It is true from our individual standpoint, but it is not a truth in the objective sense. The truth, in an objective sense, is that we live on a planet which spins on its axis and it orbits the Sun. Based on our use of language in the majority of situations, an alien may then well judge us to be very ignorant, and that our truth is self-serving.
Everyone knows perfectly well what truth is — everyone except Pontius Pilate and philosophers. Truth is the quality of being true, and being true is what some statements are. That is to say, truth is a quality of the propositions which underlie correctly-used statements.
Truth and Truthfulness
What does that mean? Nonetheless, it is perfectly natural to say that a statement itself is true; people who think this would say that the above statement, as uttered by the man who thinks Gordon Brown is PM, is false even though what he meant by it is true.
However, to generalise, it is not really the statement itself that is true or false , but what is meant by it. I dilute my solution, place it into a cuvette, and take a reading with the spectrophotometer: 0. I repeat the procedure once more and get 0. From this I get the average of 0. The variation is probably based upon tiny inconsistencies in how I am handling the equipment, so three readings should be sufficient for my purposes.
Have I discovered the truth? Well yes — I have a measurement that seems roughly consistent, and should, assuming that my notes are complete and my spectrophotometer has been calibrated, be repeatable in many other labs around the world. The spectrophotometer is set at nm, which — so I have been taught — is the wavelength used to measure protein concentration. So my experiment has determined the truth of how much protein is in the cuvette. But again, a wider context is needed. What is a protein, how do spectrophotometers work, what is albumin, why do I want to know the concentration in the first place?
Observations are great, but really rather pointless without a reason to make them, and without the theoretical knowledge for how to interpret them. Truth, even in science, is therefore highly contextual. What truth is varies not so much with different people, but rather with the narrative they are living by.
In the end, even in an entirely materialistic world, truth is just the word we use to describe an observation that we think fits into our narrative. Truth is unique to the individual. What we consider to be true, whether in morality, science, or art, shifts with the prevailing intellectual wind, and is therefore determined by the social, cultural and technological norms of that specific era. In the end, humans are both fallible and unique, and any knowledge we discover, true or otherwise, is discovered by a human, finite, individual mind.
The closest we can get to objective truth is intersubjective truth, where we have reached a general consensus due to our similar educations and social conditioning. So our definition of truth needs to be much more flexible than Plato, Descartes and other philosophers claim. This is a theory Nietzsche came close to accepting. The lack of objective truth leaves us free to carve our own truths.
Truth is mine. My truth and your truth have no necessary relevance to each other. Because truth is subjective, it can play a much more unique and decisive role in giving life meaning; I am utterly free to choose my truths, and in doing so, I shape my own life. Without subjective truth, there can be no self-determination.
Truth is interpersonal. We tell each other things, and when they work out we call them truths. What we take as truth depends on what others around us espouse. Nobody bothered to count because everyone assumed it was true. And when they finally counted, it was because everyone agreed on the result that the real truth became known. Even when we are alone, truth is interpersonal. We express these truths or errors or lies to others and to ourselves in language; and, as Wittgenstein pointed out, there can be no private language.
But the most essential truth, the truth by which we all live our lives, is intensely personal, private. Even though each of us lives our life by Truth, it can be different for each person. Or none of the above: shall I find my own Truth in my own way? But even in such a community, some beliefs would be acceptable, and others not: my belief that I am exceptional and deserve preferential treatment, perhaps because I alone have received a special revelation, is not likely to be shared by others. From within the in-group we look with fear and revulsion on those who deny the accepted beliefs.
From outside, we admire those who hold aloft the light of truth amidst the darkness of human ignorance. And in every case it is we who judge, not I alone. Even the most personal Truth is adjudicated within a community and depends on the esteem of others. Such ambiguity facilitates equivocation — useful to politicians, etc, who can be economical with the truth.
One function of language is to conceal truth. In an experiment by Solomon Asch, subjects were given pairs of cards.
A variant of redundancy theory is the disquotational theory which uses a modified form of Tarski 's schema : To say that '"P" is true' is to say that P. A version of this theory was defended by C. Williams in his book What is Truth? Yet another version of deflationism is the prosentential theory of truth, first developed by Dorothy Grover, Joseph Camp, and Nuel Belnap as an elaboration of Ramsey's claims.
