Symbolic Logic

Question 1

Define: simple statement.

Answer 1

A statement that does not contain any other statement as a compound.

Question 2

Define: compound statement.

Answer 2

A statement that contains two or more statements as components.

Question 3

Define: component.

Answer 3

A part of a compound statement that is itself a statement, and is of such a nature that, if replaced in the larger statement by any other statement, the result will be meaningful.

Question 4

Define: conjunction.

Answer 4

A truth-functional connective meaning “and,” symbolized by this dot ·. A statement in the form p · q is true if and only if p is true and q is true.

Question 5

Define: conjunct.

Answer 5

Each one of the component statements connected in a conjunctive statement.

Question 6

Define: truth value.

Answer 6

The status of any statement as true or false (T or F).

Question 7

Define: truth-functional component.

Answer 7

Any component of a compound statement whose replacement there by any other statement having the same truth value would leave the truth value of the compound statement unchanged.

Question 8

Define: truth-functional compound statement.

Answer 8

A compound statement whose truth value is determined wholly by the truth values of its components.

Question 9

Define: truth-functional connective.

Answer 9

Any logical connective (e.g. conjunction, disjunction, material implication and material equivalence) between the components of a truth-functionally compound statement.

Question 10

Define: truth table.

Answer 10

An array on which all possible truth values of compound statements are displayed, through the display of all possible combinations of the truth values of their simple components. A truth table may be used to define truth-functional connectives; it may also be used to test the validity of many deductive arguments.

Question 11

Define: negation.

Answer 11

Denial; symbolized by th tilde or curl, ~p simply means “it is not the case that p,” and may be read as “not-p.”

Question 12

Define: curl or tilde.

Answer 12

The symbol for negation, ~. It appears immediately before (to the left of) what is negated or denied).

Question 13

Define: disjunction.

Answer 13

A truth-functional connective meaning “or”; components so connected are called “disjuncts.” There are two types of disjunction: inclusive and exclusive.

Question 14

Define: inclusive disjunction.

Answer 14

A truth-functional connective between two components called disjuncts. A compound statement asserting inclusive disjunction is true when at least one of the disjuncts (that is, one or both) is true. Normally called simply “disjunction,” it is also called “weak disjunction” and is symbolized by the wedge, v.

Question 15

Define: exclusive disjunction or strong disjunction.

Answer 15

A logical relation meaning “or” that may connect two component statements. A compound statement asserting exclusive disjunction says that at least one of the disjuncts is true and that at least one of the disjuncts is false. It is contrasted with an “inclusive” (or “weak”) disjunction, which says that at least one of the disjuncts is true and that they may both be true.

Question 16

Define: wedge.

Answer 16

The symbol for weak (inclusive) disjunction, v. Any statement of the form p v q is true if p is true, or if q is true, or if both p and q are true.

Question 17

Define: punctuation.

Answer 17

The parentheses, brackets, and braces used in mathematics and logic to eliminate ambiguity.

Question 18

Define: conditional statement.

Answer 18

A hypothetical statement; a compound proposition or statement of the form “if p then q.”

Question 19

Define: antecedent.

Answer 19

In a conditional statement (“if . . . then . . . “), the component that immediately follows the “then.” Sometimes called the implicate, or the apodosis.

Question 20

Define: consequent.

Answer 20

In a conditional statement (“if . . . then . . .”), the component that immediately follows the “then.” Sometimes called the implicate, or the apodosis.

Question 21

Define: implication.

Answer 21

The relation that holds between the antecedent and the consequent of a true conditional or hypothetical statement.

Question 22

Define: horseshoe.

Answer 22

The symbol for material implication, ⊃.

Question 23

Define: material implication.

Answer 23

A truth-functional relation (symbolized by the horseshoe, ⊃) that may connect two statements. The statement “p materially implies q” is true when either p is false, or q is true.

Question 24

Define: refutation by logical analogy.

Answer 24

A method that shows the invalidity of an argument by presenting another argument that has the same form, but whose premises are known to be true and whose conclusion is known to be false.

Question 25

Define: variable or statement variable.

Answer 25

A place-holder; a letter (by convention, any of the lower case letters, beginning with p, q, etc.) for which a statement may be substituted.

