Logic Final Exam Review PDF

**[Chapter 1: What Logic Studies ]** **[Summary of Deductive Arguments]** **Valid argument**: deductive argument in which, assuming the premises are true, it is *impossible* for the conclusion to be false. **Invalid Argument**: a deductive argument in which, assuming the premises are true, it is...

**[Chapter 1: What Logic Studies ]** **[Summary of Deductive Arguments]** **Valid argument**: deductive argument in which, assuming the premises are true, it is *impossible* for the conclusion to be false. **Invalid Argument**: a deductive argument in which, assuming the premises are true, it is *possible* for the conclusion to be false. **Sound Argument**: a deductive argument is sound when both of the following requirements are met: 1. The argument is valid (logical analysis). 2. All the premises are true (truth analysis). **Unsound Argument**: A deductive argument is unsound if either or both of the following conditions hold: 1. The argument is invalid (logical analysis). 2. The argument has at least one false premise (truth value analysis). **[Summar of Inductive Arguments]** **Strong inductive argument**: An argument such that if the premises are assumed to be true, then the conclusion is probably true. In other words, the probable truth of the conclusion follows from the truth of the premises. **Weak Inductive Argument**: An argument such that either (a) if the premises are assumed to be true, then the conclusion is probably not true, or (b) a probably true conclusion does not follow from the premises. **Cogent argument**: an inductive argument is cogent when both of the following requirements are met: 1. The argument is strong (logical analysis). 2. All the premises are true (truth value analysis). **Uncogent argument**: An inductive argument is uncogent if either or both of the following conditions hold: 1. The argument is weak (logical analysis). 2. The argument has at least one false premise (truth value analysis). **[Full Summary for Chapter 1]** **Argument**: A group of statements of which one (conclusion) is claimed to follow from the others (the premises). **Statement**: A sentence that is either true or false. **Premise(s)**: The information intended to provide support for a conclusion. - Logic is the systematic use of methods and principles to analyze, evaluate and construct arguments. - Every statement is either true or false; these two possibilities are called "truth values." **Proposition**: The information content imparted by a statement, or, simply put, its meaning. **Inference**: The term used by logicians to refer to the reasoning process that is expressed by an argument. - In order to help recognize arguments, we rely on premise indicator words and phrases, and conclusion indicator words and phrases. - If a passage expresses a reasoning process \-\-- that the conclusion follows from the premises \-\-- then we say that it makes an inferential claim. - If the passage does not express a reasoning process (explicit or implicit), then it does not make an inferential claim (it is a non-inferential passage). **Explanation**: Provides reasons for why or how an event occurred. By themselves, explanations are not arguments; however, they can form part of an argument. - **Truth value analysis** determines the strength with which the premises support the conclusion. - **Logical analysis** determines the strength with which the premises support the conclusion. **Deductive argument**: An argument in which the inferential claim is that the conclusion follows *necessarily* from the premises. In other words, under the *assumption* that the premises are true it is *impossible* for the conclusion to be false. **Inductive argument**: An argument in which the inferential claim is that the conclusion is *probably true* if the premises are true. In other words, under the assumption that the premises are true it is *improbable* for the conclusion to be false. Moreover, the *probable truth* of the conclusion *follows from* the premises. **Valid deductive argument**: An argument in which, assuming the premises are true, it is impossible for the conclusion to be false. In other words, the conclusion **follows** necessarily from the premises. **Invalid deductive argument**: An argument in which, assuming the premises are true, it is possible for the conclusion to be false. In other words, the conclusion **does not follow** necessarily from the premises. - When logical analysis shows that a deductive argument is valid, and when truth value analysis of the premises shows that they are all true, then the argument is sound. - If a deductive argument is invalid, or if at least one of the premises is false (truth value analysis), then the argument is unsound. - In categorical logic, an argument form is an arrangement of logical vocabulary and letters that stand for class terms such that a uniform substitution of class terms for the letters results in an argument. - In categorical logic, a statement form is an arrangement of logical vocabulary and letters that stand for class terms such that a uniform substitution of class terms for the letters results in a statement. - A substitution instance of a *statement* occurs when a uniform substitution of class terms for the letters results in a statement. A substitution instance of an *argument* occurs when a uniform substitution of class terms for the letters results in an