Sentential Logic

1
Sentential Logic

• Semantics

2
Semantics of SL

• Review
• Truth-functional truth
• Truth-functional falsity
• Truth-functional indeterminacy
• Truth-functional equivalence

3
Truth-Functional Consistency

• Defn A set of sentences of SL is
  truth-functionally consistent iff there is at
  least one truth-value assignment where all the
  members of the set are true. A set of sentences
  is truth-functionally inconsistent iff it is not
  truth-functionally consistent.
• We can use a shortened truth-table to test for tf
  consistency
• But we must use a full table to test for tf
  inconsistency.

4
Truth-functional Consistency

• Use a full truth-table to establish that the
  following set of sentences of SL is
  truth-functionally consistent.
• A É B, B É C, C É A

5
Truth-functional Consistency

• Use a shortened truth-table to establish that the
  following set of sentences of SL is
  truth-functionally consistent.
• U v (W H), W (U v H), H v H

6
Truth-functional Consistency

• The following set is not truth-functionally
  consistent. Use a truth-table to demonstrate
  this.
• ( J É J) É H, J, H

7
Truth-Functional Entailment

• Defn A set G of sentences of SL
  truth-functionally entails a sentence P iff there
  is no truth-value assignment on which all the
  members of G are true and P is false.
• When G truth-functionally entails P, we can write
  G P.

8
Truth-Functional Entailment

• We need to use a full truth-table to establish G
  P.
• But we can use a shortened table to establish
  that G does not truth-functionally entail P.

9
Examples

• Use a full truth-table to demonstrate that
• A, (A v B), B É C C v A

10
Examples

• Use a shortened truth-table to show that the
  following truth-functional entailment does not
  hold
• A, B v C B

11
Truth-Functional Validity

• Defn An argument of SL is truth-functionally
  valid iff there is no truth-value assignment on
  which all of the premises are true and the
  conclusion is false. An argument of SL is
  truth-functionally invalid iff it is not
  truth-functionally valid.

12
Truth-Functional Validity

• As with tf entailment, we need a full truth-table
  to establish truth-functional validity.
• But a shortened table could be used to show that
  an argument is truth-functionally invalid.

13
Examples

• Show that the following argument is
  truth-functionally valid.
• (A G) v (B É G)
• G v B___________
• B v G

14
Examples

• Use a shortened truth-table to show that the
  following argument is not truth-functionally
  valid.
• A B
• B v A
• A

15
Examples

• Determine whether the following argument is
  truth-functionally valid or invalid.
• Computers can think if and only if they can have
  emotions. If computers can have emotions, then
  they can have desires as well. But computers
  cant think if they have desires. Therefore,
  computers cant think.

16
Examples

• Determine whether the following argument is
  truth-functionally valid or invalid.
• Certain non-human primates have been known to
  exhibit grief at the loss of a family member.
  But if they do that, then they are capable of
  abstract thought and must have a sense of self.
  And if they have those kinds of capabilities,
  then they are demonstrating some of the key
  indicators of personhood. Therefore, certain
  non-human primates are moral agents, since if
  they exhibit indicators of personhood they are
  moral agents.