CSCI 2824: Lecture

These 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:

  1. Raymond Smullyan's liar and truth teller puzzles.

  2. The truth table to analyze situations.

  3. Propositional Logic, Connectives and Truth Tables.

  4. Equivalence of Formulae, Tautologies and Fallacies.

This lecture corresponds to section 1.3 in Ensley and Crawley's book.

Logic

In a very crude sense, logic is the assembly language of mathematics (or philosophy). Logic can be defined as the study of reasoning itself or the study of techniques for drawing valid conclusions from premises. Understanding Logic is very important to computing at many levels. For instance, propositional Logic and propositional connectives such as AND, OR and NOT form the basic building block of digital circuits from which we build modern computers. Logic is also key to areas of CS such as automata theory, programming language semantics and artificial intelligence.

Smullyan's Liar/Truth-Teller Puzzles

Raymond Smullyan has written many books with puzzles that illustrate the beauty behind logic and formal systems (Instructor's favorite: To Mock A Mockingbird, a great way of thinking about combinatory Logic and functional programming).

The liar and truth teller puzzles all assume that we are on some island and there are two types of people on this island:

  1. (Compulsive) Truth Tellers: They always tell the truth.

  2. (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:

Puzzle

Suppose I asked a person in this island the question: “Are you a liar?”. What would be their answer?

Puzzle

We meet two people in the island A and B 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 truth tables. The idea is that we are going to evaluate every possible situation and see which ones “make sense”.

Let us do this informally.

A is truthteller? B is truthteller? A's statement is a lie B's statement is a lie Is this situation viable?
True True True False No, because, we assume that A is a ’'TruthTeller’’ when her statement is a lie
False True False False No, because A is assumed ’'liar’’ but is telling the truth.
True False False False No, because B is assumed ’'liar’’ but is telling the truth.
False False True True Yes! They are both assumed ’'liar’’ and their statements are both lies!!

Puzzle

Let 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”.

Puzzle

This 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?

Note The question or questions that you ask have to be explicitly about the castle and whether to take left or right. The instructors gut reaction was to ask A if Socrates is still alive (or the sun rose in the east yesterday) and B which is the way to the castle, but that does not count apparently!!

Propositional Logic

What 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 true or it is false.

Therefore, we simply use propositional variables (also called Boolean variables) to represent propositions. Why this? Just makes it nice and algebra like.

We use letters like p,q,r and so on to represent propositions.

Propositional Logic Formulae

Formulae in propositional logic are defined as follows:

  • Any propositional variable is a formula.

  • If f_1 and f_2 are formulas, then f_1 land f_2 (read as f_1 AND f_2) is a formula.

  • If f_1 and f_2 are formulas, then f_1 lor f_2 (read as f_1 OR f_2) is a formula.

  • If f is a formula then neg f (read as NOT f) is a formula.

Let us try some examples. Read the following propositional formulae aloud:

  • p

  • p land q

  • p lor r

  • p lor (q land neg r)

  • neg( r lor (p lor q lor neg r) ) .

We evaluate propositional formulae using truth tables.

Truth Table for AND

p q  p land q
T T T
T F F
F T F
F F F

Each row represents some kind of a situation. For example, the top most row represents the situation when propositions p and q are both ’'true’’. Then we conclude that the formula p land q in this situation is also ’'true’’.

Logicians call these situations models. In this case, a Logician would say that the truth assignment {p rightarrow true, q rightarrow true } is a model of the formula (p land q). In other words, if you imagine a situation wherein the proposition p is true and the proposition q is true then in that situation formula p land q is true.

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

p q  p lor q
T T T
T F T
F T T
F F F

Can you write down all the models of p lor q (read p OR q)?

Does this correspond to your conception of lor (the logical connective OR)?

Truth Table for NOT

p neg p
T F
F T

Truth Table for Compound Formulae

XOR is an important derived connective that is defined in terms of land, lor, neg. It has the interpretation of “either-or”: i.e, either p or q, but not both.

p XOR q is the formula  (p land neg q) lor (neg p land q).

Its truth table can be written as below:

p q p land neg q neg p land q p XOR q
T F T F T
T T F F F
F T F T T
F F F F F

We will go through few more examples of truth tables in the book.

Other examples of derived connectives are:

  • Implication p Rightarrow q is defined as  neg p lor q (we will study this in detail next week).

  • Equivalence  p Leftrightarrow q is defined as  ( p land q) lor (neg p land neg q).

  • Nand (a.k.a not of and) p mbox{NAND} q is defined as neg (p land q).

  • NOR (a.k.a not of or ) p mbox{NOR}  q is defined as neg (p lor q)

Let us write the truth table for the Leftrightarrow connective (equivalence connective):

p q p land q neg p land neg q p Leftrightarrow q
T F F F F
T T T F T
F T F F F
F F F T T

Tautology, Fallacies and Equivalence

Tautology

A 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 p lor neg p.

p neg p p lor neg p
T F T
F T T

No matter what you value one gives p, the formula p lor neg p is always true.

Other examples of tautology are

  •  (neg (p land q)) Leftrightarrow  (neg p lor neg q) (De Morgan's Law).

  •  (p land neg q) lor p (also written as (p Rightarrow q) Rightarrow p).

  •  ((p Rightarrow q) Rightarrow r) Rightarrow ((p Rightarrow r) Rightarrow (q Rightarrow r)) (do not even try expanding this :-)

Fallacies

Fallacies are the opposite of tautologies. These are formulae that are false no matter what the truth values of the propositions in them.

Example: p land neg p.

p neg p p land neg p
T F F
F T F

If we take a tautology and negate it then it becomes a fallacy.

Therefore neg (p lor neg p) is a fallacy.

Logical Equivalence

Two formulae are logically equivalent if and only if they have the same truth value in each row of the (joint) truth table.

Example:

The formulae F_1: neg(p land q) and F_2: neg p lor neg q are logically equivalent. To see why let us write their truth tables (we tack them together for convenience).

p q F_1: neg(p land q) F_2: neg p lor neg q
T T F F
F T T T
T F T T
F F T T

Notice that for all the truth table rows, F_1,F_2 coincide.

Examples of equivalent formulae include

  • p and neg neg p.

  • neg p and neg neg neg neg neg p.

  • neg (p lor q) and neg p land neg q (De-Morgan's law).

Technically, you need not assume that the formulae have the same set of propositions. For example, p land q and neg p are not logically equivalent. However, p land (q lor neg p) and p are logically equivalent.

Similarly, q land neg q and r land neg r are logically equivalent since they are both fallacies.

Theorem:

Statement of Theorem: Whenever two formulae F_1 and F_2 are equivalent then F_1 Leftrightarrow F_2 is a tautology.

Proof: Consider the truth table for the formula F_1 Leftrightarrow F_2 generated from a truth table that has a column for F_1 and a column for F_2 (as shown below).

p_1 p_2 cdots p_n F_1 F_2  F_1 Leftrightarrow F_2
T T cdots F F F T
vdots  vdots ddots vdots vdots vdots vdots
F F cdots T T T T

Note that for F_1 Leftrightarrow F_2 column to have a false entry at some row, F_1 and F_2 must have different values at that row. But since F_1 is logically equivalent to F_2, they always have the same value at each row.

As a result, the F_1 Leftrightarrow F_2 column must be all true, for all the rows in the truth table. Therefore it is a tautology.