User:Phlsph7/Rule of inference - Systems of logic

Systems of logic

Propositional logic

Main article: Propositional logic

Propositional logic examines the inferential patterns of simple and compound propositions. It uses letters, such as $P$ and $Q$, to represent simple propositions. Compound propositions are formed by modifying or combining simple propositions with logical operators, such as $\lnot$ (not), $\land$ (and), $\lor$ (or), and $\to$ (if ... then ...). For example, if $P$ stands for the statement "it is raining" and $Q$ stands for the statement "the streets are wet", then $\lnot P$ expresses "it is not raining" and $P\to Q$ expresses "if it is raining then the streets are wet". These logical operators are truth-functional, meaning that the truth value of a compound proposition depends only on the truth values of the simple propositions composing it. For instance, the compound proposition $P\land Q$ is only true if both $P$ and $Q$ are true; in all other cases, it is false. Propositional logic is not concerned with the concrete meaning of propositions other than their truth values. Key rules of inference in propositional logic are modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism. Further rules include conjunction introduction, disjunction introduction, constructive dilemma, destructive dilemma, double negation elimination, and De Morgan's laws.

Notable rules of inference

Rule of inference	Form	Example
Modus ponens	${\begin{array}{l}P\to Q\\P\\\hline Q\end{array}}$	${\begin{array}{l}{\text{If Kim is in Seoul, then Kim is in South Korea.}}\\{\text{Kim is in Seoul.}}\\\hline {\text{Therefore, Kim is in South Korea.}}\end{array}}$
Modus tollens	${\begin{array}{l}P\to Q\\\lnot Q\\\hline \lnot P\end{array}}$	${\begin{array}{l}{\text{If Koko is a koala, then Koko is cuddly.}}\\{\text{Koko is not cuddly.}}\\\hline {\text{Therefore, Koko is not a koala.}}\end{array}}$
Hypothetical syllogism	${\begin{array}{l}P\to Q\\Q\to R\\\hline P\to R\end{array}}$	${\begin{array}{l}{\text{If Leo is a lion, then Leo roars.}}\\{\text{If Leo roars, then Leo is fierce.}}\\\hline {\text{Therefore, if Leo is a lion, then Leo is fierce.}}\end{array}}$
Disjunctive syllogism	${\begin{array}{l}P\lor Q\\\lnot P\\\hline Q\end{array}}$	${\begin{array}{l}{\text{The book is on the shelf or on the table.}}\\{\text{The book is not on the shelf.}}\\\hline {\text{Therefore, the book is on the table. }}\end{array}}$

First-order logic

Main article: First-order logic

Classical first-order logic also employs the logical operators from propositional logic but includes additional devices to articulate the internal structure of propositions. Basic propositions in first-order logic consist of a predicate, symbolized with uppercase letters like $P$ and $Q$, which is applied to singular terms, symbolized with lowercase letters like $a$ and $b$. For example, if $a$ stands for "Aristotle" and $P$ stands for "is a philosopher", the formula $P(a)$ means that "Aristotle is a philosopher". Another innovation of first-order logic is the use of the quantifiers $\exists$ and $\forall$, which express that a predicate applies to some or all individuals. For instance, the formula $\exists xP(x)$ expresses that philosophers exist while $\forall xP(x)$ expresses that everyone is a philosopher. The rules of inference from propositional logic are also valid in first-order logic. Additionally, first-order logic introduces new rules of inference that govern the role of singular terms, predicates, and quantifiers in arguments. Key rules of inference are universal instantiation and existential generalization. Other rules of inference include universal generalization and existential instantiation.

Notable rules of inference

Rule of inference	Form	Example
Universal instantiation	${\begin{array}{l}\forall xP(x)\\\hline P(a)\end{array}}$	${\begin{array}{l}{\text{Everyone must pay taxes.}}\\\hline {\text{Therefore, Wesley must pay taxes.}}\end{array}}$
Existential generalization	${\begin{array}{l}P(a)\\\hline \exists xP(x)\end{array}}$	${\begin{array}{l}{\text{Socrates is mortal.}}\\\hline {\text{Therefore, someone is mortal.}}\end{array}}$

