Logic

What is Logic?

Logic is the study of reasoning. It deals primarily with inference, but also entailment, causality, consequence, induction, deduction, truth, falsity, belief, fallacies, paradoxes, probabilities, analysis, tense, modality, necessity, sufficiency, possibility, identity, vagueness, existence, description, justification, tolerance, obligation, permission, relevance, assertion, validity, contradiction, provability, and argumentation.

One of the central questions explored in the study of logic is:

Does the conclusion follow from the premises?

The phrase “follow from” can have different interpretations, including, but not limited to:

Deduction

There is no possible situation in which the all of the premises are true but the conclusion is false.

Induction

The premises give really strong evidence for the truth of the conclusion.

A Pretest

Have some fun! Take this logic pre-test.

Notation

In order to understand the form of arguments and not get bogged down in long, confusing, and often ambiguous text, we usually write statements in a very compact form. We often use common abbreviations such as the following:

It takes quite a bit of practice to translate English (or other natural language) sentences into logical notation. In fact, the translation might not always be exact; natural language is pretty fluid (see, for example, The SEP article on Conditionals for the many readings of “if...then”). We’ll just give examples, rather than rules.

We were pretty liberal with parentheses, but you don’t have to be. $\neg$ has the highest precedence, then $\wedge$, then $\vee$ then $\supset$ then $\equiv$.

A logical argument is written as a sequence of formulas. We check the validity of the argument by checking whether each formula is either (1) a premise or (2) follows from previous formulas. Arguments can also be given graphically, with premises above the line and conclusions below. Example:

The meaning of “follows from” depends on the particular system of logic you are working under.

So let’s look at different kinds of logic.

Kinds of Logic

How can there be many different kinds of logic? Well, for one there could be different ways to understand “truth.” Sometimes we aren’t concerned with exact truth, but rather “degree of belief,” or whether something is “justified,” or “achievable,” or “knowable.”

Here are some types of logics. The list is incomplete. The types are also overlapping.

Bivalent Logic

A bivalent system has exactly two truth values, usually called true and false. What kinds of systems are not bivalent? Those with a third value (sometimes called “unknown”), and those where a truth value is represented as a degree of certainty or belief (e.g. a value ∈ 0.0..1.0).

Propositional Logic

Propositional Logic deals only with atomic sentences, known as prepositions, such as “Pigs fly”, “I am late”, “The moon is square”, and “The baby wants to sleep”. By “atomic” we mean variables can only refer to complete sentences, never to the actors or the actions within the sentences. In other words, prepositions are never decomposed. Sentences can be combined with truth operators to form larger sentences. The traditional operators are:

Note: The $\wedge$ and $\vee$ operators are meant to be material, not causal. They cannot distinguish the sentence “I called her and found out about the problem” from “I found out about the problem and called her.”

Syllogistic Logic

Systems of logic featuring syllogisms were studied thousands of years ago; they include reasoning about “all” and “some”, but do not allow general propositions nor connectives like “and” and “or”. Traditionally, only the following eight kinds of statements are considered (capital letters are classes, small letters are instances):

Syllogistic logic has been completely superseded by first-order predicate logic.

Predicate Logic

A predicate logic adds objects, functions on objects, predicates, descriptions, and quantifiers (such as $\exists$ and $\forall$) to propositional logic. A first-order logic allows quantification over objects only; In a second-order logic you can quantify over first-order predicates. You can go on forever with these orders.

Classical Logic

Classical Logic is probably the most widely used logic. It is characterized by five properties (so say one of the authors of the Wikipedia article on the subject):

• Excluded middle: $A \vee \neg A$
• Non-contradiction: $\neg (A \wedge \neg A)$
• Monotonicity of entailment: If $B$ follows from $A$, then it follows from $A$ and $C$ (in other words, go ahead and add assumptions!)
• Commutativity of conjunction: $A \wedge B$ has the same truth value as $B \wedge A$
• DeMorgan dualities for all operators: $\neg P(A_1,...,A_n)$ has the same truth value as $Q(\neg A_1,...,\neg A_n)$, where $P$ and $Q$ are duals of each other. (Examples: $\wedge$ and $\vee$ are duals; $\forall$ and $\exists$ are duals. There are many others.)

The term “classical” here does mean old or ancient. This was not the logic of the Ancient Greeks, the Ancient Chinese, or the Ancient anyone else. Classical Logic dates from the late 1800s. It’s relatively new.

