Question

The Wikipedia article on thecorresponding conditionalcontains the following sentence: An argument is valid if and only if its corresponding conditional is alogical truth. Some sources use "tautology" in place of "logical truth": An argument is valid if and only if its corresponding conditional is atautology. This got me thinking about what "tautology" and "logical truth" actually mean, because a tautological corresponding conditional does not seem to be a tautology in the same way thatP v ~Pis a tautology.P v ~Pseems to be a tautology by virtue of the definition of "v". It will always be tautology, regardless of what sentencePrepresents. However, whether or not a corrsponding conditional is a tautology depends on the truth values of the premises and conclusion of the argument that the conditional represents.So, in what sense is a tautological corresponding conditional a tautology and does this differ to the sense in which P v ~P is a tautology? My confusion might be stemming from my very hazy understanding of the concepts of "logical truth", "tautology" and "necessary truth": "Tautology" seems to be a term of propositional logic which describes a sentence that is true on every possible valuation/truth-value assignment."Logical truth" seems to be a term of first-order logic, but when used within the context of propositional logic, it is synonymous with tautology (I'm not sure why, as I haven't studied FOL yet)."Necessary truth" seems to be something that is fundamentally true. All tautologies are necessary truths, but not all necessary truths are tautologies, e.g. the statement "1 = 1" is a necessary truth, but, in propositional logic, it can only be expressed using a single sentence letter, which cannot be a tautology on its own. I also came acrossthis page, which draws a distinction between1)tautologies which are true by virtue of the logical terms they contain (e.g. "every", "some" and "is") and are synonymous with logical truths, and2)truth-functionaltautologies, which are true by virtue of the connectives they contain (so, something likeP v ~P?). However, the paragraph is missing citations and I can't find any other sources that distinguish between tautologies/logical truths and truth-functional tautologies. We say in general that a formula isvalidwhen it is true in every interpretation. Atautologyis a valid formula of propositional logic [sense 2) above: "truth-functionaltautologies, which are trueonlyby virtue of the logical connectives"]. Thus,P ∨ ¬Pis a tautology (valid propositional formula), while∀x (x=x)is a valid formula of predicate logic. If we agree on this, the informal concept of "logical truth" is formalized withvalid formula[sense 1) above: "true by virtue of the logical terms they contain, e.g. thelogical connectives, 'every', 'some' and 'is' "]. We can extend the use of "valid" toarguments: a formal argument isvalidwhen there is no interpretation where the premises are true and the conclusion is false. In most logical systems we can prove the so calledDeduction (meta-)theoremthat links valid arguments with valid formulas. In this case, we have thatA ⊢ Bimplies⊢ A→Bthat means that if the argument deriving conclusionBfrom premiseAis valid, then the formulaA→Bis valid. A. Exploring Tautologies and Logical Truths B. Distinguishing Truth-Functional Tautologies C.Applying Deduction Theorem D.Comparing Logical Truths and Necessary Truths

Answer 1

A tautology in propositional logic is a statement that is true under all circumstances due to its structure, such as 'P or not P'. 'Logical truth' includes tautologies and can also encompass quantified statements in first-order logic. The validity of an argument is determined by whether, given true premises, the conclusion is necessarily true, which is a property of the argument's logical form, not of the actual truth values involved.

Step-by-step explanation:

Understanding Tautology and Logical Truth

Within the realm of logical analysis, a statement is considered a tautology if it is true in every possible circumstance due to its logical structure, which is composed solely of its logical connectives. The classic example of a tautology in propositional logic is P ∨ ¬P (‘P or not P’), which is true regardless of the truth value of P because either P is true, or it is not true, thus satisfying the condition of the disjunction. The concept of ‘tautology’ highlights statements that are true under all possible valuations in propositional logic, making them necessary truths within this system.

Meanwhile, the term logical truth is more broadly used within both propositional and first-order logics. In first-order logic, logical truths may include the logical connectives of propositional logic in addition to quantifiers like ‘for all’ (∀) and ‘there exists’ (∃), as seen in statements like ∀x (x=x), which asserts that everything is identical to itself. An argument's corresponding conditional being a tautology means that, following the argument's logical form, the conclusion necessarily follows from the premises—a concept formalized by the Deduction Theorem.

Validity in the context of deductive reasoning refers to the property of an argument such that if the premises are true, the conclusion is necessarily true. Thus, an argument with valid inference, like a disjunctive syllogism, guarantees the truth of the conclusion provided that the premises are true. This is distinct from factual truth, as validity pertains to the structure of the argument itself and not the actual truth value of the premises.