Chapter Two

Symbolizing in Sentential Logic

This chapter is a preliminary to the project of building a model of validity for sentential

arguments. We need to be able to translate English statements into symbols.

The syntax of a language shows how to formulate correct sentences using its vocabulary. The syntax is specified in formation rules.

The semantics of a language shows the meaning of the symbols and under what conditions their combinations are true and under what conditions they are false.

1. Atomic and Compound Sentences

• In English longer sentences can be built up of shorter sentences using sentence connectives such as “and” and “or”.

• Sentences built up of shorter sentences by means of sentence connectives are compound sentences.

• All other sentences are said to be atomic, or simple, sentences.

2. Truth Functions

• With truth functions we have only two truth-values: “true” and “false”.

• We use symbols to represent common mathematical functions. These are called operators.

Truth Functions, continued

Our system of logic has five truth-functional operators:

“~” (not) takes only one input.

“.” (and), “v” (or), “ ” (if…then), and “≡” (“if and only if”)⊃take two.

3. Conjunctions

Compound sentences formed by use of the connective “and” are called conjunctions, and the two sentences joined by

“and” are called conjuncts.

Conjunctions, continued

• The different truth values of compound sentences that are the products of the different truth values of their conjuncts can be represented in a truth table.

• Sentences can be used to make different statements, depending on time, and, in some cases, place.

4. Non-Truth-Functional Connectives

Many connectives in English are not truth-functional, e.g., “before”.

5. Variables and Constants• A statement variable has no truth-value; what does have

truth-value is a statement we substitute for it, and the truth-value varies according to what statement that happens to be.

• This notion of substitution is analogous to that used in algebra.

• It is conventional to use small letters, p, q, r as sentence variables, and capital letters, A, B, C as sentence constants.

6. Negations

• Some logical operators generate a new sentences out of just one starting sentences.

• Only one operator—negation—is used in standard sentential logic.

• Negation is symbolized by the tilde symbol, “~”.

7. Parentheses and Brackets• By using parentheses we can build up complex sentences

out of shorter sentences.• The shorter sentences that are combined to make longer

sentences are component sentences.• Parentheses are sued to indicate the scope of each logical

operator in any sentence; the sentences over which it operates.

Parentheses and Brackets, continued

• The main connective of a sentence is the truth-functional connective whose scope encompasses the entire remainder of the sentence.

• A sentence is well-formed if it is clear which operator is the main operator for the sentence and for each component sentence contained within the sentence.

Parentheses and Brackets, continued

Two conventions help eliminate unnecessary parentheses:

1) It is not necessary to place an outermost pair of parentheses entirely surrounding a sentence.

2) The scope of the “~” operator is always the shortest complete sentence that follows it.

8. Use and Mention• We must distinguish between using a word, phrase, or

statement, and talking about that word, phrase, or statement—that is, mentioning it.

• The language in which we speak about the logical language is the metalanguage.

• The language that we are talking about is the object language.

9. Disjunctions

• Two sentences connected by the word “or” form a compound sentence called a disjunction.

• The two sentences so connected are called disjuncts.

Disjunctions, continued

There are two different senses of the connective “or”:

1) Exclusive: If the disjunction is true one or other of the disjuncts is true, but not both.

2) Inclusive: If the disjunction is true either one of the disjuncts is true, or both are true.

Disjunctions, continued

Disjunction is symbolized by the wedge, “V”, which is a truth-functional logical connective.

10. “Not Both” and “Neither… Nor”

• All it takes to make a “not both” sentence true is for at least one of the two components to be false.

• Sentences built around the connective “neither…nor” should not be symbolized as disjunctions, but as conjunctions with two negated conjuncts.

11. Material Conditionals

• A compound sentence of the form “If… then…” is called a conditional.

• The sentence between the “if” and the “then” is called its antecedent.

• The sentence after the “then” is called its consequent.• The truth functional connective for conditionals is the

horseshoe, “ ”.⊃

Material Conditionals, continued

• A sentence whose main connective is the horseshoe is called a material conditional.

• The truth function represented by the horseshoe is called material implication.

12. Material Biconditionals• Two sentences are materially equivalent when

they have the same truth-value.• The symbol “≡” is called the tribar and stands for

material equivalence.• Compound sentences formed by the tribar are

called material equivalences, or biconditionals.

13. “Only If” and “Unless”• “Only if” sentences indicate necessary conditions,

but not sufficient conditions.

• “You will pass the class only if you pay attention” can be symbolized as C A.⊃

• A simple way of symbolizing “unless” sentences is as ‘or” sentences.

14. Symbolizing Complex Sentences

• The first step in symbolizing complex sentences is to identify the main connective of the sentence.

• The second step is to look for punctuation. Parentheses often mirror commas and semicolons.

• Be careful to determine the correct scope of negations.

15. Alternative Sentential Logic Symbols

• Negation: -, ¬• Conjunction: ,&∧• Disjunction: (almost always used as in this text)∨• Conditional: →• Biconditional: ↔

Key Terms• Antecedent• Atomic sentence• Biconditional• Component sentence• Compound sentence• Conditional• Conjunct• Conjunction

Key terms, continued

• Consequent• Disjunct• Disjunction• Dot• Exclusive disjunction• Exclusive “or”• Formation rules• Horseshoe

Key terms, continued

• Inclusive disjunction• Inclusive “or”• Main connective• Material biconditional• Material conditional• Material equivalence• Material impication• Metalanguage

Key terms, continued

• Negation• Object language• Polish notation• Scope• Semantics• Sentence• Sentence connective• Sentence constant

Key terms, continued

• Sentential logic• Statement• Statement variable• Substitution• Symbolic logic• Syntax• Tilde• Tribar

Key terms, continued

• Truth-function• Truth-functional operator• Truth table• Truth-value• Wedge• Well-formed