Question 26

Define: argument form.

Answer 26

An array of symbols exhibiting logical structure; it contains no statements but it contains statement variables. These statement variables are arranged in such a way that when statements are consistently substituted for the statement variables, the result is an argument.

Question 27

Define: substitution instance.

Answer 27

Any argument that results from the substitution of statements for the statement variables of a given argument form.

Question 28

Define: specific form.

Answer 28

When referring to a given argument, the argument form from which the argument results when a different simple statement is substituted consistently for each different statement variable in that form.

Question 29

Define: invalid.

Answer 29

Not valid; characterizing a deductive arguments that fails to provide conclusive grounds for the truth of its conclusion. Every deductive argument is either valid or invalid.

Question 30

Define: valid.

Answer 30

A deductive argument is said to be valid when its premises, if they were all true, would provide conclusive grounds for the truth of its conclusion. Validity is a form characteristic; it applies only to arguments, as distinguished from truth, which applies to propositions.

Question 31

Define: disjunctive syllogism.

Answer 31

A valid argument form in which one premise is a disjunction, another premise is the denial of one of the two disjuncts, and the conclusion is the truth of the other disjunct. Symbolized as p v q, ~ p, therefore q.

Question 32

Define: modus ponens.

Answer 32

An elementary valid argument form according to which, if the truth of a hypothetical premise is assumed, and the truth of the antecedent of that premise of also assumed, we may conclude that the consequent of that premise is true. Symbolized as: p ⊃ q, p, therefore q.

Question 33

Define: modus tollens.

Answer 33

An elementary valid argument form according to which, if the truth of a hypothetical premise is assumed, and the falsity of the consequent of that premise is also assumed, we may conclude that the antecedent of that premise is false. Symbolized as p ⊃ q, ~q, therefore ~p.

Question 34

Define: hypothetical syllogism.

Answer 34

A syllogism that contains a hypothetical proposition as a premise. If the syllogism contains hypothetical propositions exclusively, it is called a “pure” hypothetical syllogism; if the syllogism contains one conditional and one categorical premise, it is called a “mixed” hypothetical syllogism.

Question 35

Define: statement form.

Answer 35

A sequence of symbols containing no statements, but containing statement variables connected in such a way that when statements are consistently substituted for the statement variables, the result is a statement.

Question 36

Define: disjunctive statement form.

Answer 36

A statement form symbolized as p v q; it substitution instances are disjunctive statements.

Question 37

Define: specific form.

Answer 37

When referring to a given statement, the statement form from which the statement results when a different simple statement is substituted consistently for each different statement variable in that form.

Question 38

Define: tautology.

Answer 38

A statement form all of whose substitution instances must be true.

Question 39

Define: contradiction.

Answer 39

A statement form all of whose substitution instances are false.

Question 40

Define: contingent.

Answer 40

Being neither tautologous nor self-contradictory. A contingent statement may be true or false; a contingent statement form has some true and some false substitution instances.

Question 41

Define: material equivalence.

Answer 41

A truth-functional relation (symbolized by the three-bar sign, ≡) that may connect two statements. Two statements are materially equivalent when they are both true, or when they are both false–that is, when they have the same truth value. Materially equivalent statements always materially imply one another.

Question 42

Define: De Morgen’s theorems.

Answer 42

Two expressions of logical equivalence. The first states that the negation of a disjunction is logically equivalent to the conjunction of the negations of its disjuncts: ~(p v q) ≡T (~p • ~q). The second states that the negation of a conjunction is logically equivalent to the disjunction of the negations of its conjuncts: ~(p • q) ≡T (~p v ~q).

Question 43

Define: logical equivalence.

Answer 43

When referring to truth-functional compound propositions, the relationship that holds between two propositions when the statement of their material equivalence is a tautology. A very strong relation; statements that are logically equivalent must have the same meaning, and may therefore replace one another wherever they occur.

Question 44

Define: double negation.

Answer 44

An expression of the logical equivalence of any symbol and the negation of the negation of that symbol. Symbolized as p ≡T ~~p.

Question 45

Define: principle of identity.

Answer 45

The principle that asserts that is any statement is true then it is true.