Semantics I Facultad de Filosofía y Letras Universidad

Semantics I Facultad de Filosofía y Letras Universidad de Cádiz On meaning denotation: Truth and truth-conditions Bárbara Eizaga Rebollar April 2015

Truth & truth-conditions Propositions do not always correspond to one sentence. Synonymous sentences express the same proposition: 1. Paris is the capital of France. 2. Paris is France’s capital. Ambiguous sentences can express different propositions: 3. Mary said that John saw her dress. The semantics of propositional logic is described in terms of truth values (correspondence theory of truth). Every proposition can take 1 of 2 truth values: 1(T) or 0 (F). A proposition is T if and only if (iff) it correctly describes some state of affairs (Tarski, 1944). Thus, It’s raining is T iff it is raining.

Truth & truth-conditions Propositions are expressed by declarative sentences. But some sentences cannot be said to be T or F (questions, imperatives & exclamatives); they don’t have a truth-value: 4. Who opened the door? 5. Stop talking! 6. What a mess! Some propositions will always be T (tautologies): 7. A bachelor is unmarried. Bachelor already bears the feature [-MARRIED], so it is redundant to predicate this as a property of the individual. Other propositions are always F (contradictions): 8. A bachelor is married. This contradiction is the result of the lexical-semantic properties: bachelor bears the feature [-MARRIED], so it is contradictory to predicate the feature [+MARRIED].

Truth & truth-conditions Most sentences in discourse will be neither tautologies nor contradictions: they will be T in some situations, F in others (contingent statements): 9. It is raining 10. Noam is a linguist. In most places, it is likely to rain on some days, but not on others information dependent on what the real world looks like, but independent of the meaning of the words & the sentence itself. Semanticists are not interested in whether it is raining or not today, & whether Noam is really a linguist or a plumber. The distinction between truth & meaning implies that we are not so much interested in truth, but in truth conditions.

Truth & truth-conditions To know the meaning of a sentence, we need to know what the world must look like if the sentence is to have the value T. Truth conditions are the conditions needed for there to be an appropriate correspondence between the proposition expressed by the sentence & the state of affairs in the world. We usually evaluate the T of a sentence with respect to some model of the world it provides a description of the denotation of the lexical items of the language (properties every individual has, what is going on, etc. )

Truth & truth-conditions So : Ø it’s raining is T iff it’s raining in our model. Ø Noam is a linguist is T iff the person named Noam in our universe of discourse has the property of being a linguist. Assigning truth values to propositions depends on a validation function V (= binary relation between a set of propositions & a set of truth values): tion corresponds V with facts in the model. V correspond with facts in the model.

Basics of propositional logic: negation, conjunction, disjunction, implication, reduction and equivalence

Propositional calculus: a sketch It establishes the truth conditions between the propositions (symbolized p, q, r) joined by logical connectives, whose function is to define a truth value (truth functors). CONNECTIVE TRUTH FUNCTION (MEANING) NAME ∧ ∨ ⊻ → ↔ ¬ “and” Conjunction “and/or” Inclusive disjunction “or else, only one of” Exclusive disjunction “ if p then q” Semantic implication (conditional) “p if and only if (=iff) q” Semantic equivalence (biconditional) “not” Negation

Sentential connectives 1. Negation: reverses the truth value of the proposition: p ¬ p If it’s raining is T 1 0 it isn’t raining is F 0 1 2. Conjunction: T iff both conjuncts are T: p q p ∧ Ann is rich and Mason is poor T iff T q that Ann is rich & T that Mason is poor. 1 1 1 Logical conjunction ≠ conjunction in lang. : 1 0 0 0 0 1. Jenny got married & pregnant temporal 2. Jenny got pregnant & married order absent from logical connective

Sentential connectives The connective conveys less information than its natural language counterpart: but & even though often considered as conjunction. 3. Jane snores, but John sleeps well (p ∧ q) 4. Though Jane snores, John sleeps well (p ∧ q) 3. Disjunction: T whenever at least 1 of the disjuncts is T (inclusive). P q p ∨ q 1. We invite all passengers who need some extra help or who are travelling 1 1 1 with children to board the aircraft. 1 0 1 2. A doctor or a dentist can write 0 1 1 prescriptions (both… and). 0 0 0

Sentential connectives English or often has an exclusive sense, which excludes the possibility of both disjuncts being T (exclusive disjunction): p q p⊻q 1 1 0 0 1 1 0 3. Do you want tea or coffee? 4. All entrees are served with soup or salad 4. Conditional & bi-conditional: Conditional or semantic implication is F only if the antecedent is T & the consequent is F. p q p→q If it snows, it is cold 1 1 0 0 1 1 0 1 F if there is snow, but it isn’t cold. 0 T if it doesn’t snow, but it’s cold. 0

Sentential connectives The T of bi-conditional or semantic equivalence requires both parts to have the same truth-value: Jane will go to the party if and only if (iff) Joan goes T if both of them go or neither of them goes. p q p↔q 1 1 0 0 0 1 5. Complex truth tables: More complex formulas can be represented in truth tables: a. The number of columns depends on the connectives present in the formula. b. The number of rows is determined by all possible combinations of truth values.

Sentential connectives Truth tables allow us to determine whether a formula is: Øa tautology if the last column in the truth table contains p ¬ p (p ∨ only ones, i. e. the statement is always T. (p ∨ ¬p) It is raining or it is not raining. ¬p) 1 0 1 1 Ø a contradiction if the last column in the truth table contains nothing but zeros; i. e. , it’s always F regardless of the truth values assigned to the statements. p ¬ p (p ∧ ¬p) It is raining and it is not raining 1 0 0 0 1 0 Øa contingency if the last column contains both ones & zeros: T or F depending on the truth value assigned to the statements. (p → q) ↔ ¬ (p ∧ ¬q)

Sentential connectives If it snows then it is cold if and only if it is false that it snows and it is not cold. p q ¬q (p → q) (p∧¬q ¬(p∧¬q (p→q) ) ) ↔¬(p∧¬q) 1 1 0 1 0 1 1 0 0 1 1 Thus, propositional logic is a good tool to describe valid arguments between sentences. Argument = premises (assumed to be T) + a conclusion Its truth follows from its premises if valid reasoning

Sentential connectives Valid inference rules used in logic to explain the validity of reasoning patterns in natural language: Modus ponens P→Q P Q Modus tollens P→Q ¬Q ¬P Hypothetical syllogism P→Q Q→R P→R Disjunctive syllogism P∨Q ¬P Q Rules of inference are used to prove that an argument is (or isn’t) valid

Sentential connectives These reasoning patterns are natural language examples of the logical schemata above: 1. If John loves Mary, Mary is happy. John loves Mary is happy (modus ponens) 2. If John loves Mary, Mary is happy. Mary is not happy. John doesn’t love Mary (modus tollens) 3. If Fred lives in Paris, then Fred lives in France. If Fred lives in France, then Fred lives in Europe. If Fred lives in Paris, Fred lives in Europe(hypothetic syllogism) 4. Fred lives in Paris or Fred lives in London. Fred does not live in Paris. Fred lives in London (disjunctive syllogism)

References Ø Chierchia, G. and Mc. Connel-Ginet, S. 2000. Meaning and Grammar; An Introduction to Semantics. Cambridge, Mass. and London, England: MIT Press. Ø Cruse, A. 2004. Meaning in language. Oxford: O. U. P. Ø De Swart, H. 1998. Introduction to Natural Language Semantics. Standford: CSLI Publications. Ø Saeed, J. I. 1997. Semantics. Oxford: Blackwell Publishing.