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Logical Consequence

We say that tex2html_wrap_inline322 logically implies tex2html_wrap_inline324 , or that tex2html_wrap_inline326 is a logical consequence of tex2html_wrap_inline328 , if tex2html_wrap_inline330 is true whenever tex2html_wrap_inline332 is true.

Example: p logically implies tex2html_wrap_inline336 .



Note that logical consequence is a weaker condition than logical equivalence.

Theorem: A formula tex2html_wrap_inline344 logically implies tex2html_wrap_inline346 if and only if tex2html_wrap_inline348 is a tautology.

This gives us a tool to infer truths!

A rule of inference is a rule of the form: ``From premises tex2html_wrap_inline350 , infer conclusion tex2html_wrap_inline352 ''

A rule of inference is sound or valid if the conclusion tex2html_wrap_inline354 is a logical consequence of the conjunction tex2html_wrap_inline356 of all premises.

A rule of inference is unsound or bogus if it isn't!

Modus Ponens

If we assume that (1) if it is raining then today must be Monday, and (2) it is raining, then we can conclude that (3) today must be Monday.

But why?

Let r denote the proposition "it is raining" and m denote "today must be Monday".

Then tex2html_wrap_inline362 is a tautology, so m is a logical consequence of tex2html_wrap_inline366 and r.

Modus Ponens (Latin for ``method of affirming'') is the following inference rule:

From tex2html_wrap_inline370 and tex2html_wrap_inline372 , infer tex2html_wrap_inline374 .

Modus Tollens

If we assume that (1) if it is raining then today must be Monday, and (2) if it is not Monday, then we can conclude that (3) it is not raining.

Why? Because tex2html_wrap_inline376 is a tautology.


Modus Tollens (Latin for ``method of denying''):

From tex2html_wrap_inline392 and tex2html_wrap_inline394 , infer tex2html_wrap_inline396 .

Cognative Psychologists have shown that under 60% of college students have a solid intuitive understanding of Modus Tollens (versus almost 100% for Modus Ponens).

Invalid Reasoning Example

If we assume that (1) if it is raining, then today must be Monday, and (2) it is not raining, can we conclude that (3) today is not Monday?

In other words, is the following inference rule sound? ``From tex2html_wrap_inline398 and tex2html_wrap_inline400 , infer tex2html_wrap_inline402 .''

No! - because tex2html_wrap_inline404 is not a tautology.


This fails when both tex2html_wrap_inline420 and tex2html_wrap_inline422 are true.

Transitivity of Implication

From tex2html_wrap_inline424 and tex2html_wrap_inline426 , infer tex2html_wrap_inline428 :


This inference rule is also called ``hypothetical syllogism''.

Sherlock Holmes in Action

``And now we come to the great question as to the reason why. Robbery has not been the object of this murder, for nothing was taken. Was it politics, or was it a woman? That is the question confronting me. I was inclined from the first to the latter supposition. Political assassins are only too glad to do their work and fly. This murder had, on the contrary, been done most deliberately and the perpetrator had left his tracks all over the room, showing he had been there all the time.'' - A. Conan Doyle, A Study in Scarlet

What did Sherlock Holmes conclude?

Propositions and Premises

We can break the story into the following propositions:

Holmes identifies the following premises defined on these propositions:

Conclusion: P4!

Valid or Invalid Arguments?

Proof by Contradiction

Suppose that we want to prove tex2html_wrap_inline470 , and we know tex2html_wrap_inline472 is true. Instead of proving tex2html_wrap_inline474 directly, we may instead show that assuming tex2html_wrap_inline476 leads to a contradiction.

Formally, from tex2html_wrap_inline478 and tex2html_wrap_inline480 , infer tex2html_wrap_inline482 .


Example: ``Suppose I did shoot him. Well then I had to be in the room at the time of the crime to shoot him. However, at that exact time, I was teaching CSE113 before over a hundred witnesses. Therefore I couldn't have been the one to shoot him.''

There are beautiful examples of proof by contradiction in number theory.

Proofs by contradiction are very susceptible to reasoning errors unless you do them formally.

Make sure that your contradiction depends upon your assumption - otherwise it means that either your assumption was unnecessarily strong or (more likely) that you botched your reasoning.

Proof by contradiction is sometime also called ``reductio ad absurdum'' or ``indirect proof''.

From Inspector Craig's Files

A large sum of money has been stolen from the store. The criminal(s) were seen driving off from the scene. From questioning criminals A, B, and C we know:

  1. No one other than A, B, or C were involved in the robbery.
  2. C never pulls a job without using A.
  3. B does not know how to drive.
Is A innocent or guilty?
Let a, b, and c represent the respective propositions that A or B or C is guilty.

From the story, we have the following assumptions:

  1. tex2html_wrap_inline528
  2. tex2html_wrap_inline530
  3. tex2html_wrap_inline532

Proof: (by contradiction)

Assume a is innocent, i.e tex2html_wrap_inline536 .

By the contrapositive of (2) and modus ponens, we may infer tex2html_wrap_inline538 .

We thus have tex2html_wrap_inline540 , which by De Morgan's Law is logically equivalent to tex2html_wrap_inline542 .

By the contrapositive of (3) and modus ponens, we may infer tex2html_wrap_inline544 .

We now have tex2html_wrap_inline546 and tex2html_wrap_inline548 and tex2html_wrap_inline550 , which contradict assumption (1).

Thus we may conclude that a is true, which means that A is guilty!

Generalizing and Particularizing

Any man or woman is a person. Suppose I am a man. Therefore I am a person.

The inference rule disjunctive addition is a tool for generalizing: From p, infer tex2html_wrap_inline558 .

Suppose that it is raining and snowing. Therefore it is raining.

The inference rule conjunctive simplification is a tool for particularizing: From tex2html_wrap_inline560 , infer p.

Case Analysis

Suppose that I either a man or a woman. I am not a woman. Therefore I am a man.

The inference rule disjunctive syllogism is a tool for case analysis: from tex2html_wrap_inline564 and tex2html_wrap_inline566 , infer q.

Now suppose that I am either a man or a woman. Being a man means I am a person. Being a Woman means that I am person. Therefore I am a person.

The inference rule of division into cases represents a more sophisticated case analysis: From tex2html_wrap_inline570 , tex2html_wrap_inline572 , and tex2html_wrap_inline574 , infer r.

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Next: About this document Up: My Home Page

Steve Skiena
Tue Aug 24 14:51:35 EDT 1999