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The semantics of SL

The semantics of SL. Defining logical notions (validity, logical equivalence, and so forth) in terms of truth-value assignments A truth-value assignment : the assignment of T or F to each of the atomic sentences included in a sentence, or a set of sentences, or a group of sentences.

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The semantics of SL

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  1. The semantics of SL • Defining logical notions (validity, logical equivalence, and so forth) in terms of truth-value assignments • A truth-value assignment: the assignment of T or F to each of the atomic sentences included in a sentence, or a set of sentences, or a group of sentences.

  2. The semantics of SL • Truth tables: an effective procedure for establishing the logic status of individual sentences, sets of sentences, arguments, and so forth. • Each row of a truth table contains a truth value assignment. • Taken together the rows that include truth value assignments represent all the ways the world might be relevant to the sentence(s) involved.

  3. The semantics of SL • Defining logical notions in terms of truth-value assignments: the case of sentences • A sentence is truth-functionally true IFF it is true on every TVA (or IFF there is no TVA on which it is false). • A sentence is truth-functionally false IFF it is false on every TVA (or IFF there is no TVA on which it is true. • A sentence is truth-functionally indeterminate IFF it is neither truth-functionally true nor truth functionally false (of IFF it is true on at least one TVA and false on at least one TVA).

  4. Truth table conventions continued

  5. Truth table shortcuts On any TVA: • If one conjunct is false, the conjunction is false. • If one disjunct is true, the disjunction is true. • If the antecedent of a material conditional is false, the conditional is true. • If the consequent of a material conditional is true, the conditional is true. • There are no shortcuts for establishing the truth value of a biconditional.

  6. Truth table to establish the truth functional status of individual sentences • Only an entire truth table can establish that a sentence is truth functionally true. • Only an entire truth table can establish that a sentence is truth functionally false. • A two row truth table can establish that a sentence is truth functionally indeterminate. • A one row truth table can establish that a sentence is not truth functionally true. • A one row truth table can establish that a sentence is not truth functionally false.

  7. Proving that a sentence is truth-functionally indeterminate using a shortened truth table

  8. Proving that a sentence is truth-functionally indeterminate using a shortened truth table

  9. Proving that a sentence is truth-functionally indeterminate using a shortened truth table

  10. Proving that a sentence is truth-functionally indeterminate using a shortened truth table

  11. Truth functional validity • Defining logical notions in terms of truth-value assignments: the case of arguments. • An argument is truth functionally valid IFF there is no truth value assignment on which all the premises are true and the conclusion is false. • An argument is truth functionally invalid IFF there is a truth value assignment on which all the premises are true and the conclusion is false.

  12. If you studied hard, you did well in PHIL 120. You studied hard. --------------------------------- You did well in PHIL 120. S  W S ---------- W If you studied hard, you did well in PHIL 120. You did well in PHIL 120. ---------------------------------- You studied hard. S  W W -------- S Compare (again!)

  13. Establishing truth functional validity: the first argument

  14. Establishing truth functional invalidity:the second argument

  15. Using a one row truth table to prove that an argument is truth functionally invalid:A v BB_____~A

  16. Using a one row truth table to prove that an argument is truth functionally invalid:A v BB_____~A

  17. Using a one row truth table to prove that an argument is truth functionally invalid:A v BB_____~A

  18. Truth functional equivalence • Sentences P and Q are truth functionally equivalent IFF there is no TVA on which P and Q have different truth values. • Members of a pair of sentences are truth functionally non-equivalent IFF there is a TVA on which P and Q have different truth tables. • Only an entire truth table can prove that 2 sentences are truth functionally equivalent. • A one row truth table can prove that 2 sentences are not truth functionally equivalent.

  19. Proving truth functional equivalence

  20. Proving truth functional equivalence

  21. Proving truth functional equivalence

  22. Proving that 2 sentences are not truth-functionally equivalent 

  23. Truth functional consistency • A set of sentences is truth functionally consistent IFF there is at least one TVA on which all the members of the set are true. • A set of sentences is truth functionally inconsistent IFF there is no TVA on which all the members of the set are true. • A one row truth table can prove a set of sentences is truth functionally consistent. • Only an entire table can prove a set of sentences is truth functionally inconsistent

  24. Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 

  25. Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 

  26. Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 

  27. Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 

  28. Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 

  29. Truth functional entailment • Conventions: •  are used to indicate sets, with individual sentences separated by comas •  (gamma) is used as a meta variable for a set of sentences • ╞ (double turnstile) symbolizes the relationship of entailment that can obtain between a set of sentences of SL and an individual sentence of SL

  30. Truth functional entailment •  ╞ P (a formula in the meta language) Is read as “A set of sentences truth functionally entails a sentence P” • A set  of sentences truth functionally entails a sentence P IFF there is no truth value assignment on which all the members of  are true and P is false. • In SL: M  (A v B), ~A v ~B, ~A  M} ╞ ~M

  31. Truth functional entailment • ╞ with a line drawn through it from top right to bottom left symbolizes that the relationship of truth functional entailment does not hold. • A set  of sentences does not truth functionally entail a sentence P IFF there is one truth value assignment on which all the members of  are true and P is false.

  32. Using a truth table to prove entailment:{A v C, ~C╞ A

  33. Using a truth table to prove entailment:{A v C, ~C╞ A

  34. Using a truth table to prove entailment:{A v C, ~C╞ A

  35. Using a truth table to prove entailment:{A v C, ~C╞ A

  36. Using a truth table to prove a set does not entail a sentence (that the following is false:{A v C, ~C╞ ~A

  37. Using a truth table to prove a set does not entail a sentence (that the following is false):{A v C, ~C╞ ~A

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