# What is the difference between a logical truth and a tautology?

Some papers I read seem to be referring to a distinction between logical truths and tautologies.

At first I thought something was wrong since I thought they are the same by definition. I checked the Wikipedia article, "Logical truth", and I noticed that it, too, refers to a distinction between them when it says, "Logical truths (including tautologies)", but I still do not understand what the difference is.

Are both of them not simply statements that are necessarily true?

• Truth values and tautologies section of that Wikipedia article explains the difference:"In one sense of the term "tautology", it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms (may also be called a valuation or assignment depending upon the context). This is synonymous to logical truth... However, the term "tautology" is also commonly used to refer to what could more specifically be called truth-functional tautologies." Oct 16, 2017 at 20:50
• A logical truth is a unique logical statement (independently of it being the result of many others): the pencil is blue. A tautology is a logical statement that involves TWO or more parts with identical logical value: the blue pencil is blue. Since the parts of a tautology have identical logical value, the whole will always have the same value of (logical) truth as each part. Dec 28, 2021 at 11:17

1. There are usages in which 'tautology' and 'logical truth' are interchangeable.

2. However, from the literature (Quine, Tarski et al.) I'd venture that 'logical truth' is a more theoretical concept, open to highly contestable interpretations. For instance, it would be commonly agreed (I think !) that 'not (p & not-p)' is a tautology, true by virtue of its logical form and regardless of its non-logical components. However, if the following gives the core of Tarksi's view of logical truth :

For every language L and all sentences S of L, S is (logic ally true)t if and only if S is true in all the (set-theoretic) interpretations of L (Mario Gómez-Torrente, 'Logical Truth and Tarskian Logical Truth', Synthese, Vol. 117, No. 3 (1998/1999), pp. 375-408 : 376)

then this Tarskian definition, which differs from Quine's in The Ways of Paradox, is a theory of logical truth in a way - or rather, to a degree - that the definition of 'tautology' isn't. The same would be true if we adopted Quine's definition ('The logical truths . . . are those true sentences which involve only logical words essentially', Ways of Paradox, 103).

1. My objection to 'Logical truths (including tautologies) are truths which are considered to be necessarily true' is that I can see no way of interpreting 'necessarily' here other than as 'logically'. What else in context can 'necessarily' mean ? We are no further forward.
• Quine (Mathematical Logic, p 50) writes, "All tautologous statements are logically true, but not all logically true statements are tautologous. The method of showing a statement to be tautologous consists merely of constructing a table under it in the usual way.and observing that the column under the main connective is composed entirely of 'T's." +1 Sep 25, 2018 at 20:09

As Conifold pointed out in a comment quoting the Wikipedia article cited by the OP, the difference between logical truth and tautology may be described as saying a logical truth "turns out to be true under any possible interpretation of its terms" while a tautology may be limited to "truth-function tautologies".

That Wikipedia article states the following:

Logical truths (including tautologies) are truths which are considered to be necessarily true.

Understanding that statement requires understanding three concepts: necessary truth, tautology and logical truth. The authors of forall x: Calgary Remix keep these three concepts separate which may help to clarify them.

On page 13, the authors illustrate necessary truth with this English language sentence:

Either it is raining here, or it is not.

And observe

You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or it is not. That is a NECESSARY TRUTH.

Tautologies are defined for truth-function logic (TFL) or propositional logic. They write (page 70)

...we explained necessary truth and necessary falsity. Both notions have surrogates in TFL. We will start with a surrogate for necessary truth.

A is a TAUTOLOGY if it is true on every valuation.

We can determine whether a sentence is a tautology just by using truth tables. If the sentence is true on every line of a complete truth table, then it is true on every valuation, so it is a tautology.

One can think of a tautology as a truth dependent only on logical connectives such as "and", "or" and "not" between sentences.

