# 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." – Conifold Oct 16 '17 at 20:50

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 – Frank Hubeny Sep 25 '18 at 20:09

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.

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. – Logikal Sep 26 '18 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. – Frank Hubeny Sep 26 '18 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. – Logikal Sep 26 '18 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. – Frank Hubeny Sep 26 '18 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. – Logikal Sep 26 '18 at 17:30