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= Introduction =
'''Logic''' is the study of reasoning: how a conclusion follows from reasons, and how to tell a reliable inference from a misleading one. You use it when checking a news claim, explaining a scientific result, writing a program, or deciding whether an argument really proves its point.

This course is designed for '''learners aged about 14–18 and adult beginners'''. No previous symbolic logic is needed. By the end, you can identify premises and conclusions, distinguish validity from truth, work with basic logical symbols and truth tables, and challenge common reasoning errors. A good argument is not simply one that sounds persuasive; its support must be examined.

[[File:Venn diagram.png|500px|frameless|center|Overlapping sets help visualize logical relationships. Wikimedia Commons: A bag of many boiled potatoes, CC0 1.0.]]

'''Watch first:''' This short introduction places logic within philosophy and critical thinking.
{{#ev:youtube|1A_CAkYt3GY|500|center}}

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= Arguments, Premises and Conclusions =
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== How arguments work ==
An [[English:Argument (logic)|argument]] consists of one or more '''premises''' offered to support a '''conclusion'''. For example: ''Every square is a rectangle. Every rectangle has four sides. Therefore, every square has four sides.'' The word ''therefore'' often signals a conclusion, while ''because'' often introduces a premise. These clues help, but the structure matters more than the wording.

A '''deductively valid''' argument is one in which it is impossible for all the premises to be true while the conclusion is false. '''Soundness''' requires both validity and actually true premises. A valid argument can have false premises. To test validity, imagine a counterexample with true premises but a false conclusion: if such a situation is possible, the argument is invalid.

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== A reliable rule and a tempting mistake ==
'''Modus ponens''' has the pattern: ''If P, then Q. P. Therefore, Q.'' For example: ''If a number is divisible by four, it is even. Twelve is divisible by four. Therefore, twelve is even.'' By contrast, concluding P merely because Q is true commits '''affirming the consequent'''; the result might have another cause.

[[File:Modus ponens.svg|400px|frameless|center|A visual representation of modus ponens. Wikimedia Commons: Phlsph7, CC0 1.0.]]

'''Video lesson:''' Identify the stated reasons and conclusion in the examples, then decide whether the inference is valid.
{{#ev:youtube|NKEhdsnKKHs|500|center}}

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= Statements and Logical Connectives =
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== From everyday sentences to symbols ==
In basic [[English:Propositional calculus|propositional logic]], P and Q stand for statements that can be true or false. A connective builds a new statement from them. Classical logic uses two truth values; this useful model does not capture every ambiguity of everyday language.

{| class="wikitable"
! Expression !! Reading !! Truth condition
|-
| ¬P || not P || True exactly when P is false
|-
| P ∧ Q || P and Q || True when both statements are true
|-
| P ∨ Q || P or Q || True when at least one statement is true
|-
| P → Q || if P then Q || False only when P is true and Q is false
|-
| P ↔ Q || P if and only if Q || True when P and Q have the same truth value
|}

In this table, '''or''' is inclusive: ''tea or coffee'' allows both unless the situation specifies otherwise. The conditional P → Q describes a truth-functional relationship; ordinary language may also suggest causation, timing or an explanation, which the symbol alone does not express.

[[File:Logical connectives table.svg|480px|frameless|center|The sixteen binary truth functions. Wikimedia Commons: Watchduck / Tilman Piesk, public domain.]]

[[File:Venn-AND-sketch.png|500px|frameless|center|The overlap represents conjunction or intersection. Wikimedia Commons: Marble machine, CC BY-SA 4.0.]]

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= Truth Tables and Logical Equivalence =
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== Test all possible cases ==
A [[English:Truth table|truth table]] checks every assignment of truth values. With two basic statements there are four combinations. The following table shows why P → Q is false in just one case. Use '''T''' for true and '''F''' for false.

{| class="wikitable"
! P !! Q !! P ∧ Q !! P ∨ Q !! P → Q
|-
| T || T || T || T || T
|-
| T || F || F || T || F
|-
| F || T || F || T || T
|-
| F || F || F || F || T
|}

A '''tautology''' is true under every assignment, as in P ∨ ¬P. A '''contradiction''' is false under every assignment, as in P ∧ ¬P. Two expressions are '''logically equivalent''' if their final columns always agree. For example, ¬(P ∧ Q) and ¬P ∨ ¬Q are equivalent by one of [[English:De Morgan's laws|De Morgan's laws]].

