Logical truth
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| - | '''Future contingent''' propositions (or simply, '''future contingents''') are statements about states of affairs in the future that are neither [[logical truth|necessarily true]] nor [[Contradiction|necessarily false]]. | + | '''Logical truth''' is one of the most fundamental [[concept]]s in [[logic]], and there are different theories on its nature. A logical truth is a [[Statement (logic)|statement]] which is true and remains true under all [[interpretation (logic)|reinterpretations]] of its components other than its [[logical constant]]s. It is a type of [[Analytic–synthetic distinction|analytic statement]]. All of [[philosophical logic]] can be thought of as providing accounts of the nature of logical truth, as well as [[logical consequence]]. |
| - | The '''problem of future contingents''' seems to have been first discussed by [[Aristotle]] in chapter 9 of his ''[[On Interpretation]]'' (''[[De Interpretatione]]''), using the famous sea-battle example. Roughly a generation later, [[Diodorus Cronus]] from the [[Megarian school of philosophy]] stated a version of the problem in his notorious Master Argument. The problem was later discussed by [[Gottfried Wilhelm Leibniz|Leibniz]]. [[Deleuze]] used it to oppose a "logic of the event" to a "logic of [[signification]]". | + | Logical truths (including [[Tautology (logic)|tautologies]]) are truths which are considered to be '''necessarily true'''. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. However, it is not universally agreed that there are any statements which are ''necessarily'' true. |
| - | The problem can be expressed as follows. Suppose that a sea-battle will not be fought tomorrow. Then it was also true yesterday (and the week before, and last year) that it will not be fought, since any true statement about what will be the case was also true in the past. But all past truths are necessary truths, therefore it was necessarily true in the past that the battle will not be fought, and thus that the statement that it will be fought is necessarily false. Therefore it is not possible that the battle will be fought. In general, if something will not be the case, it is not possible for it to be the case. This conflicts with the idea of our own [[free will]]: that we have the power to determine the course of events in the future, which seems impossible if what happens, or does not happen, was ''necessarily'' going to happen, or not happen. | + | A logical truth was considered by [[Ludwig Wittgenstein]] to be a statement which is true in all [[possible world]]s. This is contrasted with [[fact]]s (which may also be referred to as ''contingent claims'' or ''synthetic claims'') which are true in ''this'' world, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded. The [[proposition]] "If p and q, then p" and the proposition "All married people are married" are logical truths because they are true due to their inherent meanings and not because of any facts of the world. |
| - | + | Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. | |
| - | == Aristotle's solution == | + | |
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| - | Aristotle solved the problem by asserting that the [[principle of bivalence]] found its [[wikt:exception|exception]] in this paradox of the sea battles: in this specific case, what is impossible is that both alternatives can be possible at the same time: either there ''will'' be a battle, or there won't. Both options can't be simultaneously taken. Today, they are neither true nor false; but if one is true, then the other becomes false. According to Aristotle, it is impossible to say today if the proposition is correct: we must wait for the contingent realization (or not) of the battle, logic realizes itself afterwards: | + | |
| - | :''One of the two propositions in such instances must be true and the other false, but we cannot say determinately that this or that is false, but must leave the alternative undecided. One may indeed be more likely to be true than the other, but it cannot be either actually true or actually false. It is therefore plain that it is not necessary that of an affirmation and a denial, one should be true and the other false. For in the case of that which exists potentially, but not actually, the rule which applies to that which exists actually does not hold good.'' ([http://etext.library.adelaide.edu.au/a/aristotle/interpretation/ §9]) | + | |
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| - | For Diodorus, the future battle was either [[wikt:impossible|impossible]] or [[wikt:necessary|necessary]]. Aristotle added a third term, [[contingency]], which saves logic while in the same time leaving place for indetermination in reality. What is necessary is not that there will or that there won't be a battle tomorrow, but the dichotomy itself is necessary: | + | |
| - | :''A sea-fight must either take place to-morrow or not, but it is not necessary that it should take place to-morrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place to-morrow. ([[De Interpretatione]]'', [http://etext.library.adelaide.edu.au/a/aristotle/interpretation/ 9, 19 a 30.]) | + | |