argument. - A counterexample to a statement is evidence that shows the statement is false, and it concerns truth value analysis. A counterexample to an argument shows the possibility that premises assumed to be true do not make the conclusion necessarily true. A single counterexample to a deductive argument is enough to show that an argument is invalid. **Conditional statement**: In English, the word "if" typically precedes the antecedent of a conditional statement, and the world "then" typically precedes the consequent. **Fallacy of affirming the consequent**: An invalid argument form; it is a formal fallacy. **Modus ponens**: A valid argument. (remember this) **Fallacy of denying the antecedent**: An invalid argument form; it is a formal fallacy. **Modus tollens**: A valid argument form. **Hypothetical syllogism**: A valid argument form. **Disjunction**: A compound statement that has two distinct statements, called disjuncts, connected by the word "or." **Disjunctive syllogism**: A valid argument form. **Strong inductive argument**: An argument such that if the if the premises are assumed to be true, then the conclusion follows from the truth of the premises. **Weak inductive arguments**: An argument such that either (a) if the premises are *assumed* to be true, then the conclusion is *probably not true*, or (b) a probably true conclusion *does not follow from the premises.* - An inductive argument is cogent when the argument is strong, and the premises are true. An inductive argument is uncogent when either or both of the following conditions hold: the argument is weak, or the argument has at least one false premise. **Enthymemes**: Arguments with missing premises, missing conclusions, or both. **Principle of charity**: we should choose the reconstructed argument that gives the benefit of the doubt to the person presenting the argument. **Rhetorical language**: When we speak or write for dramatic or exaggerated effects. When the language we employ may be implying things that are not explicitly said. **Rhetorical question**: Occurs when a statement is disguised in the form of a question. **Rhetorical conditional**: a conditional statement that is used to imply an argument. **[Chapter 3: Diagramming Arguments]** **Summary of Diagramming Arguments** - Diagramming premises and conclusions displays the relationship between all the parts of an argument. - The first step in diagramming an argument is to number the statements as they appear in the argument. The next step is to diagram the relationship by connecting the premise to the conclusion with an arrow. **Simple diagram**: A diagram consisting of a single premise and a single conclusion. - Premises are independent when the falsity of anyone would not nullify the support the others give to the conclusion. **Convergent diagram**: Reveals the occurrence of independent premises. - Dependent premises work together to support a conclusion. In other words, the falsity of one dependent premise weakens the support that the other dependent premises give to the conclusion. **Linked diagram**: Reveals the occurrence of dependent premises. **Divergent diagram**: Shows a single premise used to support independent conclusions. **Serial diagram**: Shows a conclusion from one argument that becomes a premise in a second argument. **[Chapter 4: Informal fallacies]** **Summary of Fallacies Based on [Personal Attacks]** - Fallacies based on personal attacks occur when someone's argument is rejected based solely on an attack against the person making the argument, not the merits of the argument itself. **Ad hominem abusive**: The fallacy is distinguished by an attack on alleged character flaws of a person instead of the person's argument. **Ad hominem circumstantial:** The fallacy occurs when someone's argument is rejected based on the circumstances of the person's life. **Poisoning the well**: The fallacy occurs when a person is attacked before she has a chance to present her case. **Tu quoque**: The fallacy is distinguished by the specific attempt of one person to avoid the issue at hand by claiming the other person is a hypocrite. **Summary of Fallacies Based on [Emotional Appeals]** - Fallacies based on emotional appeals occur when an argument relies solely on the arousal of a strong emotional state or psychological reaction to get a person to accept the conclusion. **Appeal to the people**: The fallacy occurs when an argument manipulates a psychological need or desire, such as the desire to belong to a popular group, or the need for group solidarity, so that the reader or listener will accept the conclusion. **Appeal to pity**: The fallacy results from an exclusive reliance on a sense of pity or mercy for support of a conclusion. **Appeal to fear or force**: The fallacy occurs when a threat of harmful consequences (physical or otherwise) is used to force acceptance of a course of action that would otherwise be unacceptable. **Summary of Weak Inductive Argument Fallacies** - Generalization fallacies occur when an argument relies on a mistaken use of the principle behind making a generalization. There are five individual fallacies in this group. **Rigid application of a generalization**: When a generalization or rule is inappropriately applied to the case at hand. The fallacy results from the mistaken belief that the generalization or rule is universal (meaning it has no exceptions). **Hasty Generalization**: An argument that relies on a small sample that is unlikely to represent the population. **Composition**: There are two forms of the fallacy: 1. The mistaken transfer of an attribute of the individual parts of *an object* to the *object as a whole*; and 2. The mistaken transfer of an attribute of the individual *members of a class* to the *class itself.