Intuitionistic (Constructive) Logic

In intuitionistic logic we are not concerned with the truth of a statement as much as we are about its justification. Every theorem is something justifiable, constructively. So $A$ means “$A$ is provable” and $\neg A$ means “$A$ is refutable” (i.e., it is provable that no proof of $A$ exists—in other words, assuming A allows you to derive False). Therefore:

• You can only prove $A \vee \neg A$ by proving $A$ or refuting $A$. (The inability to do neither means you can’t say anything, so shut up.)
• You can only prove $\exists x. P$ by demonstrating an object for which $P$ is true. You cannot prove it by saying “it just can’t be the case that $\neg P$ holds for every $x$.”
• You can derive $\neg \neg A$ from $A$, but you cannot derive $A$ from $\neg \neg A$. The former says “if I can prove $A$, then I can refute that $A$ is refutable” but the latter is saying “if there’s no proof that $A$ is not provable then $A$ must be provable.” That’s not good enough: you have to demonstrate the proof of $A$ in order for $A$ to be a theorem!

Relevance Logic

Classical logic has the principle of explosion: false implies everything. Or in other words, “from a contradiction, everything follows.” The rationale is: if things are only true or false, and you assume false is true, then everything is true because true is already true and now you say that false is true also.

Let’s demonstrate with a proof of $(A \wedge \neg A) \supset B$:

This means the following statements are true in classical logic:

• If the moon is six inches in diameter, then 2 + 2 = 4.
• If the moon is six inches in diameter, then 2 + 2 = 5.
• If the moon is six inches in diameter, then the moon is seven inches in diameter.

Relevance logic rejects explosion, and requires the antecedent and consequent to be related to each other. There must be some real causality. We say in classical logic, implication is material but in relevance logic it is strict. There are many different ways to encode relevance.

Read about Relevance Logic in the SEP.

Free Logic

Descriptions can be tricky. What if no object satisfies the description?

Russell regarded $P(\iota x.Ax)$ as an abbreviation for $\exists x.((\forall y.(Ay \equiv y=x)) \wedge Px)$. But this means the statement is false unless there is a unique object satisfying the description $A$ and that has the property $P$.

A free logic allows the description of terms that do not denote anything. Read about Free Logic in the SEP.

Non-monotonic Logic

A non-monotonic logic rejects the monotonicity of entailment. If a logic is monotonic, then adding new information can change the set of known facts, causing previously known truths to become falsehoods. These kind of logics are good for cases where you have to retract previous knowledge when new facts are known, as in

• default reasoning
• abductive reasoning (infer the most likely cause)
• reasoning about knowledge (I used to not know this, now I do)
• belief revision

See the SEP article on non-monotonic logic.

Paraconsistent Logic

A paraconsistent logic is one that allows contradictions, by throwing out the principle of explosion.

See the SEP article on Paraconsistent Logic.

Many-Valued Logics

A many-valued logic has, you got it, many truth-values. Many such systems exist. What’s useful about them is they get rid of the those principles of excluded middle and non-contradiction.

You might be interested in four-valued logics.

Catuṣkoṭi

The Catuṣkoṭi is also known as the “Four Corners.” Propositions take on four values :

• Being
• Not Being
• Being and Not Being
• Neither Being nor Not Being

Find out more at Wikpedia and the SEP and this lecture by Graham Priest.

Fuzzy Logic

A fuzzy logic is one in which facts have a degree of truth between 0 and 1. It is useful for statements like “$X$ is tall” or “$X$ is an adult” or “The bike is new.”

Do not confuse fuzzy logic with paraconsistent logic.

Do not confuse fuzzy logic with probability theory.

Read about Fuzzy Logic in the SEP.

Modal Logic

Modal Logic deals with modalities, which qualify statements somehow. There are many different kinds of modalities, giving rise to different kinds of logics. Each can be based on classical or non-classical logics, and use numeric truth values, too.

See the SEP article on modal logic.

Alethic Logic (Necessity and Possibility)

The basic alethic modal operators are defined as:

In classical modal logic, the two operators are duals:

Various systems of modal logic are characterized by the axioms and inference rules they add to the underlying logic.