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| - | Thus, the event always comes in the form of the future, undetermined event; logic always comes afterwards. [[Hegel]] would say the same thing by claiming that wisdom came at dusk. For Aristotle, this is also a practical, [[ethic]]al question: to pretend that the future is determined would have unacceptable consequences for man. | + | |
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| - | == Leibniz == | + | |
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| - | [[Gottfried Wilhelm Leibniz|Leibniz]] gave another response to the paradox in [http://www.anselm.edu/homepage/dbanach/Leibniz-Discourse.htm#VI §6] of ''[[Discourse on Metaphysics]]'': "That God does nothing which is not orderly, and that it is not even possible to conceive of events which are not regular." Thus, even a [[miracle]], the Event by excellence, does not break the regular order of things. What is seen as irregular is only a default of perspective, but does not appear so in relation to universal order. Possible exceeds human logics. Leibniz encounters this paradox because according to him: | + | |
| - | :''Thus the quality of king, which belonged to Alexander the Great, an abstraction from the subject, is not sufficiently determined to constitute an individual, and does not contain the other qualities of the same subject, nor everything which the idea of this prince includes. God, however, seeing the individual concept, or [[haecceity]], of Alexander, sees there at the same time the basis and the reason of all the predicates which can be truly uttered regarding him; for instance that he will conquer Darius and Porus, even to the point of knowing a priori (and not by experience) whether he died a natural death or by poison,- facts which we can learn only through history. When we carefully consider the connection of things we see also the possibility of saying that there was always in the soul of Alexander marks of all that had happened to him and evidences of all that would happen to him and traces even of everything which occurs in the universe, although God alone could recognize them all.'' ([http://www.anselm.edu/homepage/dbanach/Leibniz-Discourse.htm#VIII §8]) | + | |
| - | If everything which happens to Alexander derives from the haecceity of Alexander, then [[fatalism]] threatens Leibniz's construction: | + | |
| - | :''We have said that the concept of an individual substance includes once for all everything which can ever happen to it and that in considering this concept one will be able to see everything which can truly be said concerning the individual, just as we are able to see in the nature of a circle all the properties which can be derived from it. But does it not seem that in this way the difference between contingent and necessary truths will be destroyed, that there will be no place for human liberty, and that an absolute fatality will rule as well over all our actions as over all the rest of the events of the world? To this I reply that a distinction must be made between that which is certain and that which is necessary.'' ([http://www.anselm.edu/homepage/dbanach/Leibniz-Discourse.htm#VI §13]) | + | |
| - | Against Aristotle's separation between the [[subject (grammar)|subject]] and the [[Predicate (grammar)|predicate]], Leibniz states: | + | |
| - | :"Thus the content of the subject must always include that of the predicate in such a way that if one understands perfectly the concept of the subject, he will know that the predicate appertains to it also." ([http://www.anselm.edu/homepage/dbanach/Leibniz-Discourse.htm#VIII §8]) | + | |
| - | The predicate (what happens to Alexander) must be completely included in the subject (Alexander) "if one understands perfectly the concept of the subject". Leibniz henceforth distinguish two types of necessity: necessary necessity and contingent necessity, or universal necessity vs singular necessity. Universal necessity concerns universal truths, while singular necessity concerns something necessary which could not be (it is thus a "contingent necessity"). Leibniz hereby uses the concept of [[compossible]] worlds. According to Leibniz, contingent acts such as "Caesar crossing the Rubicon" or "Adam eating the apple" are necessary: that is, they are singular necessities, contingents and accidentals, but which concerns the [[principle of sufficient reason]]. Furthermore, this leads Leibniz to conceive of the [[subject (philosophy)|subject]] not as a universal, but as a singular: it is true that "Caesar crosses the Rubicon", but it is true only of ''this'' Caesar at this ''time'', not of any dictator nor of Caesar at any time (§8, 9, 13). Thus Leibniz conceives of [[Substance theory|substance]] as plural: there is a plurality of singular substances, which he calls [[Monad (philosophy)|monads]]. Leibniz hence creates a [[concept]] of the [[individual]] as such, and attributes to it events. There is a universal necessity, which is universally applicable, and a singular necessity, which applies to each singular substance, or event. There is one proper noun for each singular event: Leibniz creates a logic of singularity, which Aristotle thought impossible (he considered that there could only be knowledge of generality). | + | |
