Formal Semantics of S

1. Formal Semantics of S

2. Semantics and Interpretations • There are two kinds of interpretation we can give to wffs: • Assigning natural language sentences to sentence letters via a translation key (as we do for purposes of translation) • Truth value assignments…

3. Truth Value Assignments Truth Value Assignment:A truth value assignment (or tva) is an assignment of the value true (abbreviated by a ‘T’) or the value false (‘F’), but not both, to each of the atomic wffs of S. A tva is a function from the set of atomic wffs into the set {T, F}. Also called an interpretation. Model:A tva is a model for (or models) a wff P (or set of wffs G) iff the wff P (or all wffs in G) are true on that tva.

4. Constructing Truth Tables • List the n atomic components in alphabetical order at the top left of the table (these are the heads of the basis columns), then a vertical line, then the wff(s) in question; underline all • Fill in the rightmost basis column by alternating Ts and Fs for 2n rows. Move one column to the left and fill in its 2n rows by alternating pairs of Ts and Fs. Repeat this process of moving one column to the left and doubling the number of consecutive Ts and Fs, until you have filled each basis column, at which point you stop. • Fill in the columns under the wff in question as follows: • Copy the basis columns to the columns under the corresponding atomic components of the wff • Calculate (based on the values in the columns under their immediate well-formed components) the values in the columns of all the smallest well-formed components yet to be calculated. Repeat this process until the column under the main connective is complete. • Box the column under the main connective.

5. Characteristic Truth Tables

6. Semantic Properties of Individual Wffs Truth-FunctionallyTrue Wff:A wff P of S is truth-functionally true iff P is true on everytva (every tva is a model).Test:P is T-Fly true iff, in its truth table, only Tsappear in the column under the main connective. Truth-Functionally False Wff:A wff P of S is truth-functionally false if and only if P is false on every tva (no tva is a model). Test:P is T-Fly false iff, in its truth table, only Fs appear in the column under the main connective.

7. Semantic Properties of Individual Wffs Truth-FunctionallyContingent Wff:A wff P of S is truth-functionally contingent iff P is neither truth-functionally true nor truth-functionally false; i.e., it is false on at least one tva and true on at least one tva.Test:P is T-Fly contingent (has a model) iff, in its truth table, at least one F and at least one T appear in the column under the main connective.

8. Semantic Properties of Individual Wffs Truth-FunctionallySatisfiable Wff:A wff P of S is truth-functionally satisfiable iff P is not truth-functionally false; i.e., it is true on at least one tva. We also say the wff has a model or is modeled.Test:P is T-Fly satisfiable (has a model) iff, in its truth table, at least one T appears in the column under the main connective. Truth-FunctionallyFalsifiable Wff:A wff P of S is truth-functionally falsifiable iff P is not truth-functionally true; i.e., it is false on at least one tva. Test:P is T-Fly falsifiable iff, in its truth table, at least one F appears in the column under the main connective.

9. Semantic Properties of Individual Wffs

10. Semantic Properties of Sets of Wffs Truth-FunctionallyEquivalent Pairs of Wffs:Wffs P and Q of S are truth-functionally equivalent iff there is no tva on which they differ in truth value (every model of P is a model of Q, and every model of Q is a model of P). Test: Wffs P and Q are T-Fly equivalent iff there is no row in their joint truth table in which the values under their main connectives differ. Truth-FunctionallyContradictory Pairs of Wffs:Wffs P and Q of S are truth-functionally contradictory iff there is no tva on which they have the same truth value.Test: Wffs P and Q are T-Fly contradictory iff there is no row in their joint truth table in which the values under their main connectives are the same.

11. Semantic Properties of Sets of Wffs Truth-Functionally Consistent Set of Wffs:A set Gof wffs of S is truth-functionally consistent iff there is at least one tva on which all members of G are true. We also say that G has a model or is modeled. Gis truth-functionally inconsistent iff it is not truth-functionally consistent.Test: A set G of wffs is T-Fly consistent iff there is at least one row in the set’s joint truth table in which all the members of G have a T under their main connectives. G is T-Fly inconsistent iff there is no such row.* *Strictly speaking, while the definition applies to finite or infinite G, the test applies only to finite G.

12. Semantic Properties of Sets of Wffs Truth-Functional Entailment:A set Gof wffs of S truth-functionally entails a wff P iff there is no tva on which all the members of G are true and P is false. (Every model of Gis a model of P.)Test: A set GT-Fly entails a wff P iff there is no row of their joint truth table on which all the members of Ghave a T under their main connectives and P has an F under its main connective.*G‘PG entails P, P follows from G G—PG does not entail P, P does not follow from G ?‘P or ‘PP follows from the empty set, P is T-Fly true *Strictly speaking, while the definition applies to finite or infinite G, the test applies only to finite G.

13. Semantic Properties of Sets of Wffs Argument of S:An argument of S is a finite set of two or more wffs of S, one of which is the conclusion, while the others are the premises. Truth-Functionally Valid Argument:An argument in S is truth-functionally valid iff there is no tva on which all the premisesare true and the conclusion is false. (Every model of the premises is a model of the conclusion.) An argument of S is truth-functionally invalid iff it is not truth-functionally valid.Test: An argument is T-Fly valid iff there is no row of the joint truth table on which all the premiseshave a T under their main connectives and the conclusion has an F under its main connective.

14. Truth-FunctionallyTrue Wff:A wff P of S is truth-functionally true iff P is true on every tva (every tva is a model). Truth-Functionally False Wff:A wff P of S is truth-functionally false if and only if P is false on every tva (no tva is a model). Truth-FunctionallyContingent Wff:A wff P of S is truth-functionally contingent iff P is neither truth-functionally true nor truth-functionally false. Truth-FunctionallySatisfiable Wff:A wff P of S is truth-functionally satisfiable iff P is not truth-functionally false (at least one tva is a model). Truth-FunctionallyFalsifiable Wff:A wff P of S is truth-functionally falsifiable iff P is not truth-functionally true. Truth-FunctionallyEquivalent Pairs of Wffs:Wffs P and Q of S are truth-functionally equivalent iff there is no tva on which they differ in truth value (every model of P is a model of Q, and every model of Q is a model of P). Truth-FunctionallyContradictory Pairs of Wffs:Wffs P and Q of S are truth-functionally contradictory iff there is no tva on which they have the same truth value. Truth-Functionally Consistent Set of Wffs:A set  of wffs of S is truth-functionally consistent iff there is at least one tva on which all members of  are true. We also say that  has a model or is modeled.  is truth-functionally inconsistent iff it is not truth-functionally consistent. Truth-Functional Entailment:A set  of wffs of S truth-functionally entails a wff P iff there is no tva on which all the members of  are true and P is false. (Every model of  is a model of P.) Truth-Functionally Valid Argument:An argument in S is truth-functionally valid iff there is no tva on which all the premises are true and the conclusion is false. (Every model of the premises is a model of the conclusion.) Otherwise it is truth-functionally invalid.