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  1. Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument?

My answer: Valid, because this argument is impossible to have true premises and false conclusion.

  1. What if the conclusion is a contradiction?

My answer: Invalid when an argument is possible to have true premises and false conclusion; otherwise, Valid.

  1. What if one of the premises is either a tautology or a contradiction?

My answer: Invalid when an argument is possible to have true premises and false conclusion; otherwise, Valid. (It is the form that makes arguments valid. Plus we could asume a premise were true even this premise is a contradiction.)

ADDENDUM 1 to add my new answers after reading the accepted answer

  1. Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument?

My new answer: Valid. Because $\forall$x C(x), it is impossible to have $\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ].

  1. What if the conclusion is a contradiction?

My new answer:It depends on the premises.

Valid when $\forall$x [$\forall$yP(x,y) $\to$ C(x)]

Invalid when $\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ]

  1. What if one of the premises is either a tautology or a contradiction?

My new answer:

Valid when $\forall$x [$\forall$yP(x,y) $\to$ C(x)]

Invalid when $\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ]

ADDENDUM 2

Valid

$\forall$x [$\forall$yP(x,y) $\to$ C(x)]

Invalid

meaning: Negate the above formula

$\neg$ { $\forall$x [$\forall$yP(x,y) $\to$ C(x)] }

$\iff$

$\exists$x[ $\forall$y P(x,y) $\land$ $\neg$ C(x) ]

x $\in$ {1,...,$\mathrm{2}^{n}$}

y $\in$ {1,...,n}

P(x, y) A premise in (x,y) enty of the truth table.

C(x) The conclusion in the row x of the truth table.

ryang
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Stats Cruncher
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  • Do you mean Example3.e.? – Stats Cruncher Feb 15 '23 at 13:53
  • Do you mean Example1? – Stats Cruncher Feb 15 '23 at 13:56
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    Rules: 1) a tautology follows from every premise. 2) from a contradiction everything follows. – Mauro ALLEGRANZA Feb 15 '23 at 13:58
  • @MauroALLEGRANZA I don't follow that. I read similar aritcles you write about. – Stats Cruncher Feb 15 '23 at 13:59
  • Also, why provide several examples one one suffices your post is unnecessarily long… – Vivaan Daga Feb 15 '23 at 14:00
  • I am confused by this exercise and I spend a long day surfing the internet. Try to find some answer which is easily understood. – Stats Cruncher Feb 15 '23 at 14:02
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    Rule 1) a tautology is always True; thus if we have a tautology $\top T$ as conclusion, whatever proposition $P$ we use as premise we have that $P ∴ \top$ is valid because in every case where $P$ is True also $\top$ is. – Mauro ALLEGRANZA Feb 15 '23 at 14:03
  • @MauroALLEGRANZA Do you mean the cases when premises are tautologies or contradictions? – Stats Cruncher Feb 15 '23 at 14:03
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    Rule 2) a contradiction is always False; thus if we have a contradiction $\bot$ as conclusion, whatever proposition $P$ we use as premise we have that $P∴\bot$ can be valid in only one case: when $P$ is never True, i.e. when it is a contradiction. In this case, when the premises is a contradiction $\bot$, we have that $\bot ∴ Q$ holds for a proposition $Q$ whatever. – Mauro ALLEGRANZA Feb 15 '23 at 14:05
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    @MauroALLEGRANZA "only one case: when P is never True, i.e. when it is a contradiction." But how about two or more premises? – Stats Cruncher Feb 15 '23 at 14:17
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    In this case we consider a set $\Gamma$ of premises; we have that $\Gamma ∴ \bot$ holds only when $\Gamma$ is unsatisfiable, i.e. it includes a contradiction. Simple example with ${ P, \lnot P }$ as $\Gamma$. – Mauro ALLEGRANZA Feb 15 '23 at 15:02

1 Answers1

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  1. Suppose the conclusion of an argument is a tautology. What can you conclude about the validity of the argument?

Every conditional with a tautological consequent is a logical validity.

In other words, every argument with a tautological conclusion is valid.

  1. What if the conclusion is a contradiction?

A conditional with a contradiction as consequent is a logical validity precisely when its antecedent is unsatisfiable.

In other words, an argument with a contradiction as conclusion is valid precisely when its premises are inconsistent.

3a. What if one of the premises is a tautology?

A conjunction's truth value is not altered by eliminating one of its tautological conjuncts; therefore, a tautological premise confers no information about the argument's validity.

3b. What if one of the premises is a contradiction?

Every conjunction with a contradiction as conjunct is a contradiction; therefore, every argument with a contradiction as one of its premises is valid.

ryang
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