They argue that sentences like "That's true", when said in response to "It's raining", are prosentences , expressions that merely repeat the content of other expressions. In the same way that it means the same as my dog in the sentence My dog was hungry, so I fed it , That's true is supposed to mean the same as It's raining —if you say the latter and I then say the former. These variations do not necessarily follow Ramsey in asserting that truth is not a property, but rather can be understood to say that, for instance, the assertion "P" may well involve a substantial truth, and the theorists in this case are minimizing only the redundancy or prosentence involved in the statement such as "that's true.
Deflationary principles do not apply to representations that are not analogous to sentences, and also do not apply to many other things that are commonly judged to be true or otherwise. Consider the analogy between the sentence "Snow is white" and the character named Snow White, both of which can be true in some sense.
To a minimalist, saying "Snow is white is true" is the same as saying "Snow is white," but to say "Snow White is true" is not the same as saying "Snow White. Philosophical skepticism is generally any questioning attitude or doubt towards one or more items of knowledge or belief which ascribe truth to their assertions and propositions. Philosophical skepticism comes in various forms. Radical forms of skepticism deny that knowledge or rational belief is possible and urge us to suspend judgment regarding ascription of truth on many or all controversial matters.
More moderate forms of skepticism claim only that nothing can be known with certainty, or that we can know little or nothing about the "big questions" in life, such as whether God exists or whether there is an afterlife. Religious skepticism is "doubt concerning basic religious principles such as immortality, providence, and revelation ". Several of the major theories of truth hold that there is a particular property the having of which makes a belief or proposition true.
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Pluralist theories of truth assert that there may be more than one property that makes propositions true: ethical propositions might be true by virtue of coherence. Propositions about the physical world might be true by corresponding to the objects and properties they are about.
Some of the pragmatic theories, such as those by Charles Peirce and William James , included aspects of correspondence, coherence and constructivist theories. In some discourses, Wright argued, the role of the truth predicate might be played by the notion of superassertibility.
Logic is concerned with the patterns in reason that can help tell us if a proposition is true or not. However, logic does not deal with truth in the absolute sense, as for instance a metaphysician does. Logicians use formal languages to express the truths which they are concerned with, and as such there is only truth under some interpretation or truth within some logical system.
A logical truth also called an analytic truth or a necessary truth is a statement which is true in all possible worlds  or under all possible interpretations, as contrasted to a fact also called a synthetic claim or a contingency which is only true in this world as it has historically unfolded. A proposition such as "If p and q, then p" is considered to be a logical truth because of the meaning of the symbols and words in it and not because of any fact of any particular world.
They are such that they could not be untrue. Degrees of truth in logic may be represented using two or more discrete values, as with bivalent logic or binary logic , three-valued logic , and other forms of finite-valued logic. There are two main approaches to truth in mathematics. They are the model theory of truth and the proof theory of truth. Historically, with the nineteenth century development of Boolean algebra mathematical models of logic began to treat "truth", also represented as "T" or "1", as an arbitrary constant. In propositional logic , these symbols can be manipulated according to a set of axioms and rules of inference , often given in the form of truth tables.
Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution,  or even if they do, whether they have a finite or infinite number of solutions. More fundamentally, Hilbert's first problem was on the continuum hypothesis.
Martin Heidegger pointed out that truth may be essentially a matter of letting beings entities of any kind, which can include logical propositions  be free to reveal themselves as they are, and stated:. Rather, truth is disclosure of beings through which an openness essentially unfolds [ west ]. The more I think about language, the more it amazes me that people ever understand each other at all. The semantic theory of truth has as its general case for a given language:. Tarski's theory of truth named after Alfred Tarski was developed for formal languages, such as formal logic.
Here he restricted it in this way: no language could contain its own truth predicate, that is, the expression is true could only apply to sentences in some other language. The latter he called an object language , the language being talked about. It may, in turn, have a truth predicate that can be applied to sentences in still another language. The reason for his restriction was that languages that contain their own truth predicate will contain paradoxical sentences such as, "This sentence is not true".
As a result, Tarski held that the semantic theory could not be applied to any natural language, such as English, because they contain their own truth predicates. Donald Davidson used it as the foundation of his truth-conditional semantics and linked it to radical interpretation in a form of coherentism. Bertrand Russell is credited with noticing the existence of such paradoxes even in the best symbolic formations of mathematics in his day, in particular the paradox that came to be named after him, Russell's paradox.
