Truth, deduction, computation lecture 8
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This document discusses connectives and conditionals in logic. It introduces the material conditional operator "→" and shows its truth table. The conditional expresses "P only if Q" or "Q if P", where Q is necessary and P is sufficient. A biconditional "↔" is also discussed, which means "A if and only if B". The document explores expressing any truth-valued function using connectives like "∧" and "¬". It introduces the Peirce arrow "↓" and Sheffer stroke "↑" as alternatives to the standard connectives for representing logical operations.
Truth, deduction, computation lecture 8
- 1. Truth, Deduction, Computation Lecture 8 Conditionals and Other Connectives Vlad Patryshev SCU 2013
- 2. Examples in Plain English 1. 2. 3. 4. 5. It rains because we prayed causation, correlation... It rains after we prayed We go to school unless it rains If we go to school, it rains We don’t go to school only if it rains Any logic in these sentences? How about truth tables? Some of the sentences are not truth-functional
- 3. Conditional Symbol → Material conditional P Q P→Q T T T T F F F T T F F T Looks familiar? How about DNF?
- 4. Conditional Symbol → Material conditional P Q P→Q ¬PvQ T T T T T F F F F T T T F F T T
- 5. Necessary and Sufficient Conditions ● ● ● ● P only if Q - meaning if P, then Q Q if P - same thing Q is necessary P is sufficient
- 6. Conditions in Deduction P1∧P2∧...Pi∧...∧Pn→Q is a logical truth if and only if P1 … Pn Q
- 7. Biconditional Symbol ↔ ● ● ● ● A A A A ↔B if and only if B iff B “just in case” B (in math only) ○ Math: n is even just in case n2 is even ○ Real life: We took umbrellas just in case it rains
- 8. Biconditional Symbol ↔ P Q P↔Q T T T T F F F T F F F T Looks familiar? How about DNF?
- 10. Completeness Given a truth-valued function, can it be expressed via the connectives we know? E.g. via ∧v¬? Easy for n=1: P f1 f2 f3 f4 T T T F F F T F T F General case? f(P1, P2, …, Pn)
- 11. Completeness ∧v¬ is enough.Actually,one of ∧v, and ¬ Other solutions? Actually...
- 12. Peirce’s Arrow NOR, aka ↓ A ↓ B ⇔ ¬(AvB) ¬A ⇔ A↓A AvB ⇔ ¬¬(AvB) ⇔ ¬(A↓B) ⇔ (A↓B)↓(A↓B) “A or B” is “neither (neither A or B) nor (neither A or B) Other solutions?
- 13. Sheffer Stroke NAND, aka ↑ A ↑ B ⇔ ¬(A∧B) ¬A ⇔ A∧A A∧B ⇔ ¬¬(A∧B) ⇔ ¬(A↑B) ⇔ (A↑B)↑ (A↑B)
- 14. Exercise
- 15. That’s it for today