A logical truth is a similar situation but in first order logic (FOL) rather than truth-functional logic where one has in addition to the logical connectives of truth-functional logic quantifiers and equality. Rather than valuations found in a truth table one now refers to interpretations: (page 225)

An FOL sentence A is a LOGICAL TRUTH if A is true in every interpretation.

In summary: The concept "necessary truth" refers to a natural language sentence such as found in English which is always true. A "tautology" refers to a sentence of truth-functional logic where every valuation, every row of a complete truth table, evaluates to true. A "logical truth" is a sentence in first order logic where every interpretation is true.

Reference

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

Wikipedia, "Logical truth" https://en.wikipedia.org/wiki/Logical_truth

• Does your answer cover semantic tautology? I.e., all circles are round shaped; all women are human beings; all dogs are animals, etc. You seem to cover only Mathematical logic. Sep 26, 2018 at 13:19
• @Logikal I am using the definitions provided in forall x. For that text a tautology applies only to propositional calculus, or as they call it truth-functional logic because every sentence can be valued as "true" or "false". They do call this a "semantic" approach rather than the "syntactic" approach that a derivation would use. This would also be like Quine's description in Mathematical Logic page 50. So it is mainly mathematical logic. What I think you are referring to is what they would call "necessarily true" and these apply to some natural language sentences. Sep 26, 2018 at 13:42
• This will lead to confusion because this is not the context of the OP. He does not indicate a Mathematical logic only context. You volunteered to use only Mathematical logic without stating the fact until now. Sep 26, 2018 at 16:34
• @Logikal The semantic concept of tautology in propositional calculus based on true-false valuation in truth tables and the concept of logical truth based generally on syntactic concepts of derivations are both formal logical concepts and can be related to mathematical logic. According to forall x the concept of necessary truth comes from natural language sentences. I also presented that term with the forall x definition. My answer does not just cover mathematical logic. Sep 26, 2018 at 17:14
• Prepositional calculus is Mathematical logic. You seem to think otherwise. For instance Aristotelian logic does not define tautology in a Mathematical way. A tautology is a necessary truth. In this way all bachelors are married is a tautology as well as all women are human beings is a tautology. What you consider natural sentences is not the same a live English sentences. Context plays a role in a natural language and I do t see how Mathematical logic handles the same terms with a different context. Perhaps you can show this. Sep 26, 2018 at 17:30

I was quite confused when I read in Carnap's "Introduction to Symbolic Logic and its Applications" (in chapter A section 5) all tautologies are L-true but many L-true sentences are not tautologies. He referenced chapter A section 14, but does not there explicitly say "here is an L-true sentence, yet it is not a tautology". This discussion has clarified that this is not a profound, subtle logical distinction. If tautology is defined to be a necessary truth which can be established by a finite number of rows in a truth table then L-true statements with a universal quantifier and infinite number of individuals is L-true but not a tautology. If tautology just means true for all (not necessarily finite) assignments to value bearing signs, then they appear to be the same thing. Nothing profound here. When Carnap wrote, examples were not appreciated.

I am following along the course "Language, Proof, and Logic" from Stanford on EdX.

I was confused by the question of the distinction between tautology and logical truth (aka logical necessity).

One particular explanation from a lecture, and then a lecture from the textbook cleared up a lot for.

I'm going to try to give some intuitive ideas here, that may be vague in certain respects (because I am not an expert in logic), but which helped me to understand the distinctions of the terms we are discussing.

logically possible claim: a claim is logically possible if it could be (or could have been) true, at least on logical grounds. By "logical grounds" is meant "that satisfies basic logical axioms."

For example, it is logically possible for a space ship to travel faster than the speed of light. It is not physically possible, but there is no logical axiom that is broken by something traveling faster than the speed of light.

An object not being identical to itself is not, however, logically possible. This would violate the meaning of identity, which is part of some basic axiom of logic.

Another way to think about a claim being logically possible is: a claim is logically possible if there is some logically possible circumstance (or situation or world) in which the claim is true.