Russell and Whitehead attempted to solve these problems in Principia Mathematica by putting statements into a hierarchy of types , wherein a statement cannot refer to itself, but only to statements lower in the hierarchy. This in turn led to new orders of difficulty regarding the precise natures of types and the structures of conceptually possible type systems that have yet to be resolved to this day.
Kripke's theory of truth named after Saul Kripke contends that a natural language can in fact contain its own truth predicate without giving rise to contradiction. He showed how to construct one as follows:. Notice that truth never gets defined for sentences like This sentence is false , since it was not in the original subset and does not predicate truth of any sentence in the original or any subsequent set. In Kripke's terms, these are "ungrounded. This contradicts the principle of bivalence : every sentence must be either true or false.
Since this principle is a key premise in deriving the liar paradox , the paradox is dissolved. In fact, this idea—manifested by the diagonal lemma —is the basis for Tarski's theorem that truth cannot be consistently defined. While there is still a debate on whether Tarski's proof can be implemented to every similar partial truth system, none have been shown to be consistent by acceptable methods used in mathematical logic. The truth predicate " P is true" has great practical value in human language, allowing us to efficiently endorse or impeach claims made by others, to emphasize the truth or falsity of a statement, or to enable various indirect Gricean conversational implications.
Even four-year-old children can pass simple " false belief " tests and successfully assess that another individual's belief diverges from reality in a specific way;  by adulthood we have strong implicit intuitions about "truth" that form a "folk theory" of truth. These intuitions include: . Like many folk theories, our folk theory of truth is useful in everyday life but, upon deep analysis, turns out to be technically self-contradictory; in particular, any formal system that fully obeys Capture and Release semantics for truth also known as the T-schema , and that also respects classical logic, is provably inconsistent and succumbs to the liar paradox or to a similar contradiction.
The ancient Greek origins of the words "true" and "truth" have some consistent definitions throughout great spans of history that were often associated with topics of logic , geometry , mathematics , deduction , induction , and natural philosophy. Socrates ', Plato 's and Aristotle 's ideas about truth are seen by some as consistent with correspondence theory. In his Metaphysics , Aristotle stated: "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true".
Most influential is his claim in De Interpretatione 16a3 that thoughts are "likenesses" homoiosis of things. Although he nowhere defines truth in terms of a thought's likeness to a thing or fact, it is clear that such a definition would fit well into his overall philosophy of mind. Very similar statements can also be found in Plato Cratylus b2, Sophist b. In Hinduism , Truth is defined as "unchangeable", "that which has no distortion", "that which is beyond distinctions of time, space, and person", "that which pervades the universe in all its constancy".
The human body, therefore is not completely true as it changes with time, for example. There are many references, properties and explanations of truth by Hindu sages that explain varied facets of truth, such as the national motto of India : " Satyameva Jayate " Truth alone wins , as well as "Satyam muktaye" Truth liberates , "Satya' is 'Parahit'artham' va'unmanaso yatha'rthatvam' satyam" Satya is the benevolent use of words and the mind for the welfare of others or in other words responsibilities is truth too , "When one is firmly established in speaking truth, the fruits of action become subservient to him patanjali yogasutras, sutra number 2.
Unveil it, O Pusan Sun , so that I who have truth as my duty satyadharma may see it! Combined with other words, satya acts as modifier, like " ultra " or " highest ," or more literally " truest ," connoting purity and excellence. For example, satyaloka is the "highest heaven' and Satya Yuga is the "golden age" or best of the four cyclical cosmic ages in Hinduism, and so on. Christianity has somewhat a different and a more personal view of truth. According to the Bible in John , Jesus is quoted as having said "I am the way, the truth and the life: no man cometh unto the Father, but by me".
What corresponds in the mind to what is outside it. The truth of a thing is the property of the being of each thing which has been established in it. However, this definition is merely a rendering of the medieval Latin translation of the work by Simone van Riet. Truth is also said of the veridical belief in the existence [of something]. A natural thing, being placed between two intellects, is called true insofar as it conforms to either.
It is said to be true with respect to its conformity with the divine intellect insofar as it fulfills the end to which it was ordained by the divine intellect With respect to its conformity with a human intellect, a thing is said to be true insofar as it is such as to cause a true estimate about itself. Thus, for Aquinas, the truth of the human intellect logical truth is based on the truth in things ontological truth.
