structure for 'Theory of Logic'    |     alphabetical list of themes    |     unexpand these ideas

5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth

[statements held to be true because of a logic system]

24 ideas
Basic truths of logic are not proved, but seen as true when they are understood [Frege, by Burge]
     Full Idea: In Frege's view axioms are basic truth, and basic truths do not need proof. Basic truths can be (justifiably) recognised as true by understanding their content.
     From: report of Gottlob Frege (works [1890]) by Tyler Burge - Frege on Knowing the Foundations 1
     A reaction: This is the underpinning of the rationalism in Frege's philosophy.
Logical truths are known by their extreme generality [Russell]
     Full Idea: A touchstone by which logical propositions may be distinguished from all others is that they result from a process of generalisation which has been carried to its utmost limits.
     From: Bertrand Russell (The Theory of Knowledge [1913], p.129), quoted by J. Alberto Coffa - The Semantic Tradition from Kant to Carnap 7 'What'
Logical truths are just 'by-products' of the introduction rules for logical constants [Wittgenstein, by Hacking]
     Full Idea: Wittgenstein's by-product theory is that the meanings of the logical constants are conveyed by their introduction rules, and these rules have as a by-product the class of logical truths.
     From: report of Ludwig Wittgenstein (Tractatus Logico-Philosophicus [1921]) by Ian Hacking - What is Logic? §03
     A reaction: I find this approach highly plausible. All the truths about chess openings are just a by-product of the original rules.
A sentence is logically true if all sentences with that grammatical structure are true [Quine]
     Full Idea: A sentence is logically true if all sentences with that grammatical structure are true.
     From: Willard Quine (Philosophy of Logic [1970], Ch.7)
     A reaction: Quine spends some time on the tricky question of deciding which parts of a sentence are grammatical structure ('syncategorematic'), and which parts are what he calls 'lexicon'. I bet there is a Quinean argument which blurs the boundary.
Having a valid form doesn't ensure truth, as it may be meaningless [Putnam]
     Full Idea: I don't think all substitution-instances of a valid schema are 'true'; some are clearly meaningless, such as 'If all boojums are snarks and all snarks are egglehumphs, then all boojums are egglehumphs'.
     From: Hilary Putnam (Philosophy of Logic [1971], Ch.3)
     A reaction: This seems like a very good challenge to Quine's claim that it is only form which produces a logical truth. Keep deductive and semantic consequence separate, with two different types of 'logical truth'.
Logical truths and inference are characterized either syntactically or semantically [Dummett]
     Full Idea: There are two ways of characterizing logical truths and correct inference. Proof-theoretic or syntactic characterizations, if the formalization admits of proof or derivation; and model-theoretic or semantic versions, being true in all interpretations.
     From: Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
     A reaction: Dummett calls this distinction 'fundamental'. The second one involves truth, and hence meaning, where the first one just responds to rules. ..But how can you have a notion of correctly following a rule, without a notion of truth?
Restrict 'logical truth' to formal logic, rather than including analytic and metaphysical truths [Shoemaker]
     Full Idea: I favour restricting the term 'logical truth' to what logicians would count as such, excluding both analytic truths like 'Bachelors are unmarried' and Kripkean necessities like 'Gold is an element'.
     From: Sydney Shoemaker (Causal and Metaphysical Necessity [1998], I)
     A reaction: I agree. There is a tendency to splash the phrases 'logical truth' and 'logical necessity around in vague ways. I take them to strictly arise out of the requirements of formal systems of logic.
A logical truth or tautology is a logical consequence of the empty set [Enderton]
     Full Idea: A is a logical truth (tautology) (|= A) iff it is a semantic consequence of the empty set of premises (φ |= A), that is, every interpretation makes A true.
     From: Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.3.4)
     A reaction: So the final column of every line of the truth table will be T.
A statement is logically true if it comes out true in all interpretations in all (non-empty) domains [Musgrave]
     Full Idea: The standard modern view of logical truth is that a statement is logically true if it comes out true in all interpretations in all (non-empty) domains.
     From: Alan Musgrave (Logicism Revisited [1977], §3)
Logical truths may contain non-logical notions, as in 'all men are men' [Musgrave]
     Full Idea: Containing only logical notions is not a necessary condition for being a logical truth, since a logical truth such as 'all men are men' may contain non-logical notions such as 'men'.
     From: Alan Musgrave (Logicism Revisited [1977], §3)
     A reaction: [He attributes this point to Russell] Maybe it is only a logical truth in its general form, as ∀x(x=x). Of course not all 'banks' are banks.
A logical truth is true in virtue of the nature of the logical concepts [Fine,K]
     Full Idea: One wants to define a logical truth as one that is true in virtue of the nature of the logical concepts.
     From: Kit Fine (Senses of Essence [1995], §3)
     A reaction: This is part of Fine's project to give a revised account of essence, which includes the essence of concepts as well as the essence of objects. Everyone should pay close attention to this project.
Logic holding between indefinite sentences is the core of all language [Fine,K]