• System K, the weakest form of modal logic, adds the following to an underlying logic:
  • From $\Box(A \supset B)$ infer $\Box A \supset \Box B$
  • If $A$ is a theorem of the underlying logic, infer $\Box A$
• System T is System K plus
  • From $\Box A$ infer $A$
• S4 is System T plus
  • From $\Box A$ infer $\Box \Box A$
• S5 is System T plus
  • From $\lozenge A$ infer $\Box \lozenge A$

Deontic Logic (Obligation and Permission)

The basic deontic operators are defined as:

Epistemic Logic (Knowledge)

In epistemic logic we have:

Doxastic Logic (Belief)

In doxastic logic we have:

You may sometimes see the epistemic and the doxastic distinguished in the following context. Let $\Gamma = \exists g. G g$, i.e. there exists a $g$ such that $g$ is a god, or “(at least one) god exists.” Then:

• $\mathscr{K}_x\,\Gamma \wedge \mathscr{B}_x\,\Gamma$ means $x$ is a gnostic theist
• $\mathscr{K}_x\,\Gamma \wedge \neg \mathscr{B}_x\,\Gamma$ means $x$ is a gnostic atheist
• $\neg \mathscr{K}_x\,\Gamma \wedge \mathscr{B}_x\,\Gamma$ means $x$ is an agnostic theist
• $\neg \mathscr{K}_x\,\Gamma \wedge \neg \mathscr{B}_x\,\Gamma$ means $x$ is a agnostic atheist

Temporal Logic

Temporal logic deals with the modalities of time.

Dynamic Logic

Dynamic Logic brings events into the picture:

Temporal vs. Dynamic Logic

Temporal logic [is] the modal logic of choice for reasoning about concurrent systems with its aspects of synchronization, interference, independence, deadlock, livelock, fairness, etc. These concerns of concurrency would appear to be less central to linguistics, philosophy, and artificial intelligence, the areas in which dynamic logic is most often encountered nowadays. — Wikipedia

Formal Logic

In logic we are not so much concerned with the absolute truth or falsity of individual premises themselves, but rather with the form of arguments. We can make this idea precise with formal systems.

A Formal System is a mechanism for symbolically generating a set of formulas (called theorems) via inference rules.

A logistic system (a.k.a. a system of formal logic) is a formal system in which formulas are assigned truth values.

We need examples.

Syntax

So first, what is a formula? It’s a statement, written as a string of symbols from a finite alphabet, that is formed according to formation rules which we call a grammar. The grammar defines the syntax of the formal system.

For example, here is a grammar for classical propositional logic:

var      → "p" | "q" | "r" | var "′"
formula  → "T" | "F" | var
         | "¬" formula
         | "(" formula "∧" formula ")"
         | "(" formula "∨" formula ")"
         | "(" formula "⊃" formula ")"
         | "(" formula "≡" formula ")"

which means that

    p
    (¬(p′′′ ∧ q) ∨ ¬¬p)
    (r ⊃ p′)

are formulas, but

    ¬≡pp∧))′∧¬q∨(F′′

is not. Neither, by the way is p ∧ q, since the syntax above requires parentheses around all formulae formed with binary operators. Look again, closely!

Semantics

The semantics is given by a defining a function that maps each formula to a truth value. The mapping is necessarily relative to some interpretation (or situation) φ which gives meanings to the variables in the system.

For example, the semantics of classical propositional logic is defined as follows:

We’re interested ultimately in valid reasoning—what follows from what—so let’s define some terms:

Satisfiability

If formula $A$ is true under some interpretation $\varphi$ we write $\vDash_{\varphi}A$ and say $\varphi$ satisfies $A$, or that $A$ is satisfiable.

Validity

If formula $A$ is true under all interpretations we write $\vDash A$ and say A is a tautology, or that $A$ is valid.

Entailment

If formula $A$ is true under all interpretations in which formula $B$ is true, we write $B \vDash A$ and say $B$ entails $A$. Entailment is central to reasoning; if our knowledge base contains $B$ and we know that $B$ entails $A$, we know, then, that $A$ is true.

Inference Rules

In a logistic system, we don’t necessarily compute semantic functions for the fun of it. Instead, we apply inference rules to algorithmically, (or mechanistically, typographically, symbolically) generate a formulas in such a way as to mimic a reasoning process. Define

$\{A_1, ..., A_n\} \vdash B$

to mean “$B$ given assumptions $A_1, ..., A_n$.” The inference rules of a logistic system produce these kinds of statements.

If $\varnothing \vdash A$, we say $A$ is a theorem and write $\vdash A$. The sequence of inferences producing a theorem is a proof of that theorem.

Properties of Logistic Systems

The definition of truth in a system is generally just stated in precise natural language, or with semantic functions (as in our example above). The derivation of theorems, however, is a purely mechanical process. We would like to know how well, in a logistic system, theorem derivation (proof) correlates with truth.

Soundness

A system is sound iff every theorem is true.

Completeness

A system is complete iff every true formula is a theorem.

Consistency

A system is consistent iff no two theorems have contradictory truth values (where “contradictory” depends on the type of logic being considered).

Decidability

A system is decidable iff there exists an (always-terminating) algorithm to determine whether or not a given formula is a theorem.

The reasoning about logistic systems themselves is called metalogic.

Summary