Truth is the conformity of the intellect and things. Aquinas also said that real things participate in the act of being of the Creator God who is Subsistent Being, Intelligence, and Truth. Thus, these beings possess the light of intelligibility and are knowable. These things beings; reality are the foundation of the truth that is found in the human mind, when it acquires knowledge of things, first through the senses , then through the understanding and the judgement done by reason. For Aquinas, human intelligence "intus", within and "legere", to read has the capability to reach the essence and existence of things because it has a non-material, spiritual element, although some moral, educational, and other elements might interfere with its capability.
Richard Firth Green examined the concept of truth in the later Middle Ages in his A Crisis of Truth , and concludes that roughly during the reign of Richard II of England the very meaning of the concept changes. The idea of the oath, which was so much part and parcel of for instance Romance literature ,  changes from a subjective concept to a more objective one in Derek Pearsall 's summary. Immanuel Kant endorses a definition of truth along the lines of the correspondence theory of truth.
Kant states in his logic lectures:. In consequence of this mere nominal definition, my cognition, to count as true, is supposed to agree with its object. Now I can compare the object with my cognition, however, only by cognizing it. Hence my cognition is supposed to confirm itself, which is far short of being sufficient for truth. For since the object is outside me, the cognition in me, all I can ever pass judgement on is whether my cognition of the object agrees with my cognition of the object.
The ancients called such a circle in explanation a diallelon. And actually the logicians were always reproached with this mistake by the sceptics, who observed that with this definition of truth it is just as when someone makes a statement before a court and in doing so appeals to a witness with whom no one is acquainted, but who wants to establish his credibility by maintaining that the one who called him as witness is an honest man.
The accusation was grounded, too. Only the solution of the indicated problem is impossible without qualification and for every man. This passage makes use of his distinction between nominal and real definitions. A nominal definition explains the meaning of a linguistic expression. A real definition describes the essence of certain objects and enables us to determine whether any given item falls within the definition. According to Kant, the ancient skeptics were critical of the logicians for holding that, by means of a merely nominal definition of truth, they can establish which judgements are true.
They were trying to do something that is "impossible without qualification and for every man". Georg Hegel distanced his philosophy from psychology by presenting truth as being an external self-moving object instead of being related to inner, subjective thoughts. Hegel's truth is analogous to the mechanics of a material body in motion under the influence of its own inner force. According to Hegel, the progression of philosophical truth is a resolution of past oppositions into increasingly more accurate approximations of absolute truth. The "thesis" consists of an incomplete historical movement.
To resolve the incompletion, an "antithesis" occurs which opposes the "thesis. This "synthesis" thereby becomes a "thesis," which will again necessitate an "antithesis," requiring a new "synthesis" until a final state is reached as the result of reason's historical movement. History is the Absolute Spirit moving toward a goal. This historical progression will finally conclude itself when the Absolute Spirit understands its own infinite self at the very end of history. Absolute Spirit will then be the complete expression of an infinite God. For Arthur Schopenhauer ,  a judgment is a combination or separation of two or more concepts.
If a judgment is to be an expression of knowledge , it must have a sufficient reason or ground by which the judgment could be called true. Truth is the reference of a judgment to something different from itself which is its sufficient reason ground. Judgments can have material, formal, transcendental, or metalogical truth. A judgment has material truth if its concepts are based on intuitive perceptions that are generated from sensations.
If a judgment has its reason ground in another judgment, its truth is called logical or formal. If a judgment, of, for example, pure mathematics or pure science, is based on the forms space, time, causality of intuitive, empirical knowledge, then the judgment has transcendental truth. Objective truths are concerned with the facts of a person's being, while subjective truths are concerned with a person's way of being.
Kierkegaard agrees that objective truths for the study of subjects like mathematics, science, and history are relevant and necessary, but argues that objective truths do not shed any light on a person's inner relationship to existence.
Free Thought Lives
At best, these truths can only provide a severely narrowed perspective that has little to do with one's actual experience of life. While objective truths are final and static, subjective truths are continuing and dynamic. The truth of one's existence is a living, inward, and subjective experience that is always in the process of becoming.
The values, morals, and spiritual approaches a person adopts, while not denying the existence of objective truths of those beliefs, can only become truly known when they have been inwardly appropriated through subjective experience. Thus, Kierkegaard criticizes all systematic philosophies which attempt to know life or the truth of existence via theories and objective knowledge about reality.