     Full Idea: If language is like a tree, then penumbral connection (logic holding among indefinite sentences) is the seed from which the tree grows, for it provides an initial repository of truths that are to be retained throughout all growth.
     From: Kit Fine (Vagueness, Truth and Logic [1975], 2)
     A reaction: A nice incidental insight arising from his investigation of vagueness. People accept one another's reasons even when they are confused, or hopeless at expressing themselves. Nice.
'Tautologies' are valid formulas of classical sentential logic - or substitution instances in other logics [Burgess]
     Full Idea: The valid formulas of classical sentential logic are called 'tautologically valid', or simply 'tautologies'; with other logics 'tautologies' are formulas that are substitution instances of valid formulas of classical sentential logic.
     From: John P. Burgess (Philosophical Logic [2009], 1.5)
'Logically true' (|= φ) is true for every truth-assignment [Zalabardo]
     Full Idea: A propositional logic sentence is 'logically true', written |= φ, if it is true for every admissible truth-assignment.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §2.4)
Logically true sentences are true in all structures [Zalabardo]
     Full Idea: In first-order languages, logically true sentences are true in all structures.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §3.5)
The semantical notion of a logical truth is validity, being true in all interpretations [Sider]
     Full Idea: The semantical notion of a logical truth is that of a valid formula, which is true in all interpretations. In propositional logic they are 'tautologies'.
     From: Theodore Sider (Logic for Philosophy [2010], 2.3)
     A reaction: This implies that there is a proof-theoretic account of logical truth as well. Intuitively a logical truth is a sequent which holds no matter which subject matter it refers to, so the semantic view sounds OK.
It is hard to say which are the logical truths in modal logic, especially for iterated modal operators [Sider]
     Full Idea: It isn't clear which formulas of modal propositional logic are logical truths, ...especially for sentences that contain iterations of modal operators. Is □P→□□P a logical truth? It's hard to say.
     From: Theodore Sider (Logic for Philosophy [2010], 6.3)
     A reaction: The result, of course, is that there are numerous 'systems' for modal logic, so that you can choose the one that gives you the logical truths you want. His example is valid in S4 and S5, but not in the others.
A logical truth is the conclusion of a valid inference with no premisses [Read]
     Full Idea: Logical truth is a degenerate, or extreme, case of consequence. A logical truth is the conclusion of a valid inference with no premisses, or a proposition in the premisses of an argument which is unnecessary or may be suppressed.
     From: Stephen Read (Thinking About Logic [1995], Ch.2)
Modern logical truths are true under all interpretations of the non-logical words [Potter]
     Full Idea: In the modern definition, a 'logical truth' is true under every interpretation of the non-logical words it contains.
     From: Michael Potter (The Rise of Analytic Philosophy 1879-1930 [2020], 19 'Frege's')
     A reaction: What if the non-logical words are nonsense, or are used inconsistently ('good'), or ambiguously ('bank'), or vaguely ('bald'), or with unsure reference ('the greatest philosopher' becomes 'Bentham')? What qualifies as an 'interpretation'?
A 'logical truth' (or 'tautology', or 'theorem') follows from empty premises [Beall/Restall]
     Full Idea: If a conclusion follows from an empty collection of premises, it is true by logic alone, and is a 'logical truth' (sometimes a 'tautology'), or, in the proof-centred approach, 'theorems'.
     From: JC Beall / G Restall (Logical Consequence [2005], 4)
     A reaction: These truths are written as following from the empty set Φ. They are just implications derived from the axioms and the rules.
Logical truth is much more important if mathematics rests on it, as logicism claims [Beall/Restall]
     Full Idea: If mathematical truth reduces to logical truth then it is important what counts as logically true, …but if logicism is not a going concern, then the body of purely logical truths will be less interesting.
     From: JC Beall / G Restall (Logical Pluralism [2006], 2.2)
     A reaction: Logicism would only be one motivation for pursuing logical truths. Maybe my new 'Necessitism' will derive the Peano Axioms from broad necessary truths, rather than from logic.
Logically valid sentences are analytic truths which are just true because of their logical words [McGee]
     Full Idea: Logically valid sentences are a species of analytic sentence, being true not just in virtue of the meanings of their words, but true in virtue of the meanings of their logical words.
     From: Vann McGee (Logical Consequence [2014], 4)
     A reaction: A helpful link between logical truths and analytic truths, which had not struck me before.
Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley]
     Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1)
Logical truths are just the assumption-free by-products of logical rules [Rumfitt]
     Full Idea: Gentzen's way of formalising logic has accustomed people to the idea that logical truths are simply the by-products of logical rules, that arise when all the assumptions on which a conclusion rests have been discharged.
     From: Ian Rumfitt (The Boundary Stones of Thought [2015], 2.5)
     A reaction: This is the key belief of those who favour the natural deduction account of logic. If you really believe in separate logic truths, then you can use them as axioms.