# Is there any logical operator that indicates tautology (in the form of truth table)? And, if so what could be it's possible significance be?

I got to this question because I was trying to see the inconsistencies that could arise if I would approach every informal logic in a formal way, that is when I start to break down what the difference between the two is and why fallacies usually arise within informal logic but it is usually not the case in formal language and one of the reasons is because of definitions or semantics involved with language in these informal premises. In formal logic, everything depends upon the agreed use of symbols and thus making it a pure form of logic where if done right is free of fallacies. From this understanding I thought what the existing operators we currently have are and I took a probabilistic method to see if there is every possible combination of premises and operators and this way I saw that one can create his own form of operators by taking a subset from the set of combinations of premises and logical connectors through this thought process I got to the question of having an operator that is called "tautology" where regardless of the premises i.e., either True or False the conclusion would always be True. In table form:

# Tautology(*)

P1 Tau P2 Conclusion
T * T T
T * F T
F * T T
F * F T

So now my question is, is there such a logical connector and so if there is what is its significance and if not why isn't there such a logical connector? Same for a logical connector that would guarantee contradiction?

For some reason I feel like I am missing a core understanding of logic, something basic that I didn't understand while going through school. I hope anyone could explain or give any suggestions for books or any sources I can read to understand why this isn't the case?

TL;DR: Based on the answers given I understand that there seems to be some sort of connector ⊨ (turnstile), but I guess my question is asking to see if there is a logical connector that connects two premises just like (AND, OR, IF and such) but instead of different values of truth values the conclusions, in this case, always result in all trues. I have never seen such a connection between two premises and I am asking why not?

• << Based on the answers given I understand that there seems to be some sort of connector ⊨ (turnstile) >> The "turnstile" is not a connector, and that is certainly not the answer I have given... Commented Mar 1 at 17:36
• @JulioDiEgidio At the moment you gave the link to the Wiki, I couldn't find the section where it mentions Wittgenstein table. I looked again and there it was, I really appreciate your feedback and it is just what I was looking for. Follow up: how come I have never seen tautology as an operator before, is it common for logicians to work with it? Commented Mar 1 at 17:48
• Tautology is not a binary relation between formulae as indicated in your sample truth table. If such a logical operator were to be introduced it would necessarily be a unary operator.
– nwr
Commented Mar 1 at 17:48
• @Howwhye I have rather linked to "Truth table", which comes just before "Wittgenstein table": tautology is the last operator on the right of that table, denoted by the symbol `T`, and it's just the operator that returns true identically, i.e. in all cases. Commented Mar 1 at 17:53
• @nwr But why would it be unary, why wouldn't it be a binary operator as shown in my case, also it seems like in the Wikipedia page provided below it seems to be that tautology is used as a binary operator in Wittgenstein's approach. Commented Mar 1 at 18:02

is there such a logical connector and so if there is what is its significance

In the context of classical propositional logic and the truth table method: all the 2^4, i.e. 16, possible binary operators exist, and the one you are asking about indeed is commonly called "tautology": see Truth table - Wikipedia for more details.

The symbol:

⊨φ

(called double turnstile), means that formula φ is a valid formula, that in case of propositional logic menas that it is a tautology.

The symbol (called False) is a propositional constant (i.e. a nullary connective), the truth value of which being always false: thus, it can stay for contradiction.

Its negation: , is always true.

Having said that, in order to "produce" a tautology from a formula whatever, you cal use the following trick: the truth table for the formula φ ∨ ⊤ evaluates always to True.

• So negation can be used like a logical connector within applications of logic? Also can you tell me real world scenarios where such connectors would be used. Commented Mar 1 at 13:50
• @Howwhye en.wikipedia.org/wiki/Negation
– TKoL
Commented Mar 1 at 13:54
• This answer seems confusing. The first paragraph is something that, at first glance, seems to be an answer to the question, and a bit of knowledge and thought are required in order to realize that it's actually the answer to a different question which sounds similar to the one that was asked. I think this answer would be much clearer if you moved the first paragraph to the bottom and explicitly pointed out the difference between the two questions. Commented Mar 1 at 22:55
• (For posterity, the question that the first paragraph answers is "Is there a symbol which forms an assertion that its operand is a tautology?" and the question that the rest of the post answers is "Is there a logical connective which forms a tautology regardless of what its operands are?") Commented Mar 1 at 22:57

You could do something like this:

(P⊤Q):=((P&~Q)v(P&Q)v(~P&Q)v(~P&~Q)).

This is ultimately just equivalent to the Law of the Excluded Middle/Tautology operator.

Still, what this specific version of the operator is doing is just enumerating the combinations of truth values for two propositions via disjunction, negation, and conjunction. It could be generalized to enumerate the truth value combinations for any finite number of propositional atoms. This at least seems different from taking the disjunction of two arbitrary propositions and a tautology, but in terms of truth-values/satisfaction, it is no different.

A Boolean formula F is a tautology if and only if its negation “non-F” is not satisfiable, i.e. if non-F is false independently of the truth value of each variable of F, see tautology.

There exists algorithms and their implementations, named SAT-solver, which negate F and check the formula “non-F” for satisfiability, see SAT solver.

But the satisfiability problem is NP-complete, a technical term, which implies that the algorithm in the worst case has exponential run-time, see NP-completeness. Nevertheless, even the SAT-solvers with free download from the internet have amazing small run-time for small numbers of variables.

Hence in general not an operator, but an algorithm is able to check whether a given logical formula with a finite number of variables is a tautology.