As Kierkegaard claims, human truth is something that is continually occurring, and a human being cannot find truth separate from the subjective experience of one's own existing, defined by the values and fundamental essence that consist of one's way of life. Friedrich Nietzsche believed the search for truth, or 'the will to truth', was a consequence of the will to power of philosophers.
He thought that truth should be used as long as it promoted life and the will to power , and he thought untruth was better than truth if it had this life enhancement as a consequence. As he wrote in Beyond Good and Evil , "The falseness of a judgment is to us not necessarily an objection to a judgment The question is to what extent it is life-advancing, life-preserving, species-preserving, perhaps even species-breeding He proposed the will to power as a truth only because, according to him, it was the most life-affirming and sincere perspective one could have.
In this essay, Nietzsche rejects the idea of universal constants, and claims that what we call "truth" is only "a mobile army of metaphors, metonyms, and anthropomorphisms. Separately Nietzsche suggested that an ancient, metaphysical belief in the divinity of Truth lies at the heart of and has served as the foundation for the entire subsequent Western intellectual tradition : "But you will have gathered what I am getting at, namely, that it is still a metaphysical faith on which our faith in science rests—that even we knowers of today, we godless anti-metaphysicians still take our fire too, from the flame lit by the thousand-year old faith, the Christian faith which was also Plato's faith, that God is Truth; that Truth is 'Divine' Other philosophers take this common meaning to be secondary and derivative.
According to Martin Heidegger , the original meaning and essence of truth in Ancient Greece was unconcealment, or the revealing or bringing of what was previously hidden into the open, as indicated by the original Greek term for truth, aletheia. Alfred North Whitehead , a British mathematician who became an American philosopher, said: "There are no whole truths; all truths are half-truths.
It is trying to treat them as whole truths that plays the devil". The logical progression or connection of this line of thought is to conclude that truth can lie, since half-truths are deceptive and may lead to a false conclusion. Pragmatists like C. Peirce take truth to have some manner of essential relation to human practices for inquiring into and discovering truth, with Peirce himself holding that truth is what human inquiry would find out on a matter, if our practice of inquiry were taken as far as it could profitably go: "The opinion which is fated to be ultimately agreed to by all who investigate, is what we mean by the truth And even among those who accept it, there is little if any agreement about what generic criteria determine the form of an arbitrary sentence.
A remarkable fact about logical truth is that many have thought it plausible that the set of logical truths of certain rich formalized languages is characterizable in terms of concepts of standard mathematics. In particular, on some views the set of logical truths of a language of that kind is always the set of sentences of the language derivable in a certain calculus. On other, more widespread views, the set of logical truths of a language of that kind can be identified with the set of sentences that are valid across a certain range of mathematical interpretations where validity is something related to but different from the condition that all the sentences that are replacement instances of its form be true too; see below, section 2.
One main achievement of early mathematical logic was precisely to show how to characterize notions of derivability and validity in terms of concepts of standard mathematics. Sections 2. In part 1 of this entry we will describe in very broad outline the main existing views about how to understand the ideas of modality and formality relevant to logical truth. In part 2 we will describe, also in outline, a particular set of philosophical issues that arise when one considers the attempted mathematical characterizations of logical truth.
The question of whether or in what sense these characterizations are correct is bound with the question of what is or should be our specific understanding of the ideas of modality and formality. As we said above, it seems to be universally accepted that, if there are any logical truths at all, a logical truth ought to be such that it could not be false, or equivalently, it ought to be such that it must be true. But as we also said, there is virtually no agreement about the specific character of the pertinent modality.
These values may but need not be expressions. On one traditional but not uncontroversial interpretation, Aristotle's claim that the conclusion of a syllogismos must be true if the premises are true ought to be understood in this way. For this interpretation see e.
Alexander of Aphrodisias, Diodorus' view appears to have been very common in the Middle Ages, when authors like William of Sherwood and Walter Burley seem to have understood the perceived necessity of conditionals like 2 as truth at all times see Knuuttila , pp. An understanding of necessity as eternity is frequent also in later authors; see e. Many authors have thought that views of this sort do not account for the full strength of the modal import of logical truths.
