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Logical Equivalence

If two formulas evaluate to the same truth value in all situations, so that their truth tables are the same, they are said to be logically equivalent.

We write tex2html_wrap_inline352 to indicate that two formulas tex2html_wrap_inline354 and tex2html_wrap_inline356 are logically equivalent.

If two formulas are logically equivalent, their syntax may be different, but their semantics is the same. The logical equivalence of two formulas can be established by inspecting the associated truth tables.

Is tex2html_wrap_inline358 logically equivalent to tex2html_wrap_inline360 ?


Lines 2 and 3 prove that this is not the case.

Substituting logically inequivalent formulas is the source of most real-world reasoning errors.

Is tex2html_wrap_inline376 logically equivalent to tex2html_wrap_inline378 ?


Yes, tex2html_wrap_inline394 .

De Morgan's Laws

There are a number of important equivalences, including the following De Morgan's Laws:



These equivalences can be used to transform a formula into a logically equivalent one of a certain syntactic form (called a ``normal form'').

Another useful logical equivalence is double negation:




The first equivalence is by double negation (as equivalence can be used in both directions), the second by De Morgan's Law.

Note that we have just derived a new equivalence,


which shows that disjunction can be expressed in terms of conjunction and negation!

Some Logical Equivalences

You should be able to convince yourself of (i.e., prove) each of these:

Commutativity of tex2html_wrap_inline396 : tex2html_wrap_inline398

Commutativity of tex2html_wrap_inline400 : tex2html_wrap_inline402

Associativity of tex2html_wrap_inline404 : tex2html_wrap_inline406

Associativity of tex2html_wrap_inline408 : tex2html_wrap_inline410

Idempotence: tex2html_wrap_inline412

Absorption: tex2html_wrap_inline414

Distributivity of tex2html_wrap_inline416 : tex2html_wrap_inline418

Distributivity of tex2html_wrap_inline420 : tex2html_wrap_inline422

De Morgan's Law for tex2html_wrap_inline424 : tex2html_wrap_inline426

De Morgan's Law for tex2html_wrap_inline428 : tex2html_wrap_inline430

Double Negation: tex2html_wrap_inline432

Contradictions: tex2html_wrap_inline434

Identities: tex2html_wrap_inline436

Tautologies: tex2html_wrap_inline438

Tautologies and Contradictions

A tautology is a formula that is always true, no matter which truth values we assign to its variables.

Consider the proposition ``I passed the exam or I did not pass the exam,'' the logical form of which is represented by the formula tex2html_wrap_inline440 .


This is a tautology, as we get T in every row of its truth table.

A contradiction is a formula that is always false.

The logical form of the proposition ``I passed the exam and I did not pass the exam'' is represented by tex2html_wrap_inline448 .


Tautologies and contradictions are related.

Theorem: If tex2html_wrap_inline456 is a tautology (contradiction) then tex2html_wrap_inline458 is a contradiction (tautology).


Syntax: If tex2html_wrap_inline460 and tex2html_wrap_inline462 are formulas, then tex2html_wrap_inline464 (read `` tex2html_wrap_inline466 implies tex2html_wrap_inline468 '') is also a formula. We call tex2html_wrap_inline470 the premise and tex2html_wrap_inline472 the conclusion of the implication.

Semantics: If tex2html_wrap_inline474 is true and tex2html_wrap_inline476 is false, then tex2html_wrap_inline478 is false. In all other cases, tex2html_wrap_inline480 is true.

Truth table:



The semantics of implication is trickier than for the other connectives.

Implication can also be expressed by other connectives, for example, tex2html_wrap_inline506 is logically equivalent to tex2html_wrap_inline508 .

Fun example: The Case of the Stupid Defense Attorney.

Prosecutor: ``If the defendant is guilty, then he had an accomplice.''

Defense Attorney: ``That's not true!!''

What did the defense attorney just claim??


Syntax: If tex2html_wrap_inline510 and tex2html_wrap_inline512 are formulas, then tex2html_wrap_inline514 (read `` tex2html_wrap_inline516 if and only if (iff) tex2html_wrap_inline518 '') is also a formula.

Semantics: If tex2html_wrap_inline520 and tex2html_wrap_inline522 are either both true or both false, then tex2html_wrap_inline524 is true. Otherwise, tex2html_wrap_inline526 is false.

Truth table:



The biconditional may be viewed as a shorthand for a conjunction of two implications, as tex2html_wrap_inline540 is logically equivalent to tex2html_wrap_inline542 .

Necessary and Sufficient Conditions

The phrase ``necessary and sufficient conditions'' appears often in mathematics.

A proposition tex2html_wrap_inline544 is necessary for tex2html_wrap_inline546 if tex2html_wrap_inline548 cannot be true without it: tex2html_wrap_inline550 is a tautology.

Example: It is necessary for a student to have a 2.8 GPA in the core courses to be admitted to become a CSE major.

A proposition tex2html_wrap_inline552 is sufficient for tex2html_wrap_inline554 if tex2html_wrap_inline556 is a tautology.

Example: It is sufficient for a student to get A's in CSE 113, CSE 114, CSE 213, CSE 214, and CSE 220 in order to be admitted to become a CSE major.


If a proposition tex2html_wrap_inline558 is both necessary and sufficient for tex2html_wrap_inline560 , then tex2html_wrap_inline562 and tex2html_wrap_inline564 are logically equivalent (and vice versa).

Tautologies and Logical Equivalence

Theorem: A propositional formula tex2html_wrap_inline566 is logically equivalent to tex2html_wrap_inline568 if and only if tex2html_wrap_inline570 is a tautology.

Proof: (a) If tex2html_wrap_inline572 is a tautology, then tex2html_wrap_inline574 is logically equivalent to tex2html_wrap_inline576 .

Why? If tex2html_wrap_inline578 is a tautology, then it is true for all truth assignments. By the semantics of the biconditional, this means that tex2html_wrap_inline580 and tex2html_wrap_inline582 agree on every row of the truth table. Hence the two formulas are logically equivalent.

(b) If tex2html_wrap_inline584 is logically equivalent to tex2html_wrap_inline586 , then tex2html_wrap_inline588 is a tautology.

Why? If tex2html_wrap_inline590 and tex2html_wrap_inline592 are logically equivalent, then they evaluate to the same truth value for each truth assignment. By the semantics of the biconditional, the formula tex2html_wrap_inline594 is true in all situations. height6pt width4pt

Related Implications

Implication: tex2html_wrap_inline596 : If you get A's on all exams, you get an A in the course.

Converse: tex2html_wrap_inline598 . If you get an A in the course, then you got A's on all exams.

Inverse: tex2html_wrap_inline600 . If you didn't get A's on all exams, then you didn't get an A in the course.

Contrapositive: tex2html_wrap_inline602 . If you didn't get an A in the course, then you didn't get A's on all exams.


Note that implication is logically equivalent to the contrapositive, and that the inverse is logically equivalent to the converse!

Deriving Logical Equivalences

We can establish logical equivalence either symbolically or via truth tables.

Example: tex2html_wrap_inline616 is logically equivalent to tex2html_wrap_inline618 .


Symbolic proofs are much like the simplifications you did in high school algebra - trial-and-error leads to experience and finally cunning.

Example: tex2html_wrap_inline632

Proof: (which laws are used at each step?)


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Next: About this document Up: My Home Page

Steve Skiena
Tue Aug 24 13:43:25 EDT 1999