'''Animation:''' Observe how a truth table is constructed one row at a time. Pause it to predict the next result.
[[File:Construction of a logical value table.gif|500px|frameless|center|Animated construction of a truth table. Wikimedia Commons: Stephan Kulla, CC BY-SA 3.0.]]

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= Quantifiers, Sets and General Claims =
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== All, some and not all ==
[[English:First-order logic|Predicate logic]] can describe individual objects and their properties. The universal quantifier '''∀''' means ''for every'', while the existential quantifier '''∃''' means ''there exists at least one''. For example, ∀x (Cat(x) → Mammal(x)) means every cat is a mammal. The expression ∃x (Cat(x) ∧ Black(x)) means at least one cat is black. Both depend on a specified domain of objects.

The denial of ''all students passed'' is '''at least one student did not pass''; it is not ''no students passed''. Similarly, ''some'' means ''one or more'' in ordinary mathematical logic, even if casual conversation sometimes suggests ''not all''. These distinctions make policies, data claims and surveys easier to assess.

[[File:Existential quantifier.svg|350px|frameless|center|The existential quantifier is used for claims about at least one object. Wikimedia Commons: Phlsph7, CC BY 4.0.]]

'''Try it:''' Imagine a set of three objects: two red circles and one blue square. Decide whether ''all objects are circles'', ''some objects are red'', and ''no objects are squares'' are true. Explain your answers using a diagram or a short sentence.

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= Informal Reasoning, Fallacies and Applications =
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== Beyond certainty ==
[[English:Deductive reasoning|Deduction]] aims at a conclusion that must follow if the premises hold. [[English:Inductive reasoning|Induction]] uses observations to support a general expectation, such as predicting tomorrow's weather from previous measurements; its conclusion can be probable without being guaranteed. [[English:Abductive reasoning|Abduction]] looks for a plausible explanation of evidence, but the best current explanation may still be wrong.

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== Spot the reasoning error ==
A [[English:False dilemma|false dilemma]] presents only two choices when more exist. An [[English:Ad hominem|ad hominem]] attack targets a person instead of the claim. '''Affirming the consequent''' wrongly moves from ''if P then Q'' and ''Q'' to ''P''. The '''fallacy of division''' assumes that what is true of a whole must also be true of each of its parts. Not every criticism of a person is automatically a fallacy; ask whether it is actually relevant to the issue.

'''Audio (2 minutes):''' Listen to this freely licensed explanation of the fallacy of division and write one original counterexample.
[[File:Fallacy of division.ogg|center|Audio recording: Fallacy of division. Wikimedia Commons: JohnAnkerBow, CC BY-SA 4.0.]]

Logic supports [[English:Computer science|computer science]] through Boolean expressions and program conditions; it supports mathematics through proofs; and it supports everyday life through evidence-based decisions. It cannot replace checking evidence or recognizing uncertainty. A logically valid inference from false premises need not describe reality correctly.

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= Interactive Tasks =
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== Quiz: Test Your Knowledge ==
{{MC}}
'''What is a premise in an argument?'''
(A reason offered in support of a conclusion)
(!A statement that must always be false)
(!The final sentence in every paragraph)
(!An example with no relevance to the claim)
{{E}}
<br>

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'''When is a deductive argument valid?'''
(When true premises cannot lead to a false conclusion)
(!Whenever its conclusion sounds convincing)
(!Only when it contains exactly two premises)
(!Whenever the audience agrees with it)
{{E}}
<br>

{{MC}}
'''What makes a deductive argument sound?'''
(It is valid and all its premises are true)
(!It has a surprising conclusion)
(!It uses only mathematical symbols)
(!Its conclusion is popular)
{{E}}
<br>

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'''When is the conjunction P and Q true?'''
(When both P and Q are true)
(!When at least one is true)
(!Only when P is false)
(!Whenever Q is false)
{{E}}
<br>

{{MC}}
'''What does inclusive or mean in classical logic?'''
(At least one of the statements is true)
(!Exactly one statement is true)
(!Both statements must be false)
(!The statements must be unrelated)
{{E}}
<br>