A nowadays very common, but apparently late view in the history of philosophy, is that the necessity of a logical truth does not merely imply that some generalization about actual items holds, but also implies that the truth would have been true at a whole range of counterfactual circumstances. See Lewis for an introduction to the contemporary polemics in this area. However, even after Leibniz and up to the present, many logicians seem to have avoided a commitment to a strong notion of necessity as truth in all actual and counterfactual circumstances.
Tarski is even closer to the view traditionally attributed to Aristotle, for it is pretty clear that for him to say that e. Quine is known for his explicit rejection of any modality that cannot be understood in terms of universal generalizations about the actual world see especially Quine In some of these cases, this attitude is explained by a distrust of notions that are thought not to have reached a fully respectable scientific status, like the strong modal notions; it is frequently accompanied in such authors, who are often practicing logicians, by the proposal to characterize logical truth as a species of validity in the sense of 2.
On another recent understanding of logical necessity as a species of generality, proposed by Rumfitt , the necessity of a logical truth consists just in its being usable under all sets of subject-specific ways of drawing implications provided these sets satisfy certain structural rules ; or, more roughly, just in its being applicable no matter what sort of reasoning is at stake. On this view, a more substantive understanding of the modality at stake in logical truth is again not required.
It may be noted that, although he postulates a variety of subject-specific implication relations, Rumfitt rejects pluralism about logical truth in the sense of Beall and Restall see his , p. It is an old observation, going at least as far back as Plato, that some truths count as intuitively known by us even in cases where we don't seem to have any empirical grounds for them.
Truths that are knowable on non-empirical grounds are called a priori an expression that begins to be used with this meaning around the time of Leibniz; see e. The axioms and theorems of mathematics, the lexicographic and stipulative definitions, and also the paradigmatic logical truths, have been given as examples. If it is accepted that logical truths are a priori , it is natural to think that they must be true or could not be false at least partly in the strong sense that their negations are incompatible with what we are able to know non-empirically.
Assuming that such a priori knowledge exists in some way or other, much recent philosophy has occupied itself with the issue of how it is possible. See, e. See the entry on rationalism vs. Some philosophers, empiricists and otherwise, have attempted to explain a priori knowledge as arising from some sort of convention or tacit agreement to assent to certain sentences such as 1 and use certain rules.
Wittgenstein , I. Strictly speaking, Wittgenstein and Carnap think that logical truths do not express propositions at all, and are just vacuous sentences that for some reason or other we find useful to manipulate; thus it is only in a somewhat diminished sense that we can speak of a priori knowledge of them. For this reason it can be said that they explain the apriority of logical truths in terms of their analyticity.
Kant's explanation of the apriority of logical truths has seemed harder to extricate. Kant characterizes analytic truths as those where the concept of the predicate is contained in or identical with the concept of the subject, and, more fundamentally, as those whose denial is contradictory. Mill , bk. II, ch. II, pt.
This and the apparent lack of clear pronouncements of Kant on the issue has led at least Maddy and Hanna to consider though not accept the hypothesis that Kant viewed some logical truths as synthetic a priori. On an interpretation of this sort, the apriority of many logical truths would be explained by the fact that they would be required by the cognitive structure of the transcendental subject, and specifically by the forms of judgment. Capozzi and Roncaglia On an interpretation of this sort, Kant's forms of judgment may be identified with logical concepts susceptible of analysis see e.
Allison , pp. A substantively Kantian contemporary theory of the epistemology of logic and its roots in cognition is developed in Hanna ; this theory does not seek to explain the apriority of logic in terms of its analyticity, and appeals instead to a specific kind of logical intuition and a specific cognitive logic faculty. Compare also the anti-aprioristic and anti-analytic but broadly Kantian view of Maddy , mentioned below.
The early Wittgenstein shares with Kant the idea that the logical expressions do not express meanings in the way that non-logical expressions do see , 4. It is unclear how apriority is explainable in this framework. Wittgenstein calls the logical truths analytic , 6. But the extension of the idea to quantificational logic is problematic, despite Wittgenstein's efforts to reduce quantificational logic to truth-functional logic; as we now know, there is no algorithm for deciding if a quantificational sentence is valid.