* **Division**: There are two forms of the fallacy 1. The mistaken transfer of an attribute of an object as a whole to the individual *parts of the object* 2. The mistaken transfer of an attribute of a class to the individual *members of the class*. **Biased Sample**: An argument that uses a nonrepresentative sample as support for a statistical claim about the entire population. - A **false cause fallacy** occurs when a causal connection is assumed to exist between two events when none actually exists, or when they assumed causal connection is unlikely to exist. There are two individual fallacies in this group. **Post Hoc**: The fallacy occurs from the mistaken assumption that just because one event occurred before another event, the first event *must have caused* the second event. **Slippery slope**: An argument that attempts to connect a series of occurrences such that the first link in a chain leads directly to the second link, and so on, until a final unwanted situation is said to be the inevitable result. **Summary of Fallacies of Unwarranted Assumption and Diversion** - **Fallacies of unwarranted assumptions** are arguments that assume the truth of some unproved or questionable claim. **Begging the question**: In one type, the fallacy occurs when a premise is simply reworded in the conclusion. In a second type, called *circular reasoning*, a set of statements seem to support each other with no clear beginning or end point. In a third type, the argument assumes certain key information that may be controversial or not supported by facts. **Complex question**: The fallacy occurs when a single question actually contains multiple parts and an unestablished hidden assumption. **Appeal to ignorance**: An argument built on a position of ignorance claims either that 1. A statement must be true because it has not been proven to be false 2. A statement must be false because it has not been proven to be true. **Appeal to an unqualified authority**: An argument that relies on the opinions of people who either have no expertise, training, or knowledge relevant to the issue at hand or whose testimony is not trustworthy. **False Dichotomy**: The fallacy occurs when it is assumed that only two choices are possible, when in fact others exist. - A **fallacy of diversion** occurs when the meaning of terms or phrases are changed (intentionally or unintentionally) within the argument, or when our attention is purposely (or accidentally) diverted from the issue at hand. **Equivocation**: The fallacy occurs when the conclusion of an argument relies on an intentional or unintentional shift in the meaning of a term or phrase in the premises. **Straw man**: The fallacy occurs when an argument is misrepresented in order to create a new argument that can be easily refuted. The new argument is so weak that is it "made of straw." The arguer then falsely claims that his opponent's real argument has been defeated. **Red Herring**: The fallacy occurs when someone completely ignores an opponent\'s position and changes the subject, diverting the discussion in a new direction. **Misleading precision**: A claim that appears to be statistically significant but is not. **Missing the point**: When premises that seem to lead logically to one conclusion are used instead to support an unexpected conclusion. **Summary** **Formal fallacy**: A logical error that occurs in the form or structure of an argument and is restricted to deductive arguments. **Informal fallacy**: A mistake in reasoning that occurs in ordinary language and concerns the content of the argument rather than its form. **[Chapter 5: Categorical Propositions]** **Summary of Conversion, Obversion, and Contraposition** **Method of Conversion** \ [↔ ]{.math.display}\ **The Method of Obversion** **The Method of Contraposition** **Full Summary of Chapter 5** **Class**: A group of objects. **Categorical Proposition**: Relates two classes of objects. Subject term: The term that comes first in a standard-form categorical proposition. **Predicate Term**: The term that comes second in a standard-form categorical proposition. **A-proposition:** Asserts that the entire subject class is included in the predicate class ("All S are P"). **I-proposition:** Asserts that part of the subject class is included in the predicate class ("Some S are P"). **E-proposition:** Asserts that the entire subject class is excluded from the predicate class (No S are P"). **O-proposition**: Asserts that part of the subject class is excluded from the predicate class ("Some S are not P"). - "Universal" and "particular" refer to the quantity of a categorical proposition. - "Affirmative" and "negative" refer to the quality of a categorical proposition. - The words "all," "no," and "some" are called quantifiers. They tell us the extent of the class inclusion or exclusion. - The words "are" and "are not" are referred to as "cupula." They are simply forms of "to be" and serve to link (to "couple") the subject class with the predicate class. - If a categorical proposition asserts something about