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| - | == 20th century == | + | |
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| - | One of the early motivations for the study of [[many-valued logic]]s has been precisely this issue. In the early 20th century, the Polish formal logician [[Jan Łukasiewicz]] proposed three truth-values: the true, the false and the ''as-yet-undetermined''. This approach was later developed by [[Arend Heyting]] and [[L. E. J. Brouwer]]; see [[Łukasiewicz logic]]. | + | |
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| - | Issues such as this have also been addressed in various [[temporal logic]]s, where one can assert that "''Eventually'', either there will be a sea battle tomorrow, or there won't be." (Which is true if "tomorrow" eventually occurs.) | + | |
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| - | '''The Modal Fallacy''' | + | |
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| - | The error in the argument underlying the alleged "Problem of Future Contingents" lies in the assumption that “X is the case” implies that “necessarily, X is the case”. In logic, this is known as the Modal Fallacy. | + | |
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| - | By asserting “''A sea-fight must either take place to-morrow or not, but it is not necessary that it should take place to-morrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place to-morrow.''” Aristotle is simply claiming “necessarily (a or not-a)”, which is correct. | + | |
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| - | However, the next step in Aristotle’s reasoning seems to be: “If a is the case, then necessarily, a is the case”, which is a logical fallacy. | + | |
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| - | Expressed in another way: | + | |
| - | (i) If a proposition is true, then it cannot be false. | + | |
| - | (ii) If a proposition cannot be false, then it is necessarily true. | + | |
| - | (iii) Therefore if a proposition is true, it is necessarily true. | + | |
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| - | That is, there are no contingent propositions. Every proposition is either necessarily true or necessarily false. | + | |
| - | The fallacy arises in the ambiguity of the first premise. If we interpret it close to the English, we get: | + | |
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| - | (iv) P [[logical consequence|entails]] it is not possible that not-P | + | |
| - | (v) It is not possible that not-P entails it is necessary that P | + | |
| - | (vi) Therefore, P entails it is necessary that P | + | |
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| - | However, if we recognize that the original English expression (i) is potentially misleading, that it assigns a necessity to what is simply nothing more than a necessary condition, then we get instead as our premises: | + | |
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| - | (vii) It is not possible that (P and not P) | + | |
| - | (viii) (It is not possible that not P) entails (it is necessary that P) | + | |
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| - | From these latter two premises, one '''cannot''' validly infer the conclusion: | + | |
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| - | (ix) P entails it is necessary that P | + | |
| + | The existence of logical truths is sometimes put forward as an objection to [[empiricism]] because it is impossible to account for our [[knowledge]] of logical truths on empiricist grounds. | ||
| == See also == | == See also == | ||
| - | *In [[Jorge Luis Borges|Borges]]' ''[[The Garden of Forking Paths]]'', both alternatives happen, thus leading to what Deleuze calls "incompossible worlds" | + | *[[Validity]] |
| - | *[[Principle of distributivity]] | + | *[[Theorem]] |
| - | *[[Principle of plenitude]] | + | *[[Satisfiability]] |
| - | *[[Truth-value link]] | + | |
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Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.
Logical truths (including tautologies) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. However, it is not universally agreed that there are any statements which are necessarily true.
A logical truth was considered by Ludwig Wittgenstein to be a statement which is true in all possible worlds. This is contrasted with facts (which may also be referred to as contingent claims or synthetic claims) which are true in this world, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded. The proposition "If p and q, then p" and the proposition "All married people are married" are logical truths because they are true due to their inherent meanings and not because of any facts of the world. Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations.
The existence of logical truths is sometimes put forward as an objection to empiricism because it is impossible to account for our knowledge of logical truths on empiricist grounds.
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