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Glossary

Glossary for SL

2-place connectives See binary connective.
Affirming the consequent A fallacious argument of the form : If α then β. β. Therefore, α.
Antecedent α in any statement of the form "If α then β" or a WFF (α→β). Not to be confused with the premises of an argument.
Argument A list of statements made up of a conclusion and one or more premises.
Assignment A specification of truth-values to sentence letters, e.g. P is T, Q is F, etc.
Biconditional Any statement of the form "α if and only if β", "α when and only when β" or any WFF of the form (α↔β).
Binary connective A sentential connective that connects two sentences or WFFs to form a new sentence or WFF.
Conditional Any statement of the form "If α, then β", or any WFF of the form (α→β).
Conjunct Given a WFF of the form (α&β). α is the left conjunct, and β the right conjunct.
Conjunction Any WFF of the form (α&β).
Consequent β in any statement of the form "If α then β" or a WFF (α→β). Not to be confused with the premises of an argument. Not to be confused with the conclusion of an argument.
Consistent (1) A single statement or WFF is consistent if and only if it is logically possible for it to be true. (2) A set of one or more statements or WFFs is consistent if and only if it is logically possible for all the statements or WFFs in the set to be true at the same time.
Contingent Any statement or WFF that is neither inconsistent nor a tautology. A consistent WFF in SL is one where there is at least one assignment under which it is true, and at least another assignment under which it is false.
Disambiguation The process of identifying the different possible interpretations of an ambiguous piece of language.
Disjunction An WFF of the form (α∨β).
Disjuncts Given a WFF of the form (α∨β), α is the left disjunct and β the right disjunct.
Entail We say that α1 ... αn entail β if and only if it is logically impossible for α1 ... αn to be true and β to be false at the same time. In SL, this means that there is no assignment of truth-values under which cally impossible for α1 ... αn are all true and β is false.
Exclusive-or "α or β" under the exclusive interpretation is true when α is true, or when β is true, but false when α and β are both true. It is false when α and β are false.
Fallacy A mistake of reasoning. Note that not all fallacies are arguments.
Formal language A language with precisely specified syntactic rules.
Formalization The process of translating arguments or sentences in natural languages into a formal language.
Follows from To say that β follows from α1 ... αn is the same as saying that α1 ... αn entail β.
Indirect method A method for checking the validity of a sequent in SL. In this method we first assume that the argument is invalid, and proceed to find an invalidating assignment. If this turns out to be impossible, then the argument is valid. Otherwise it is not.
Invalid Not valid
Invalidating assignment An invalidating assignment for a sequent in SL is a specification of truth-values for the sentence letters that appear in the sequent, such that under such a sequent, the premises of the sequent are true, and the conclusion false.
Invalidating counterexample A real or hypothetical situation where the premises of an argument are true and the conclusion is false, showing that the argument is not valid.
Inconsistent Not consistent.
Inclusive-or "α or β" under the inclusive interpretation is true when α is true, or when β is true, and true when both α and β are true. It is false when α and β are false.
Logic The systematic study of the principles of correct reasoning
Logical consequence The set of logically consequences of α1 ... αn isthe set of WFFs or statements that follow from α1 ... αn.
Logical equivalence Two statements or WFFs are logically equivalent (with each other) when they entail each other.
Main connective The main connective in a WFF is the connective that has the widest scope.
Modus Ponens A Latin name for any valid argument of the form : If α then β. α. Therefore β.
Modus Tollens A Latin name for any valid argument of the form : If α then β. not-β. Therefore not-α.
Natural language A public language used for general communication such as English, French, etc.
Negation sign The symbol ~. In some formal systems, the negation sign is ¬
One-place connective A symbol or expression that connects to a single statement or WFF to form a longer statement or WFF.
Predicate logic A formal system of logic that includes SL.
Reductio A method for proving that a statement is false. We first assume that it is true, and shows that it leads to a false conclusion. We then conclude that the statement must be false.
Rules of proof Rules that specify how logical proofs are to be constucted.
Scope The scope of a connective in a WFF α is the shortest WFF in α that contains the occurrence of the connective.
Semantic rules Rules for interpreting the sentences in a language. They tell us what the sentences mean and the conditions under which the sentences are true or false.
Sentence letters The sentence letters in SL are capital letters such as A, B, etc. Used for translating sentences.
Sentential connective A symbol or expression that connects to one or more sentences or WFFs to form a new sentence or WFFs.
Sequent An argument in a formal system of logic.
Soundness A sound argument is a valid argument with only true premises.
Statement A declarative sentence.
Syntactic ambiguity A phrase having more than one meaning because there is more than one way to interpret its grammatical structure
Syntactic rules The rules that define the grammatical sentences of a language.
Tautology A WFF that is true under all assignments of truth-values to its sentence letters.
The full truth-table method A method for checking the validity of a sequent in SL.
Truth-functional sentential connective A special kind of sentential connective. A sentential connective α is truth-functional if and only if it has a truth-table.
Truth-table A table that shows the truth-values of a WFF under all possible assignments of truth-values to its sentence letters.
Truth-value Either T or F, i.e. truth or falsity.
Validity An argument is valid if and only if there is no logically possible situation where all the premises are true and the conclusion is false.
Well-formed Formula (WFF) Any expression constructed according to the rules of formation for a formal language.

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Men are apt to mistake the strength of their feeling for the strength of their argument. The heated mind resents the chill touch and relentless scrutiny of logic.


William Gladstone

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