every member of a class, then the term designating that class is said to be distributed. On the other hand, if the proposition does not assert something every member of a class, then the term designating that class is said to be undistributed. **Existential import**: When a proposition presupposes the existence of certain kinds of objects. **Opposition**: Occurs when two standard form categorical propositions refer to the same subject and predicate classes but differ in quality, quantity, or both. **Contradictory:** Pairs of propositions in which one is the negation of the other. A- and O-propositions are contradictories, as are E- and I-Propositions. - Venn diagrams use circles to represent categorical proposition forms. **Immediate argument**: An argument that has only one premise. **Mediate argument**: An argument that has more than one premise. **Conversion**: An immediate argument created by interchanging the subject and predicate terms of a given categorical proposition. **Complement**: The set of objects that do not belong to a given class. **Obversion**: An immediate argument formed by changing the quality of the given proposition and then replacing the predicate term with its complement. **Contraposition**: Formed by replacing the subject term of a given proposition with the complement of its predicate term and then replacing the predicate term of the given proposition with the complement of its subject term. **Contraries:** Pairs of propositions that cannot both be true at the same time but can both be false at the same time. **A-** and **E-** propositions are contraries. **Subcontraries:** Paris of propositions that cannot both be false at the same time, but both can be true; also, if one is false then the other must be true. **I-** and **O-** propositions are subcontraries. **Subalternation:** The relationship between a universal proposition (the superaltern) and its corresponding particular proposition (the subaltern). **Conversion by limitation:** When we first change a universal **A-**proposition into its corresponding particular **I-**proposition, and then we use the process of conversion on the **I-**proposition. **Contraposition by limitation:** When subalternation is used to change the universal **E-**proposition into its corresponding particular **O-**proposition. We then apply the regular process of forming a contrapositive to this **O-**proposition. **Singular Proposition:** Asserts something about a specific person, place, or thing. **Exceptive propositions:** Statements that need to be translated into compound statements containing the word "and" (for example, propositions that take the form "All except S are P" and "All but S are P"). **[Chapter 6: Categorical Syllogisms]** **Summary of Rules** ---------------------- --------------------------------------------------------------------------------------- **Rule 1** The middle term must be distributed in at least one premise. **Rule 2** If a term is distributed in the conclusion, then it must be distributed in a premise. **Rule 3** A categorical syllogism cannot have two negative premises **Rule 4** A negative premise must have a negative conclusion. **Rule 5** A negative conclusion must have a negative premise. **Rule 6** Two universal premises cannot have a particular conclusion. +-----------------------+-----------------------+-----------------------+ | **The Method of | **The Method of | **The Method of | | Conversion** | Obversion** | Contraposition** | +=======================+=======================+=======================+ | Switch the subject | **Step 1:** Change | **Step 1:** Switch | | and predicate. | the quality of the | the subject and | | | given proposition. | predicate terms. | | | | | | | **Step 2:** Replace | **Step 2:** Replace | | | the predicate term | both the subject and | | | with its complement. | predicate terms with | | | | their term | | | | complements. | +-----------------------+-----------------------+-----------------------+ **Full Summary for Chapter 6** **Syllogism:** A deductive argument that has exactly two premises and a conclusion. **Categorical syllogism:** a syllogism constructed entirely of categorical propositions. It contains three different terms, each of which is used two times. **Minor term:** The subject of the conclusion of a categorical syllogism. **Major term:** The predicate of the conclusion of a categorical syllogism. **Middle term:** The term that occurs only in the premises of a categorical syllogism. **Major premise:** The first premise of a categorical syllogism contains the major term. **Minor premise:** The second premise of a categorical syllogism contains the minor term. - In order to be a standard-form categorical syllogism, three requirements must be met: (1) All three statements must be standard-form categorical propositions. (2) The two occurrences of each term must be identical and have the same sense. (3) The major premise must occur first, the minor premise second, and the conclusion last. - The mood of a categorical syllogism consists of the type of categorical propositions involved (A, E, I, or O) and the order in which they occur. - The middle term can be arranged in the two premises in four different ways. These placements determine the figure of the categorical syllogism. - There are **six rules** for standard-form categorical syllogisms: (1) The middle term must be distributed in at least