In natural languages, an indicative conditional is a conditional sentence such as "If Leona is at home, she isn't in Paris", whose grammatical form restricts it to discussing what could be true. Indicatives are typically defined in opposition to counterfactual conditionals, which have extra grammatical marking which allows them to discuss eventualities which are no longer possible.
Indicatives are a major topic of research in philosophy of language, philosophical logic, and linguistics. Open questions include which logical operation indicatives denote, how such denotations could be composed from their grammatical form, and the implications of those denotations for areas including metaphysics, psychology of reasoning, and philosophy of mathematics.
Early analyses identified indicative conditionals with the logical operation known as the material conditional. According to the material conditional analysis, an indicative "If A then B" is true unless A is true and B is not. Although this analysis covers many observed cases, it misses some crucial properties of actual conditional speech and reasoning.
One problem for the material conditional analysis is that it allows indicatives to be true even when their antecedent and consequent are unrelated. For instance, the indicative "If Paris is in France then trout are fish" is intuitively strange since the location of Paris has nothing to do with the classification of trout. However, since its antecedent and the consequent are both true, the material conditional analysis treats it as a true statement. Similarly, the material conditional analysis treats conditionals with false antecedents as vacuously true. For instance, since Paris is not in Australia, the conditional "If Paris is in Australia, then trout are fish" would be treated as true on a material conditional analysis. These arguments have been taken to show that no truth-functional operator will suffice as a semantics for indicative conditionals. In the mid-20th century, work by H.P. Grice, Frank Cameron Jackson, and others attempted to maintain the material conditional as an analysis of indicatives' literal semantic denotation, while appealing to pragmatics in order to explain the apparent discrepancies. [1]
Contemporary work in philosophical logic and formal semantics generally proposes alternative denotations for indicative conditionals. Proposed alternatives include analyses based on relevance logic, modal logic, probability theory, Kratzerian modal semantics, and dynamic semantics. [2]
Most behavioral experiments on conditionals in the psychology of reasoning have been carried out with indicative conditionals, causal conditionals, and counterfactual conditionals. People readily make the modus ponens inference, that is, given if A then B, and given A, they conclude B, but only about half of participants in experiments make the modus tollens inference, that is, given if A then B, and given not-B, only about half of participants conclude not-A, the remainder say that nothing follows (Evans et al., 1993). When participants are given counterfactual conditionals, they make both the modus ponens and the modus tollens inferences (Byrne, 2005).
In classical logic, disjunctive syllogism is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
In propositional logic, modus ponens, also known as modus ponendo ponens, implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore Q must also be true."
In propositional logic, modus tollens (MT), also known as modus tollendo tollens and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. it is impossible for the premises to be true and the conclusion to be false.
In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises.
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion. For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic, in the sense that if the premises are true, then so is the conclusion.
Counterfactual conditionals are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood.
Conditional sentences are natural language sentences that express that one thing is contingent on something else, e.g. "If it rains, the picnic will be cancelled." They are so called because the impact of the main clause of the sentence is conditional on the dependent clause. A full conditional thus contains two clauses: a dependent clause called the antecedent, which expresses the condition, and a main clause called the consequent expressing the result.
The material conditional is an operation commonly used in logic. When the conditional symbol is interpreted as material implication, a formula is true unless is true and is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.
The Wason selection task is a logic puzzle devised by Peter Cathcart Wason in 1966. It is one of the most famous tasks in the study of deductive reasoning. An example of the puzzle is:
You are shown a set of four cards placed on a table, each of which has a number on one side and a color on the other. The visible faces of the cards show 3, 8, blue and red. Which card(s) must you turn over in order to test that if a card shows an even number on one face, then its opposite face is blue?
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
The paradoxes of material implication are a group of true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated as "if ... then ...". A material conditional formula is true unless is true and is false. If natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis had won World War Two, everybody would be happy" is vacuously true. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.
The psychology of reasoning is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory.
Formal semantics is the study of grammatical meaning in natural languages using formal tools from logic and theoretical computer science. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. It provides accounts of what linguistic expressions mean and how their meanings are composed from the meanings of their parts. The enterprise of formal semantics can be thought of as that of reverse-engineering the semantic components of natural languages' grammars.
Logical consequence is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.
In formal semantics and philosophical logic, simplification of disjunctive antecedents (SDA) is the phenomenon whereby a disjunction in the antecedent of a conditional appears to distribute over the conditional as a whole. This inference is shown schematically below:
In logic, import-export is a deductive argument form which states that . In natural language terms, the principle means that the following English sentences are logically equivalent.
