March 23, 2018

Tautologies and Contradictions

Now we care about the truth of the phrase independent of any particular \(x\)

If a predicate \(P(x)\) is TRUE for all \(x\), we say it is a TAUTOLOGY

“a statement that is true by necessity or by virtue of its logical form”

If a predicate \(P(x)\) is FALSE for all \(x\), we say it is a CONTRADICTION

Example of Tautology

NOT (P(x) AND Q(y)) EQUIVALENT (NOT P(x) OR NOT Q(y))

\[\forall x,\forall y,\quad \neg(P(x)\wedge Q(y)) \Leftrightarrow (\neg P(x) \vee \neg Q(y))\]

((NOT P(x)) OR Q(y)) EQUIVALENT (P(x) IMPLIES Q(y))

\[\forall x,\forall y,\quad (\neg P(x)\vee Q(x)) \Leftrightarrow ( P(x) \Rightarrow Q(x) )\]

To see if the predicate is a TAUTOLOGY we see what happens if we prepend the universal quantifier \(\forall\)

Understanding the complete phrase

Now we can understand better the meaning of “IMPLIES”

“If you are at Istanbul then you are in Turkey”

  • For all things \(x\), IF \(AtIstanbul(x)\) THEN \(InTurkey(x)\)
  • \(\forall x, AtIstanbul(x)\Rightarrow InTurkey(x)\)

This phrase is a TAUTOLOGY

That means that the argument is correct, in the logic sense

Valid arguments

An argument is a phrase saying that IF several predicates (called premises) are true, THEN another predicate (called conclusion) must also be true

All the premises are connected by AND

\[\forall x, (P(x)\text{ AND }Q(x))\Rightarrow R(x)\]

The argument is valid if it is a tautology

If the argument is not correct, we say it is a fallacy

Example

  • Elizabeth owns either a Honda or a Toyota.
  • Elizabeth does not own a Honda.
  • Therefore, Elizabeth owns a Toyota.

\[(Honda(e)\vee Toyota(e)) \wedge \neg Honda(e)) \Rightarrow Toyota(e)\]

\[\forall x,(P(x)\vee Q(x)) \wedge \neg P(x)) \Rightarrow Q(x)\]

Notice that the argument is valid even if Elizabeth as no car

Another example

  • All toasters are items made of gold.
  • All items made of gold are time-travel devices.
  • Therefore, all toasters are time-travel devices.

\[\begin{matrix}(\forall x, Toaster(x) \Rightarrow Gold(x)) \wedge (\forall x,Gold(x)\Rightarrow TimeMachine(x))\\ \Rightarrow (\forall x, Toaster(x) \Rightarrow TimeMachine(x))\end{matrix}\] \[(P(x)\vee Q(x)) \wedge \neg P(x)) \Rightarrow Q(x)\]

The first two propositions are not true. Nevertheless if they were true, the third proposition is necessarily true

Validity and Soundness

An argument may be valid even if the premises are never true

A argument is sound if and only if it is both valid, and all of its premises are actually true.

Reference: http://www.iep.utm.edu/val-snd/

Some important arguments

We saw that, for all x \[(P(x)\vee Q(x)) \wedge \neg P(x)) \Rightarrow Q(x)\] \[(P(x)\Rightarrow Q(x)) \wedge (Q(x)\Rightarrow R(x))\Rightarrow (P(x)\Rightarrow R(x))\] We also have “modus ponens” \[\forall x, (P(x)\Rightarrow Q(x)) \wedge P(x))\Rightarrow Q(x)\] and “modus tollens” \[\forall x, (P(x)\Rightarrow Q(x)) \wedge \neg Q(x))\Rightarrow \neg P(x)\]

Example of modus ponens

If being rich makes you happy and you are rich, then you are happy

IF you being rich IMPLIES you being happy AND you are rich THEN you are happy

IF \(Rich(x)\Rightarrow Happy(x)\) AND \(Rich(x)\), THEN \(Happy(x)\)

\(\forall x, (P(x)\Rightarrow Q(x)) \wedge P(x))\Rightarrow Q(x)\)

Example of modus tollens

If being rich makes you happy and you are unhappy, then you are not rich

IF you being rich IMPLIES you being happy AND you are not happy THEN you are not rich

IF \(Rich(x)\Rightarrow Happy(x)\) AND \(\neg Happy(x)\), THEN \(\neg Rich(x)\)

\(\forall x, (P(x)\Rightarrow Q(x)) \wedge \neg Q(x))\Rightarrow \neg P(x)\)

Instantiation

General rules apply to particular cases

  • All men are mortal
  • Socrates is a man
  • Therefore Socrates is mortal

For all things \(x\), IF \(x\) is Man, THEN \(x\) is Mortal, AND socrates is Man, THEN socrates is mortal

For all \(x\), (\(Man(x)\) IMPLIES \(Mortal(x)\)) AND \(Man(socrates)\), THEN \(Mortal(socrates)\)

\((\forall x, Man(x)\Rightarrow Mortal(x))\wedge Man(socrates) \Rightarrow Mortal(socrates)\)

Summary

IF someone is at Istanbul THEN that person is on Turkey

  • What can we deduce if we know that Ali is at Istanbul
  • What can we deduce if we know that Ali is not in Turkey