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What is a justified step?
"[A] justified step is either a premise, or an assumption, or else a step that follows from previous steps according to one of the given rules of inference." Virginia Klenk, Understanding Symbolic Logic, 5th ed. (Pearson, 2008), 201.

What is a proof?
Short answer: "a derivation (a sequence of justified steops) in which the last step is the desired conclusion."
Long answer: "a sequence of steps, each of which must be either a premise, or an assumption, or a step that follows from previous steps according to one of the given rules of inference, and such that the last step in the sequence is the desired conclusion." Klenk, Understanding Symbolic Logic, 201

Under what condition is an argument (particular instance) valid?
"An argument (a particular instance) is valid if and only if it is an instance of, or has, a valid form." Klenk, Understanding Symbolic Logic, 35

Under what condition is an argument form valid?
"An argument form is valid if and only if there is no instance of that form in which all premises are true and the conclusion is false." Klenk, Understanding Symbolic Logic, 35

Under what condition is an argument form invalid?
 "A form will be invalid . . . just in case there is an instance of that form with all true premises and a false conclusion."
 Klenk, Understanding Symbolic Logic, 29

What is a counterexample to an argument form?
"an instance of that form (a particular example) in which all the premises are true and the conclusion is false." Klenk, Understanding Symbolic Logic, 35

What are the five main sentential (or propositional) operators?
 "and": dot or ampersand: "." or "&"
 "or": wedge: v
 "not": the tilde or dash: "~" or ""
 "ifthen": the horseshoe: ">"
 "if and only if": the triple bar: "="

What is a compound sentence?
 "A sentence is compound if it logically contains another complete sentence as a component."
 Klenk, Understanding Symbolic Logic, 48

What is a simple sentence?
 "A sentence is simple if and only if it is not compound."
 Klenk, Understanding Symbolic Logic, 48

When is a sentence a component of another sentence?
 "One sentence is a component of another sentence if, whenever the first sentence is replaced by any other declarative sentence, the result is still a grammatical sentence."
 Virginia Klenk, Understanding Symbolic Logic, 49

When does a sentence logically contains another?
 "One sentence logically contains another if it either literally contains the other as a component or can be paraphrased into an explicitly compound sentence that contains the other as a component."
 Virginia Klenk, Understanding Symbolic Logic, 49

What is a sentential operator?
 "A sentential operator is an expression containing blanks such that, when the blanks are filled with complete sentences, the result is a sentence."
 Virginia Klenk, Understanding Symbolic Logic, 49

When is an operator truth functional?
"An operator is truth functional if and only if the truth value of the compound that it forms is completely determined by the truth values of the component parts [given the rules of computation for the operator]." Klenk, 66

When is a system of logic truth functional?
"A system of logic is truth functional if and only if each operator of that system is truth functional." (Klenk, 66)

What is a tautology?
"A tautology is a single statement form that is true for every substitution instance; that is, it comes out true under the major operator for every row in the truth table." (Klenk, 125)

What is a contradiction?
"A contradiction is a single statement form that is false for every substitution instance; that is, it comes out false under the major operator for every row in the truth table." (Klenk, 125)

What is a contingency?
"A contingency is a single statement form that is false for some substitution instances and true for others; that is, it has both T's and F's in its truth table under the major operator." (Klenk, 126)

When are two (or more) statement forms logically equivalent?
"Two (or more) statement forms are logically equivalent if and only if their truth tables are identical under their major operators." (Klenk, 126)

When does one statement imply another?
"One statement form logically implies another if and only if there is no row in their joint truth table in which the first comes out true and the second comes out false." (Klenk, 126)

When are two statement forms logically equivalent?
"Two statement forms are logically equivalent if and only if they logically imply each other." (Klenk, 126)

When are two statement forms logically equivalent? (using biconditional)
"Two statement forms are logically equivalent if and only if the result of joining them with a biconditional is a tautology." (Klenk, 126)

When is a set of statements consistent?
"A set of statement forms is consistent if and only if there is a row in their joint truth table in which they all come out true at once." (Klenk, 126)

When is a set of statement forms inconsistent?
"A set of statement forms is inconsistent if and only if there is no row in their joint truth table in which they all come out true at once." (Klenk, 126)

"There is obviously a very close relationship between inconsistency and contradiction; . . ." (Klenk, 123)
Given the above, what is the difference between contradiction and inconsistency?
"[T]he difference is that a contradiction is a single formula, whereas consistency is a property of sets of formulas. We can say, however, that a set of formulas is inconsistent if and only if the conjunction of all the formulas is a contradiction." (Klenk, 123)