Often this rejection has been accompanied by criticism of the other views. See Grice and Strawson and Carnap for reactions to these criticisms. Quine especially also argued that accepted sentences in general, including paradigmatic logical truths, can be best seen as something like hypotheses that are used to deal with experience, any of which can be rejected if this helps make sense of the empirical world see Putnam for a similar view and a purported example. On this view there cannot be strictly a priori grounds for any truth. Three recent subtle anti-aprioristic positions are Maddy's , , Azzouni's , , and Sher's For Maddy, logical truths are a posteriori , but they cannot be disconfirmed merely by observation and experiment, since they form part of very basic ways of thinking of ours, deeply embedded in our conceptual machinery a conceptual machinery that is structurally similar to Kant's postulated transcendental organization of the understanding.
Similarly, for Azzouni logical truths are equally a posteriori , though our sense that they must be true comes from their being psychologically deeply ingrained; unlike Maddy, however, Azzouni thinks that the logical rules by which we reason are opaque to introspection. Sher provides an attempt at combining a Quinean epistemology of logic with a commitment to a metaphysically realist view of the modal ground of logical truth.
One especially noteworthy kind of skeptical consideration in the epistemology of logic is that the possibility of inferential a priori knowledge of these facts seems to face a problem of circularity or of infinite regress. In any case, it seems clear that not all claims of this latter kind, expressing that a certain truth is a logical truth or that a certain logical schema is truth-preserving, could be given an a priori inferential justification without the use of some of the same logical rules whose correctness they might be thought to codify.
The point can again be reasonably derived from Carroll Some of the recent literature on this consideration, and on anti-skeptical rejoinders, includes Dummett , and Boghossian On most views, even if it were true that logical truths are true in all counterfactual circumstances, a priori , and analytic, this would not give sufficient conditions for a truth to be a logical truth. For philosophers who accept the idea of formality, as we said above, the logical form of a sentence is a certain schema in which the expressions that are not schematic letters are widely applicable across different areas of discourse.
There is explicit reflection on the contrast between the formal schemata or moods and the matter hyle of syllogismoi in Alexander of Aphrodisias The matter are the values of the schematic letters. The idea that the non-schematic expressions in logical forms, i. The same idea is conspicuous as well in Tarski , ch. But beyond this there is little if any agreement about what generic feature makes an expression logical, and hence about what determines the logical form of a sentence.
Most authors sympathetic to the idea that logic is formal have tried to go beyond the minimal thesis. It would be generally agreed that being widely applicable across different areas of discourse is only a necessary, not sufficient property of logical expressions; for example, presumably most prepositions are widely applicable, but they are not logical expressions on any implicit generic notion of a logical expression.
Attempts to enrich the notion of a logical expression have typically sought to provide further properties that collectively amount to necessary and sufficient conditions for an expression to be logical. One idea that has been used in such characterizations, and that is also present in Aristotle, is that logical expressions do not, strictly speaking, signify anything; or, that they do not signify anything in the way that substantives, adjectives and verbs signify something.
We saw that the idea was still present in Kant and the early Wittgenstein. It reemerged in the Middle Ages. In a somewhat different, earlier, grammatical sense of the word, syncategorematic expressions were said to be those that cannot be used as subjects or predicates in categorical propositions; see Kretzmann , pp. The idea of syncategorematicity is somewhat imprecise, but there are serious doubts that it can serve to characterize the idea of a logical expression, whatever this may be. Most prepositions and adverbs are presumably syncategorematic, but they are also presumably non-logical expressions.
They are of course categorematic in the grammatical sense, in which prepositions and adverbs are equally clearly syncategorematic. One recent suggestion is that logical expressions are those that do not allow us to distinguish different individuals. One way in which this has been made precise is through the characterization of logical expressions as those whose extension or denotation over any particular domain of individuals is invariant under permutations of that domain. See Tarski and Givant , p. A permutation of a domain is a one-to-one correspondence between the domain and itself.
Other paradigmatic logical expressions receive more complicated extensions over domains, but the extensions they receive are invariant under permutations. One problem with the proposal is that many expressions that seem clearly non-logical, because they are not widely applicable, are nevertheless invariant under permutations, and thus unable to distinguish different individuals. The simplest examples are perhaps non-logical predicates that have an empty extension over any domain, and hence have empty induced images as well.
See Kneale , Hacking , Peacocke , Hodes , among others. A necessary property of purely inferential rules is that they regulate only inferential transitions between verbal items, not between extra-verbal assertibility conditions and verbal items, or between verbal items and actions licensed by those items.
