The semantics of SL

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# The semantics of SL - PowerPoint PPT Presentation

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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Presentation Transcript
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.
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.
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).
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.
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.
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.
If you studied hard, you did well in PHIL 120.

You studied hard.

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You did well in PHIL 120.

S  W

S

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W

If you studied hard, you did well in PHIL 120.

You did well in PHIL 120.

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You studied hard.

S  W

W

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S

Compare (again!)
Using a one row truth table to prove that an argument is truth functionally invalid:A v BB_____~A
Using a one row truth table to prove that an argument is truth functionally invalid:A v BB_____~A
Using a one row truth table to prove that an argument is truth functionally invalid:A v BB_____~A
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.
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
Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 
Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 
Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 
Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 
Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F  ~D} 
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
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

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