What is a constant?
"A constant is a term that has a definite, particular value." (Klenk, 156)

What is a variable?
"A variable is a term that may represent any value." (Klenk, 156)

What is a statement variable?
"A statement variable is a letter that can take as substitution instances any particular statement, simple or complex. (We will use lowercase letters from the middle of the alphabet, p, q, r, . . . , as our statement variables.)." (Klenk, 157)

What is a statement constant?
"A statement constant is a capital letter that is used as an abbreviation for a particular truth functionally simple English sentence." (Klenk, 157)

What is a statement?
"A statement is a formula (simple or complex) that has statement constants as its smallest components." (Klenk, 157)

What is a statement form?
"A statement form is a formula (simple or complex) that has statement variables as its smallest components." (Klenk, 157)

What is a substitution instance (s.i.) of a statement form?
"A substitution instance (s.i.) of a statement form is a statement obtained by substituting (uniformly) some statement for each variable in the statement form. (We must substitute the same statement for repeated occurrences of the same variable, and we may substitute the same statement for different variables. Thus both A v B and A v A are s.i. of p v q, but A v B is not an s.i. of p v p.)

What is the scope of an assumption?
It is roughly an assumption's "extent or for how long it is operative, for how long we are assuming it. The scope of an assumption includes all (and only) the steps of the subproof and is indicated by the arrow and vertical line." (Klenk, 197)

What is a derivation?
"A derivation is a sequence of justified steps." (Klenk, 157)

What is a justified step?
"A justified step is either a premise, or an assumption, or a step that follows from previous steps according to one of the given rules of inference." Klenk, 213

What is a proof?
"A proof is a sequence of justified steps in which the last step is the desired conclusion; that is, it is a sequence of steps, each of which is either a premise, or an assumption, or a step that follows from previous steps according to one of the given rules of inference and in which the last step is the desired conclusion." Klenk, 213

What does it mean to discharge an assumption?
"To discharge an assumption is simply to cut off the scope of that assumption, to end the subproof." (Klenk, 203)

What are the three restrictions related to discharging an assumption?
1. "E]very assumption made in a proof must eventually be discharged. (Klenk, 202)
2. "[O]nce an assumption has been discharged you may not use that assumption or any step that falls within the scope of that assumption again." (ibid)
3. "[A]ssumptions made inside the scope of other assumptions must be discharged in the reverse order in which they were made." (Klenk, 204)

Can you use the method of proof (derivation) to show that an argument is invalid?
 No: "[T]he method of proofs is good only for demonstrating positive results, for concluding that the argument is valid. It can never be used to show that an argument is invalid." Virginia Klenk, Understanding Symbolic Logic 5th ed., 209

How do you show that an argument is invalid?
 "To show that an argument is invalid, you cannot use the proof method at all; you must revert to the method of counterexample. The only way to demonstrate that an argument is invalid is to show that its form is invalid, which means we must construct a counterexample, that is, an instance with true premises and a false conclusion. . . . this generally means using the short truth table method." Virginia Klen, Understanding Symbolic Logic, 5th ed., 210

What is the most important fact about the two methods: truth tables and proof?
They are, for classical logic, exactly equivalent: they give exactly the same results." Virginia Klenk, Understanding Symbolic Logic, 5th ed., 211

What is another name for the truth table method?
Semantic method. Virginia Klenk, Understanding Symbolic Logic, 211

When is a logical system complete?
When an argument is valid according to the truth table and a proof can be constructed for it. Virginia Klenk, Understanding Symbolic Logic, 211

When is a logical system consistent?
When an argument that can be proved (that can be shown valid by the proof method) can be shown to be valid by the truth table (semantic) method. Virginia Klenk, Understanding Symbolic Logic, 5th ed., 211

The equivalence between the truth table (semantic) method and the method of proof for what kind of logical system?
"[A]ll classical, twovalued logic through relational predicate logic with identity." Virginia Klenk, Understanding Symbolic Logic, 5th ed., 211

State the simplified description of Kurt Godel's (190678) incompleteness theorem.
"[I]n any formal, consistent logical system capable of describing arithmetic there is at least one sentence that can neither be proved nor disproved within the system." Julian Baggini and Peter S. Fosl, The Philosopher's Toolkit: A Compendium of Philosophical Concepts and Methods, 2nd ed. (Chichester, UK: WileyBlackwell, 2010), 252

