The material conditional can yield some unexpected truths when expressed in natural language. For example, any material conditional statement with a false antecedent is true (see vacuous truth). So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true.
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Neither conjunction, disjunction, nor material conditional has an equivalent form constructed from the other four logical connectives.
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In Wason's study, not even 10% of subjects found the correct solution. This result was replicated in 1993. Some authors have argued that participants do not read "if... then..." as the material conditional, since the natural language conditional is not the material conditional. (See also the paradoxes of the material conditional for more information.) However one interesting feature of the task is how participants react when the classical logic solution is explained: This latter comment is also controversial, since it does not explain whether the subjects regarded their previous solution incorrect, or whether they regarded the problem sufficiently vague to have two interpretations.
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For example, lazy evaluation is sometimes implemented for and , so these connectives are not commutative if either or both of the expressions , have side effects. Also, a conditional, which in some sense corresponds to the material conditional connective, is essentially non-Boolean because for `if (P) then Q;`, the consequent Q is not executed if the antecedent P is false (although a compound as a whole is successful ≈ "true" in such case). This is closer to intuitionist and constructivist views on the material conditional— rather than to classical logic's views.
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The material conditional can be considered as a symbol of a formal theory, taken as a set of sentences, satisfying all the classical inferences involving →, in particular the following characteristic rules: # Modus ponens; # Conditional proof; # Classical contraposition; # Classical reductio ad absurdum. Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated. For example, in intuitionistic logic, which rejects proofs by contraposition as valid rules of inference, is not a propositional theorem, but the material conditional is used to define negation.
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Some logicians, such as Paul Grice, have used conversational implicature to argue that, despite apparent difficulties, the material conditional is just fine as a translation for the natural language 'if...then...'. Others still have turned to relevance logic to supply a connection between the antecedent and consequent of provable conditionals.
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Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" (but only when used to denote material conditional). The following is an example of a very simple inference within the scope of propositional logic: :Premise 1: If it's raining then it's cloudy. :Premise 2: It's raining.
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While the material conditional operator used in classical logic is sometimes read aloud in the form of a conditional sentence, the intuitive interpretation of conditional statements in natural language does not always correspond to it. Thus, philosophical logicians and formal semanticists have developed a wide variety of conditional logics which better match actual conditional sentences and actual conditional reasoning.
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Also, "p implies q" is equivalent to "p is false or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or material conditional. The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not familiar with them.
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Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds where its antecedent holds. More recent formal analyses have treated them using tools such as causal models and dynamic semantics.
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According to the material conditional analysis, a natural language conditional, a statement of the form ‘if P then Q’, is true whenever its antecedent, P, is false. Since counterfactual conditionals are those whose antecedents are false, this analysis would wrongly predict that all counterfactuals are vacuously true. Goodman illustrates this point using the following pair in a context where it is understood that the piece of butter under discussion had not been heated.Goodman, N., "The Problem of Counterfactual Conditionals", The Journal of Philosophy, Vol.
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In logic, the corresponding conditional of an argument (or derivation) is a material conditional whose antecedent is the conjunction of the argument's (or derivation's) premises and whose consequent is the argument's conclusion. An argument is valid if and only if its corresponding conditional is a logical truth. It follows that an argument is valid if and only if the negation of its corresponding conditional is a contradiction. Therefore, the construction of a corresponding conditional provides a useful technique for determining the validity of an argument.
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The paradox is ultimately based on the principle of formal logic that the statement A \rightarrow B is true whenever A is false, i.e., any statement follows from a false statement (ex falso quodlibet). What is important to the paradox is that the conditional in classical (and intuitionistic) logic is the material conditional. It has the property that A \rightarrow B is true if B is true or if A is false (in classical logic, but not intuitionistic logic, this is also a necessary condition).
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Counterfactuals were first discussed by Nelson Goodman as a problem for the material conditional used in classical logic. Because of these problems, early work such as that of W.V. Quine held that counterfactuals aren't strictly logical, and do not make true or false claims about the world. However, in the 1970s, David Lewis showed that these problems are surmountable given an appropriate logical framework. Work since then in formal semantics, philosophical logic, philosophy of language, and cognitive science has built on Lewis's insight, taking it in a variety of different directions.
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Such a logical connective as converse implication "←" is actually the same as material conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in classical logic), certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the classical equivalence between and . Therefore, a classical-based logical system does not need the conditional operator "→" if "¬" (not) and "∨" (or) are already in use, or may use the "→" only as a syntactic sugar for a compound having one negation and one disjunction.
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Victor Howard ("Vic") Dudman (October 10, 1935January 10, 2009) was an Australian logician based at Macquarie University. Born in Sydney, he was greatly influenced by Willard Van Orman Quine on whose work he based his undergraduate logic courses. He is particularly noted for his views on the interpretation of the material conditional. David Lewis, a frequent visitor to Australian departments of logic, once noted "If Dudman's view is correct, and I cannot at the moment see what is wrong with it, then almost everything I have written on conditionals is mistaken".