one premise. (2) If a term is distributed in the conclusion, then it must be distributed in a premise. (3) A categorical syllogism cannot have two negative premises. (4) A negative premise must have a negative conclusion. (5) A negative conclusion must have a negative premise. (6) Two universal premises cannot have a particular conclusion. **Undistributed middle:** A formal fallacy that occurs when the middle term in a categorical syllogism is distributed in both premises of a categorical syllogism. **Illicit major:** A formal fallacy that occurs when the major term in a categorical syllogism is distributed in the conclusion but not in the major premise. **Exclusive premises:** A formal fallacy that occurs when both premises in a categorical syllogism are negative. **Affirmative conclusion/negative premise:** A formal fallacy that occurs when a categorical syllogism has a negative premise and an affirmative conclusion. **Negative conclusion/affirmative premises:** A formal fallacy that occurs when a categorical syllogism has a negative conclusion and two affirmative premises. **Existential fallacy:** A formal fallacy that occurs when a categorical syllogism has a particular conclusion and two universal premises. **Enthymemes:** Arguments with missing premises, missing conclusions, or both. **Sorites:** A special type of enthymeme in which the missing parts are intermediate conclusions each of which, in turn becomes a premise in the next link in the chain. **[Chapter 7: Propositional Logic]** ----------------------------------------------------------------------------------------------------------- **Operator** **Name** **Compound** **Used to Translate** ------------------------- ------------ ------------------------------------- ------------------------------ \ Tilde Negation Not; it is not the case that [∼ ]{.math.display}\ \ Dot conjunction And; aldo; moreover [  ]{.math.display}\ \ Wedge disjunction Or; unless [*ν* ]{.math.display}\ \ Horseshoe conditional If\...then\...; only if [⊃ ]{.math.display}\ \ Triple bar Biconditional (like two horseshoes) If and only if [≡ ]{.math.display}\ ----------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- SUMMARY OF OPERATORS AND ORDINARY LANGUAGE -------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Operator Words and Phrases in Ordinary Language \ Not; it is not the case that; it is false that; it is not true that [∼ ]{.math.display}\ \ And; both\...and\...; but; still; moreover; while; however; also; moreover; although; yet; nevertheless; whereas [  ]{.math.display}\ \ Or; unless; otherwise; either\... or [*ν* ]{.math.display}\ \ If; only if; every time; given that; each time; provided that; all cases where; in any case where; any time; supposing that; in the event of; on any occurrence of; on condition that; for every instance of [⊃ ]{.math.display}\ \ If and only if [≡ ]{.math.display}\ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The Number of Different Simple Propositions The Number of Lines in the Truth Table --------------------------------------------- ---------------------------------------- 1 2 2 4 3 8 4 16 5 32 6 64 **Full Summary of Chapter 7** **Logical operators:** Special symbols that are used to translate ordinary language statements - The basic components in propositional logic are statements **Simple statement:** One that does not have any other statement or logical operator as a component. Compound statement: A statement that has at least one simple statement and at least one logical operator as components. - The five logical operator names: tilde, dot, wedge, horseshoe, triple bar. - The word "not" and the phrase "it is not the case that" are used to deny the statement that follows them. And we refer to their use as "negation." **Conjunction:** A compound statement that has two distinct statements (called *conjuncts*) connected by the dot symbol. **Disjunction:** A compound statement that has two distinct statements (called *disjuncts*) connected by the wedge symbol. **Inclusive disjunction:** When we assert that *at least one* disjunct is true, and *possibly both* disjuncts are true. Given this, an inclusive disjunction is false when both disjuncts are false, otherwise it is true. **Exclusive disjunction:** When we assert that *at least one* disjunct is true, but *not* both. In other words, we assert that the truth of one *excludes* the truth of the other. Given this, an exclusive disjunction is true when only one of the disjuncts is true, otherwise it is false. **Conditional statement:** In ordinary language, the word "if" typically precedes the antecedent of a conditional statement, and the statement that follows the word "then" is referred to as the consequent. **Sufficient condition:** Whenever one event ensures that another event is realized. **Necessary condition:** Whenever one thing is essential, mandatory, or required in order for another thing to be realized. **Biconditional:** A compound statement made up of two conditionals -- one indicated by the word "if" and the other indicated by the phrase "only if." **Well-formed formula:** Any statement letter standing alone, or a compound statement such that an arrangement of operators symbols and statements letters results in a grammatically correct symbolic expression. **Scope:** The statement or statements that a logical operator governs. **Main operator:** The operator that has the entire well-formed