFs; iii. since predicates can be predicates of any degree, where R is a predicate letter followed by any number of individual constants or individual variables, x, y, z. . ., Rxyz is an atomic WFF; b. molecular WFFs: where and are WFFs, so are: ~, , , , ; c. general WFFs: where R is a predicate letter and x is an individual variable, the following are WFFs: ("x)Rx is a WFF ∃xRx is a WFF; d. nothing is a WFF unless it can be constructed by a finite number of applications of rules a - c.

Whole An object is treated as a whole relative to the various parts of which it is composed.