What Is Logic?

Logic is often described as the study of correct reasoning.

That is a good beginning.

It is not enough.

Logic asks what makes a conclusion follow from premises.

It studies structures of inference independently of many details of subject matter.

This abstraction makes logic useful in:

• philosophy,
• mathematics,
• computer science,
• science,
• everyday reasoning.

Reasoning

Reasoning moves from some claims to another claim.

For example:

All mammals are warm-blooded.

Whales are mammals.

Therefore whales are warm-blooded.

Logic asks whether the conclusion follows from the premises.

Form Matters

Replace the content:

All A are B.

C is A.

Therefore C is B.

The structure remains valid.

Logic studies this form.

The particular meanings can vary.

Inference

An inference is a transition from premises to conclusion.

Some inferences guarantee their conclusions if the premises are true.

Others only make conclusions probable.

Logic contains several kinds of inference.

Deduction

A deductive inference aims at necessity.

If the premises are true and the form is valid, the conclusion cannot be false.

Example:

All humans are mortal.

Socrates is human.

Therefore Socrates is mortal.

Induction

Induction moves from observed cases toward a broader generalization or prediction.

The sun has risen every observed morning.

Therefore it will rise tomorrow.

The conclusion may be strongly supported.

It is not logically guaranteed by the premises alone.

Abduction

Abduction infers a plausible explanation.

The lawn is wet.

If it rained, the lawn would be wet.

Perhaps it rained.

This can be reasonable.

But another explanation may exist.

Logic Is Not Psychology

Logic does not merely describe how people actually think.

Humans make:

• mistakes,
• shortcuts,
• biases.

Logic evaluates inference standards.

Psychology studies cognitive processes.

The two overlap but are distinct.

Logic Is Not Rhetoric

An argument can be persuasive but invalid.

Another can be valid but rhetorically unconvincing.

Logic asks:

does the conclusion follow?

Rhetoric asks:

what persuades an audience?

Persuasion and validity are different.

Logic Is Not Truth Alone

A conclusion can be true even if the reasoning is bad.

Example:

Paris is in France.

Therefore 2 + 2 = 4.

The conclusion is true.

The argument gives no logical support for it.

Truth and inference must be separated.

Validity

An argument is valid when there is no possible case in which:

• all premises are true,
• the conclusion is false.

Validity concerns structure.

It does not require the actual premises to be true.

Soundness

A deductive argument is sound if:

• it is valid,
• all its premises are true.

Soundness therefore combines form and fact.

An Invalid Argument with True Statements

Consider:

All dogs are animals.

Cats are animals.

Therefore cats are dogs.

The premises are true.

The conclusion is false.

The inference is invalid.

Shared category membership does not imply identity.

A Valid Argument with a False Premise

Consider:

All planets are made of cheese.

Earth is a planet.

Therefore Earth is made of cheese.

The argument form is valid.

The first premise is false.

So the argument is unsound.

Logic Studies Possibility

Validity asks about all possible interpretations consistent with the form.

Can the premises be true while the conclusion is false?

If yes, the inference is invalid.

This makes logic deeply connected to possibility.

Propositions

Logic typically studies propositions:

contents that can be true or false.

The sentences:

“Snow is white.”

and a translation of that sentence

may express the same proposition.

Logic often abstracts away from exact wording.

Arguments

An argument in logic is not a quarrel.

It is a structured set of claims:

• premises,
• conclusion.

The premises are offered as reasons for the conclusion.

Logical Consequence

The relation between premises and conclusion is called logical consequence.

We may write:

[ P_1, P_2, \ldots, P_n \models C ]

to express that (C) follows semantically from the premises.

Different formal systems define consequence precisely.

Proof

Another route is syntactic.

A conclusion may be derivable through formal rules.

We may write:

[ P_1, P_2, \ldots, P_n dash C ]

The distinction between semantic consequence and formal derivability will become important later.

Logic and Language

Logic requires representations.

Natural language is ambiguous.

Formal logic introduces controlled symbols to expose inference structure.

For example:

if (P), then (Q).

(P).

Therefore (Q).

The form becomes explicit.

Modus Ponens

This inference rule is called modus ponens:

[ P ightarrow Q ]

[ P ]

therefore:

[ Q ]

It is one of the simplest valid patterns.

Modus Tollens

Another is modus tollens:

[ P ightarrow Q ]

[

eg Q ]

therefore:

[

eg P ]

Again, validity depends on form.

Fallacies

Invalid but tempting reasoning patterns are called fallacies.

Example:

If it rains, the street is wet.

The street is wet.

Therefore it rained.

Maybe.

But sprinklers could explain it.

This is affirming the consequent.

Formal and Informal Logic

Formal logic

Uses symbolic systems and explicit rules.

Informal logic

Studies arguments in ordinary language.

Both matter.

Formal systems give precision.

Informal analysis handles real-world complexity.

Classical Logic

Classical logic usually assumes principles such as:

• bivalence,
• noncontradiction,
• excluded middle.

A proposition is true or false.

A proposition and its negation cannot both be true.

Many alternative logics modify some of these assumptions.

Alternative Logics

Later we will encounter:

• modal logic,
• temporal logic,
• fuzzy logic,
• deontic logic.

Logic is not one single immutable system.

Different logics formalize different kinds of reasoning.

Logic and Reality

Does logic describe reality?

Or only our representations?

This is a deep philosophical question.

Some see logical laws as features of rational thought.

Others treat them as abstract structural truths.

The series will return to this indirectly.

Logic and Computers

Digital circuits implement Boolean operations.

Programming languages use conditions and inference-like structures.

Automated theorem provers manipulate formal statements.

Logic became part of machinery.

Logic and Mathematics

Proof is central to mathematics.

Logic formalizes:

• axioms,
• derivations,
• theoremhood.

In the twentieth century, mathematicians discovered that formalization also has limits.

Gödel’s theorems made logic self-reflective.

The Philosophical Lesson

Logic is the study of relations of consequence.

It asks not merely whether claims are true, but whether one claim follows from others under specified rules.

Good reasoning requires both:

• reliable premises,
• reliable inference.

The Next Question

To analyze an argument, we first need to know its parts.

Which statements are reasons?

Which statement is being supported?

That brings us to:

arguments, premises, and conclusions.