Next we have the terms logical truth and logical necessity. As far as I understand these are the same thing. A claim is a logical truth aka logical necessity if it is true in every logically possible circumstance or situation; it is a sentence which cannot be false.

Consider the (atomic) sentence a=a. If we build a truth table for this atomic sentence, it has two rows, one for T and one for T.

By the definition of tautology, this is not a tautology: the sentence is not true for every possible truth value of its constituents (and note that there is only one constituent, the sentence itself).

However, it is not logically possible for a=a. Therefore, despite not being a tautology, a=a is a logical truth, aka a logical necessity.

Finally, let me show the diagram. The diagram includes some information which is specific to the course, in particular this thing called "Tarski's World". You can ignore it for the purposes of this discussion. You also don't have to know what the predicate Tet() is. The important thing is that any claim of the form P & ~P is a tautology.

Seems like a tautology is a claim that is true for whatever configuration of the atomic sentences it is composed of, and this means that the actual structure of the claim makes it so.

However, there are structures of claims that when you build out the truth table you see that the claim isn't true in all configurations. However, there is some other stronger constraint on what is possible than just the structure of the connectives of the claim. Such a stronger constraint is, for example, what I am conjecturing to be fundamental axioms/laws of logic.

This is what is exemplified by a=a or, as in the diagram above, also something like ~(Larger(a,b) & Larger(b,a)). If we were to use some other predicate instead of Larger(), we might get something which weren't a logical necessity. For example ~(Likes(a,b) & Likes(b,a)). It is totally conceivable that a likes b and b likes a, making the whole claim false. But it is not possible for b to be larger than a, and a to be larger than b. This particular predicate is admittedly very vague, and maybe there is some esoteric field which allows two objects to each be bigger than the other. If a and b represent physical objects, I believe it would probably be justified to say that if Larger means having more mass, then the claim in question, ~(Larger(a,b) & Larger(b,a)), is a logical truth.

In the Aristotelian tradition, a logical truth is an implication which is self-evidently true, such as for example the modus ponens, transposition, the hypothetical syllogism, or even more basic implications such as A ∧ B ⊢ B and A ⊢ A ∨ B, and indeed A ⊢ A.

Thus, logical truths are not identified through formal proofs but through our innate logical intuition, which is still our only means of justifying any logical deduction.

Tautologies is something else entirely. A tautology is defined as any logical expression (not necessarily an implication), which is true in all logical cases. For example, A ∨ ¬A is trivially true in all cases, so it is a tautology.

It became necessary to introduce the notion of tautology in mathematical logic because of its use of the material implication as proxy for the logical implication.

Intuitively a logical truth and a tautology are more or less the same thing. However, in modern logic they have become distinguished:

A logic is a language with a set of axioms and a deductive system. A logical truth is a sentence that can be deduced via the deductive system starting out from the axioms.

A model of a logic is a world where we can interpret the logic. Sometimes this is called semantics as opposed to the syntax of the logic. Then a tautology is defined to a sentence that is true in every possible model.

Sometimes tautologies or logical truths are dismissed as being empty of content as they are merely deductively true. This is of course correct - there is no empirical content in logic - but also it is not the whole story. Mario Bunge, a philosopher of science, dismissed Wittgensteins philosophy of mathematics because he reduced mathematics to logic. For example, he said:

Wittgenstein wrote in his Tractatus:

6.234 Mathematics is a method of logic

And

6.22 The logic of the world, which is shown in tautologies, by the propositions of logic, is shown in mathematics by equations

Of course, Wittgenstein was writing in the early days of model theory. And what he was writing about was not mathematics per se but how it could be modelled as a formal language. There, mathematical truths - theorems, in a word - are tautologies. Now, the content of a specific theory is which tautologies, that is theorems, are chosen to investigate. Not all tautologies, that is theorems, are created alike. But in model theory, we think of all tautologies are alike as we are interested in questions of the logic as a whole. For example, whether it is sound, complete or decidable.