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In logic, a strict conditional (symbol: \Box, or ⥽) is a conditional governed by a modal operator, that is, a logical connective of modal logic. It is logically equivalent to the material conditional of classical logic, combined with the necessity operator from modal logic. For any two propositions p and q, the formula p → q says that p materially implies q while \Box (p \rightarrow q) says that p strictly implies q.Graham Priest, An Introduction to Non-Classical Logic: From if to is, 2nd ed, Cambridge University Press, 2008, , p. 72.
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A person's dharma consists of duties that sustain him according to his innate characteristics which are both spiritual and material, generating two corresponding types: # Sanatana-dharma – duties performed according to one's spiritual (constitutional) identity as atman and are thus the same for everyone. # Varnashrama-dharma – duties performed according to one's material (conditional) nature and are specific to the individual at that particular time. According to the notion of sanatana-dharma, the eternal and intrinsic inclination of the living entity (atman) is to perform seva (service). Sanatana-dharma, being transcendental, refers to universal and axiomatic laws that are beyond our temporary belief systems.
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B. Assess exposure to MNMs # WHO suggests assessing workers' exposure in workplaces with methods similar to those used for the proposed specific occupational exposure limit (OEL) value of the MNM (conditional recommendation, low-quality evidence). # Because there are no specific regulatory OEL values for MNMs in workplaces, WHO suggests assessing whether workplace exposure exceeds a proposed OEL value for the MNM. A list of proposed OEL values is provided in an annex of the guidelines. The chosen OEL should be at least as protective as a legally mandated OEL for the bulk form of the material (conditional recommendation, low-quality evidence).
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More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if London is in France, then the Eiffel Tower is in Bolivia". Such statements are considered vacuous truths, because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. In essence, they are true because a material conditional is defined to be true when the antecedent is false (regardless of whether the conclusion is true or false).
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Zellweger’s contribution to the field of logic is best demonstrated through his development of the X-stem Logic Alphabet (XLA). The XLA notation is a highly advanced extension of both Charles Sanders Peirce’s box-X notation (1902) and Warren Sturgis McCulloch’s dot-X notation (1942). It could be said that XLA (1961–62) is the evolutionary product of the comprehensive work of Peirce, McCulloch, and Zellweger, or PMZ as an acronym. The standard notation used today (dot Logical conjunction, vee Logical disjunction, horseshoe Material conditional representing and, or, if) is a lingering, overly abstract, unsystematically selected set of symbols that was primarily developed and used by Peano, Whitehead, and Russell, or by common acronym PWR.
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So the statement "if I have a penny in my pocket then Paris is in France" is always true, regardless of whether or not there is a penny in my pocket. These problems are known as the paradoxes of material implication, though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions. These unexpected truths arise because speakers of English (and other natural languages) are tempted to equivocate between the material conditional and the indicative conditional, or other conditional statements, like the counterfactual conditional and the material biconditional. It is not surprising that a rigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwise expressed by "if … then …" sentences in natural languages.
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So-called "a priori physicalists" hold that from knowledge of the conjunction of all physical truths, a totality or that's-all truth (to rule out non-physical epiphenomena, and enforce the closure of the physical world), and some primitive indexical truths such as "I am A" and "now is B", the truth of physicalism is knowable a priori.See Chalmers and Jackson, 2001 Let "P" stand for the conjunction of all physical truths and laws, "T" for a that's-all truth, "I" for the indexical "centering" truths, and "N" for any [presumably non-physical] truth at the actual world. We can then, using the material conditional "→", represent a priori physicalism as the thesis that PTI → N is knowable a priori. An important wrinkle here is that the concepts in N must be possessed non-deferentially in order for PTI → N to be knowable a priori.
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Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems with consequents that are necessarily true (such as 2 + 2 = 4) or antecedents that are necessarily false.Roy A. Sorensen, A Brief History of the Paradox: Philosophy and the labyrinths of the mind, Oxford University Press, 2003, , p. 105. The following sentence, for example, is not correctly formalized by a strict conditional: : If Bill Gates graduated in Medicine, then 2 + 2 = 4. Using strict conditionals, this sentence is expressed as: : \Box (Bill Gates graduated in Medicine → 2 + 2 = 4) In modal logic, this formula means that, in every possible world where Bill Gates graduated in medicine, it holds that 2 + 2 = 4. Since 2 + 2 is equal to 4 in all possible worlds, this formula is true, although it does not seem that the original sentence should be.
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In both of Gettier's actual examples (see also counterfactual conditional), the justified true belief came about, if Smith's purported claims are disputable, as the result of entailment (but see also material conditional) from justified false beliefs that "Jones will get the job" (in case I), and that "Jones owns a Ford" (in case II). This led some early responses to Gettier to conclude that the definition of knowledge could be easily adjusted, so that knowledge was justified true belief that does not depend on false premises. The interesting issue that arises is then of how to know which premises are in reality false or true when deriving a conclusion, because as in the Gettier cases, one sees that premises can be very reasonable to believe and be likely true, but unknown to the believer there are confounding factors and extra information that may have been missed while concluding something. The question that arises is therefore to what extent would one have to be able to go about attempting to "prove" all premises in the argument before solidifying a conclusion.
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