## CSCI 2824: LectureThese notes are for the part of lecture alphabet sequences for roughly the first 10-15 minutes of the lecture. In this lecture, we will cover the following concepts: Raymond Smullyan's liar and truth teller puzzles. The truth table to analyze situations. Propositional Logic, Connectives and Truth Tables. Equivalence of Formulae, Tautologies and Fallacies.
This lecture corresponds to section 1.3 in Ensley and Crawley's book. ## LogicIn a very crude sense, logic is the ## Smullyan's Liar/Truth-Teller PuzzlesRaymond Smullyan has written many books with puzzles
that illustrate the beauty behind logic and formal systems (Instructor's favorite: The liar and truth teller puzzles all assume that we are on some island and there are two types of people on this island: (Compulsive) Truth Tellers: They *always*tell the truth.(Compulsive) Liars: They *never*tell the truth.
Each puzzle gives us some statements that people make and you have to analyze who is a liar and who is a truth teller. Assume that there are no physical characteristics to differentiate each type from the other. Let us do some warm up questions: ## PuzzleSuppose I asked a person in this island the question: “Are you a liar?”. What would be their answer? ## PuzzleWe meet two people in the island and make the following remarks at the same time: A says: “Exactly one of us is lying” and B says: “At least one of us is truthful”. What can you conclude about the identities of A and B are they liars or are they truth tellers?
Puzzles like these are best analyzed using Let us do this informally.
## PuzzleLet us try this in class: A says “Exactly one of us is telling the truth”. B says “We are all lying”. C says “The other two are lying”.
## PuzzleThis one is tricky but doable. Whoever first gives me the answer gets a free drink of their choice at the celestial seasonings cafe in the engineering lobby: Two people A and B in the island are either both liars or both truth tellers. But we do not know which is the case. We wish to reach the castle which is either on path to the left or the path to the right. What question should we ask one or both of them so that we can find out which path leads to the castle?
## Propositional LogicWhat are propositions? Propositions are simply statements that can receive a true or false valuation. Examples of propositions: “Socrates is mortal”. “Jenny went to lunch with Craig”. “I am telling the truth”. “The quick brown fox jumped over the lazy brown dog”.
While studying propositional logic, we do not really care about what the proposition itself “means” just
that it is either Therefore, we simply use We use letters like and so on to represent propositions. ## Propositional Logic FormulaeFormulae in propositional logic are defined as follows: Any propositional variable is a formula. If and are formulas, then (read as AND ) is a formula. If and are formulas, then (read as OR ) is a formula. If is a formula then (read as NOT ) is a formula.
Let us try some examples. Read the following propositional formulae aloud: .
We evaluate propositional formulae using truth tables. ## Truth Table for AND
Each row represents some kind of a situation. For example, the top most row represents the situation when propositions and are both ’'true’’. Then we conclude that the formula in this situation is also ’'true’’. Logicians call these situations To avoid confusion let us use the term “situations” and “models”. We will formalize models later for first-order (predicate) logic. ## Truth Table for OR
Can you write down all the models of (read OR )? Does this correspond to your conception of (the logical connective OR)? ## Truth Table for NOT
## Truth Table for Compound Formulae
XOR is the formula . Its truth table can be written as below:
We will go through few more examples of truth tables in the book. Other examples of derived connectives are: **Implication**is defined as (we will study this in detail next week).**Equivalence**is defined as .**Nand**(a.k.a not of and) is defined as .**NOR**(a.k.a not of or ) is defined as
Let us write the truth table for the connective (equivalence connective):
## Tautology, Fallacies and Equivalence## TautologyA formula is a tautology if and only if it is true no matter what value one gives to the propositions involved in the formula. Example is .
No matter what you value one gives , the formula is always true. Other examples of tautology are (De Morgan's Law). (also written as ). (do not even try expanding this :-)
## FallaciesFallacies are the opposite of tautologies. These are formulae that are false no matter what the truth values of the propositions in them. Example: .
If we take a tautology and negate it then it becomes a fallacy. Therefore is a fallacy. ## Logical EquivalenceTwo formulae are Example: The formulae and are logically equivalent. To see why let us write their truth tables (we tack them together for convenience).
Notice that for all the truth table rows, coincide. Examples of equivalent formulae include and . and . and (De-Morgan's law).
Technically, you need not assume that the formulae have the same set of propositions. For example, and are not logically equivalent. However, and are logically equivalent. Similarly, and are logically equivalent since they are both fallacies. ## Theorem:
Note that for column to have a false entry at some row, and must have different values at that row. But since is logically equivalent to , they always have the same value at each row. As a result, the column must be all true, for all the rows in the truth table. Therefore it is a tautology. |