Modal logics

Main article: Modal logic

Modal logics are formal systems that extend propositional logic and first-order logic with additional logical operators. Alethic modal logic introduces the operator $\Diamond$ to express that something is possible and the operator $\Box$ to express that something is necessary. For example, if the $P$ means that "Parvati works", then $\Diamond P$ means that "It is possible that Parvati works" while $\Box P$ means that "It is necessary that Parvati works". These two operators are related by a rule of replacement stating that $\Box P$ is equivalent to $\lnot \Diamond \lnot P$. In other words: if something is necessarily true then it is not possible that it is not true. Further rules of inference include the necessitation rule, which asserts that a statement is necessarily true if it is provable in a formal system without any additional premises, and the distribution axiom, which allows one to derive $\Diamond P\to \Diamond Q$ from $\Diamond (P\to Q)$. These rules of inference belong to system K, a weak form of modal logic with only the most basic rules of inference. Many formal systems of alethic modal logic include additional rules of inference, such as system T, which allows one to deduce $P$ from $\Diamond P$.

Non-alethic systems of modal logic introduce operators that behave like $\Diamond$ and $\Box$ in alethic modal logic, following similar rules of inference but with different meanings. Deontic logic is one type of non-alethic logic. It uses the operator $P$ to express that an action is permitted and the operator $O$ to express that an action is required, where $P$ behaves similarly to $\Diamond$ and $O$ behaves similarly to $\Box$. For instance, the rule of replacement in alethic modal logic asserting that $\Box Q$ is equivalent to $\lnot \Diamond \lnot Q$ also applies to deontic logic. As a result, one can deduce from $OQ$ (e.g. Quinn has an obligation to help) that $\lnot P\lnot Q$ (e.g. Quinn is not permitted not to help). Other systems of modal logic include temporal modal logic, which has operators for what is always or sometimes the case, as well as doxastic and epistemic modal logics, which have operators for what people believe and know.

Others

Many other systems of logic have been proposed. One of the earliest systems is Aristotelian logic, according to which each statement is made up of two terms, a subject and a predicate, connected by a copula. For example, the statement "all humans are mortal" has the subject "all humans", the predicate "mortal", and the copula "is". All rules of inference in Aristotelian logic have the form of syllogisms, which consist of two premises and a conclusion. For instance, the Barbara rule of inference describes the validity of arguments of the form "All men are mortal. All Greeks are men. Therefore, all Greeks are mortal."

Second-order logic extends first-order logic by allowing quantifiers to apply to predicates in addition to singular terms. For example, to express that the individuals Adam ($a$) and Bianca ($b$) share a property, one can use the formula $\exists X(X(a)\land X(b))$. Second-order logic also comes with new rules of inference. For instance, one can infer $P(a)$ (Adam is a philosopher) from $\forall XX(a)$ (every property applies to Adam).

Intuitionistic logic is a non-classical variant of propositional and first-order logic. It shares with them many rules of inference, such as modus ponens, but excludes certain rules. For example, in classical logic, one can infer $P$ from $\lnot \lnot P$ using the rule of double negation elimination. However, in intuitionistic logic, this inference is invalid. As a result, every theorem that can be deduced in intuitionistic logic can also be deduced in classical logic, but some theorems provable in classical logic cannot be proven in intuitionistic logic.

Paraconsistent logics revise classical logic to allow the existence of contradictions. In logic, a contradiction happens if the same proposition is both affirmed and denied, meaning that a formal system contains both $P$ and $\lnot P$ as theorems. Classical logic prohibits contradictions because classical rules of inference lead to the principle of explosion, an admissible rule of inference that makes it possible to infer $Q$ from the premises $P$ and $\lnot P$. Since $Q$ is unrelated to $P$, any arbitrary statement can be deduced from a contradiction, making the affected systems useless for deciding what is true and false. Paraconsistent logics solve this problem by modifying the rules of inference in such a way that the principle of explosion is not an admissible rule of inference. As a result, it is possible to reason about inconsistent information without deriving absurd conclusions.

Many-valued logics modify classical logic by introducing additional truth values. In classical logic, a proposition is either true or false with nothing inbetween. In many-valued logics, some propositions are neither true nor false. Kleene logic, for example, is a three-valued logic that introduces the additional truth value undefined to describe situations where information is incomplete or uncertain. Many-valued logics have adjusted rules of inference to accommodate the additional truth values. For instance, the classical rule of replacement stating that $P\to Q$ is equivalent to $\lnot P\lor Q$ is invalid in many three-valued systems.

References

Sources

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