# What can we infer about an argument's validity given the following information?

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.)

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?

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

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

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.

• Rules: 1) a tautology follows from every premise. 2) from a contradiction everything follows. Commented Feb 15, 2023 at 13:58
• 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. Commented Feb 15, 2023 at 14:03
• 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. Commented Feb 15, 2023 at 14:05
• @MauroALLEGRANZA "only one case: when P is never True, i.e. when it is a contradiction." But how about two or more premises? Commented Feb 15, 2023 at 14:17
• 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 $\Gamma$. Commented Feb 15, 2023 at 15:02

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.

• Appreciate it. The wordinig is quite readable and understood easily. Commented Feb 16, 2023 at 1:34
• The difficulty I face is that although the core of the concept is the same, but different people or article use different wording or terminology to compose sentences to convey the concept. sigh Commented Feb 16, 2023 at 3:05
• @StatsCruncher Indeed. Commented Feb 16, 2023 at 3:48
• You have open mindset to accept different users. Even some users like me can't clarify questions clearly at first. But users like you remind me of Federico Ardila-Mantilla's axioms. So I mentions Diversity and I guess that you accept different users who study mathematics even they face various learning difficulties. Commented Feb 19, 2023 at 21:06
• @StatsCruncher Thanks for your kind words. -) Commented Feb 20, 2023 at 3:45