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a) $1+1=2$ is tautology?

A formula is said to be a Tautology if every truth assignment to its component statements results in the formula being true

$1+1=2$ can be represented by single propositional variable (e.g. $P$) and $P$ can be true or false. But actually $1+1=2$ is true, so I'm confused if it is tautology or not

b) If premises are contradiction and conclusion is contradiction, is this argument valid?

An argument is valid if and only if when all premises are true conclusion must be true(cannot be false) But in this case, premises cannot be true. How can I determine this argument is valid or invalid?

c) If premises are contradiction and conclusion is tautology, is this argument valid?

d) If premise is contradiction and conclusion is $1+1=2$, is this argument valid?

e) If premise is $1+1=3$ and conclusion is tautology, is this argument valid?

f) If premise is $1+1=3$ and conclusion is contradiction, is this argument valid?

g) If premise is $1+1=3$ and conclusion is $2+2=5$, is this argument valid?

(edit)

h) If premise is $1+1=3$ and conclusion is $2+2=4$, is this argument valid?

firia2000
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    First define the symbols $1$, $2$ and the binary operation $+$ in your language $L$, then you can decide whether $1+1=2$ is a tautology in $L$. – user10354138 Jun 02 '19 at 11:21
  • @user10354138 I don't know formal language at all. For example, how can I define symbol 1, 2 and + so that $1+1=2$ to be tautology ? – firia2000 Jun 02 '19 at 13:11
  • If you define $2$ to be $1+1$, then of course $1+1=2$ is a tautology. If you define it in other ways, then well, it depends on what exactly you do. – user10354138 Jun 02 '19 at 13:25

1 Answers1

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a) $1+1=2$ is not a tautology

A tautology in propositional logic is a formula that is true in every truth assignment.

In a broad sense, we can call "tautology" also a formula of predicate logic that is valid, i.e. true in every interpretation, like e.g. $\forall x (x=x)$.

b) (and c) and d)) An argument is valid when in every interpretation where all the premises are true, also the conclusion is true.

Alternatively, we have that an argument is valid if there is no interpretation where all the premises are tue and the conclusion is false.

Thus, for an argument with contradictory premises we have that there is no interpretation where the premises are all true, and a fortiori there is no interpretation where all the premises are tue and the conclusion is false.

Conclusion : an argument with contradictory premises is valid.

Mauro ALLEGRANZA
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