A number of such conditions are postulated in the relevant literature see e. Belnap a reply to Prior , Hacking and Hodes However, even when the notion of pure inferentiality is strengthened in these ways, problems remain. Most often the proposal is that an expression is logical just in case certain purely inferential rules give its whole meaning, including its sense, or the set of aspects of its use that need to be mastered in order to understand it as in Kneale , Peacocke and Hodes However, it seems clear that some paradigmatic logical expressions have extra sense attached to them that is not codifiable purely inferentially.
A different version of the proposal consists in saying that an expression is logical just in case certain purely inferential rules that are part of its sense suffice to determine its extension as in Hacking Some philosophers have reacted even more radically to the problems of usual characterizations, claiming that the distinction between logical and non-logical expressions must be vacuous, and thus rejecting the notion of logical form altogether.
See e. Orayen , ch. These philosophers typically think of logical truth as a notion roughly equivalent to that of analytic truth simpliciter. But they are even more liable to the charge of giving up on extended intuitions than the proposals of the previous paragraph. For more thorough treatments of the ideas of formality and of a logical expression see the entry logical constants , and MacFarlane This term is usually employed to cover several distinct though related phenomena, all of them present in Frege One of these is the use of a completely specified set of artificial symbols to which the logician unambiguously assigns meanings, related to the meanings of corresponding natural language expressions, but much more clearly delimited and stripped from the notes that in those natural language expressions seem irrelevant to truth-conditional content; this is especially true of symbols meant to represent the logical expressions of natural language.
A third phenomenon is the postulation of a deductive calculus with a very clear specification of axioms and rules of inference for the artificial formulae see the next section ; such a calculus is intended to represent in some way deductive reasoning with the correlates of the formulae, but unlike ordinary deductions, derivations in the calculus contain no steps that are not definite applications of the specified rules of inference.
In first-order Fregean formalized languages, among these formulae one finds artificial correlates of 1 , 2 and 3 , things like. See the entry on logic, classical. Fregean formalized languages include also classical higher-order languages. See the entry on logic, second-order and higher-order. The logical expressions in these languages are standardly taken to be the symbols for the truth-functions, the quantifiers, identity and other symbols definable in terms of those but there are dissenting views on the status of the higher-order quantifiers; see 2. The restriction to artificial formulae raises a number of questions about the exact value of the Fregean enterprise for the demarcation of logical truths in natural language; much of this value depends on how many and how important are perceived to be the notes stripped from the natural language expressions that are correlates of the standard logical expressions of formalized languages.
But whatever one's view of the exact value of formalization, there is little doubt that it has been very illuminating for logical purposes. One reason is that it's often clear that the stripped notes are really irrelevant to truth-conditional content this is especially true of the use of natural language logical expressions for doing mathematics. Another of the reasons is that the fact that the grammar and meaning of the artificial formulae is so well delimited has permitted the development of proposed characterizations of logical truth that use only concepts of standard mathematics.
This in turn has allowed the study of the characterized notions by means of standard mathematical techniques. The next two sections describe the two main approaches to characterization in broad outline. We just noted that the Fregean logician's formalized grammar amounts to an algorithm for producing formulae from the basic artificial symbols. This is meant very literally. As was clear to mathematical logicians from very early on, the basic symbols can be seen as or codified by natural numbers, and the formation rules in the artificial grammar can be seen as or codified by simple computable arithmetical operations.
The grammatical formulae can then be seen as or codified by the numbers obtainable from the basic numbers after some finite series of applications of the operations, and thus their set is characterizable in terms of concepts of arithmetic and set theory in fact arithmetic suffices, with the help of some tricks. Exactly the same is true of the set of formulae that are derivable in a formalized deductive calculus.
But the axioms are certain formulae built by the process of grammatical formation, so they can be seen as or codified by certain numbers; and the rules of inference can again be seen as or codified by certain computable arithmetical operations. So the derivable formulae can be seen as or codified by the numbers obtainable from the axiom numbers after some finite series of applications of the inference operations, and thus their set is again characterizable in terms of concepts of standard mathematics again arithmetic suffices.
In the time following Frege's revolution, there appears to have been a widespread belief that the set of logical truths of any Fregean language could be characterized as the set of formulae derivable in some suitably chosen calculus hence, essentially, as the set of numbers obtainable by certain arithmetical operations. The idea follows straightforwardly from Russell's conception of mathematics and logic as identical see Russell , ch.
Related Philosophical Truth: or Truthful Philosophy
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