formula in its scope. **Truth-functional proposition:** The truth value of any compound proposition using one or more of the five operators is a function of (that is, uniquely determined by) the truth values of its component propositions. - The truth value of a truth-functional compound proposition is determined by the truth values of its components and the definition of the logical operators involved. Any truth-functional compound proposition that can be determined in this manner is said to be a truth function. - A statement variable can stand for any statement, simple or compound. **Statement form:** In propositional logic, an arrangement of logical operators and statement variables such that a uniform substitution of statements for the variables results in a statement. **Argument form:** In propositional logic, an argument form is an arrangement of logical operators and statements variables such that uniform substitution of statements for the variable results in an argument. **Substitution instance:** A substitution instance of a statement occurs when a uniform substitution of statements for the variables results in a statement. A substitution instance for an argument occurs when a uniform substitution of statements for the variables results in an argument. **Truth table:** An arrangement of truth values for a truth-functional compound proposition is determined by the truth values of its simple components. **Order of operations:** The order of handling the logical operators within a truth-functional proposition; it is a step-by-step method of generating a complete truth table. **Contingent statements:** Statements that are neither necessarily true nor necessarily false (they are sometimes true, sometimes false). **Noncontingent statements:** Statements such that the truth values in the main operator column do not depend on the truth values of the component\'s parts. **Tautology:** A statement that is necessarily true. **Self-contradiction:** A statement that is necessarily false. **Logically equivalent statements:** Two truth-functional statements that have identical truth tables under the main operator. **Contradictory statements:** Two statements that have opposite truth values under the main operator on every line of their respective truth tables. **Consistent statements:** Two (or more) statements that have at least one line on their respective truth tables where the main operators are true. **Inconsistent statements:** Two (or more) statements that do not have even one line on their respective truth tables where the main operators are true (but they can be false) at the same time. ***Modus ponens:*** A valid argument form (also referred to as affirming the antecedent). **Fallacy of affirming the consequent:** An invalid argument form; it is a formal fallacy. ***Modus tollens:*** A valid argument form (also referred to as denying the consequent). **Fallacy of denying the antecedent:** An invalid argument form; it is a formal fallacy. **[Chapter 8: Natural Deduction]** **Full Summary of Chapter 8** **Natural deduction:** A proof procedure by which the conclusion of an argument is validly derived from the premises through the use of rules of inference. **There are two types of rules of inference:** implication rules and replacement rules. The function of rules of inference is to justify the steps of a proof. **Proof:** A sequence of steps in which each step either is a premise or follows from earlier steps in the sequence according to the rules of inference.\ **Impliocation rules are valid argument forms**: They are validly applied only to an entire line. **Replacxement rules**: Pairs of logically equivalent statement forms. **Modus ponens (MP):** A rule of inference (implication rule). **Substitutions instance:** In propositional logic, a substitution instance of an argument occurs when a uniform substitution of statements for the variables results in an argument. **Modus tollens (MT):** A rule of inference (implication rule). **Hypothetical syllogism (HS):** A rule of inference (implication rule). **Disjunctive Syllogism (DS):** A rule of inference (implication rule). **Justification:** refers to the rule of inference that is applied to every validly derived step in a proof. **Tactics:** The use of small-scale maneuvers or devices. **Strategy:** Typically understood as referring to a greater, overall goal. **Simplification (Simp):** A rule of inference (implication rule). **Conjunction (Conj):** A rule of inference (implication rule). **Addition (Add):** A rule of inference (implication rule). **Constructive dilemma (CD):** A rule of inference (implication rule). **Principle of replacement:** logically equivalent expression may replace each other within the context of a proof. **De Morgan (DM):** A rule of inference (replacement rule). **Double negation (DN):** A rule of inference (replacement rule). **Commutation (Com):** A rule of inference (replacement rule). **Association (Assoc):** A rule of inference (replacement rule). **Distribution (Dist):** A rule of inference (replacement rule). **Transposition (Trans):** A rule of inference (replacement rule). **Material implication (Impl):** A rule of inference (replacement rule). **Material equivalence (Equiv):** A rule of inference (replacement rule). **Exportation (Exp):** A rule of inference (replacement rule). **Tautology (Taut):** A rule of inference (a replacement rule).