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'''In which case is if P then Q false in a truth table?'''
(P is true and Q is false)
(!P and Q are both true)
(!P is false and Q is true)
(!P and Q are both false)
{{E}}
<br>

{{MC}}
'''Which reasoning pattern is modus ponens?'''
(If P then Q and P therefore Q)
(!If P then Q and Q therefore P)
(!If P then Q and not P therefore not Q)
(!P or Q and P therefore not Q)
{{E}}
<br>

{{MC}}
'''What follows from the denial of all birds can fly?'''
(At least one bird cannot fly)
(!No birds can fly)
(!Exactly one bird can fly)
(!All animals can fly)
{{E}}
<br>

{{MC}}
'''Which description best fits inductive reasoning?'''
(Observations support a conclusion that may remain uncertain)
(!Premises guarantee a conclusion in every case)
(!A statement is true only because it is repeated)
(!All conclusions are chosen at random)
{{E}}
<br>

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'''What is a false dilemma?'''
(Presenting two options while ignoring real alternatives)
(!Checking whether a conclusion follows from premises)
(!Writing a complete truth table)
(!Using an example to clarify a concept)
{{E}}
<br>

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== Memory Game ==
Match each distinct concept with its definition.
<div class="memo-quiz">
{|
|-
| Premise || A reason supplied in support of a claim
|-
| Conclusion || The claim an argument attempts to establish
|-
| Validity || Impossibility of true premises with a false conclusion
|-
| Conjunction || A connective that requires both components to be true
|-
| Tautology || A statement true under every truth assignment
|-
| Quantifier || An expression indicating all or at least one
|-
| Fallacy || A recognizable error in reasoning
|}
{{E}}
<br>

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== Drag and Drop ==
Match each concept to its explanation.
<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Logic concepts
|-
| '''Negation''' || Reverses the truth value of a statement
|-
| '''Disjunction''' || True if at least one component is true
|-
| '''Biconditional''' || True when both statements have matching truth values
|-
| '''Existential claim''' || Asserts that at least one example exists
|-
| '''Counterexample''' || A case showing that a proposed rule or inference fails
|}
{{E}}
<br>

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== Crossword Puzzle ==
Use one-word answers to complete the clues.
<div class="kreuzwort-quiz">
{|
|-
| PREMISE || What do you call a statement offered as a reason in an argument?
|-
| VALIDITY || What property rules out true premises and a false conclusion?
|-
| NEGATION || What connective reverses a truth value?
|-
| TAUTOLOGY || What is a statement true under every possible assignment called?
|-
| QUANTIFIER || What logical expression indicates all or at least one?
|-
| INDUCTION || What type of inference generalizes from observations without guaranteeing certainty?
|}
{{E}}
<br>

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== LearningApps ==
Explore additional practice activities related to logic. Check the quality of an activity before using it as evidence of learning.
<iframe> https://learningapps.org/index.php?s=Logic </iframe>

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== Cloze Text ==
<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
An argument offers reasons called { premises } in support of a conclusion.
An argument is deductively { valid } when its premises cannot all be true while its conclusion is false.
A valid argument with true premises is { sound }.
The connective AND is also called { conjunction }.
A systematic chart of truth values is a truth { table }.
A statement true under every assignment is a { tautology }.
The symbol for the existential quantifier means that something { exists }.
Reasoning from observations to a probable generalization is { induction }.
A misleading choice between only two options may be a false { dilemma }.
</quiz>
<br>

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= Open Tasks =
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=== Easy ===
# [[English:Argument (logic)|Spot the argument]]: Find three everyday claims in fictional examples, underline each premise and circle its conclusion.
# [[English:Deductive reasoning|Write a deduction]]: Write one valid argument about shapes and explain why no counterexample works.
# [[English:Propositional calculus|Make symbol cards]]: Create illustrated cards for NOT, AND, OR, and IF THEN with one correct everyday example each.
# [[English:Truth table|Build a small table]]: Fill in the four truth-value combinations for P AND Q and check with a partner.

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=== Standard ===
# [[English:Logical consequence|Test a counterexample]]: Construct an invalid argument and exhibit true premises with a false conclusion.
# [[English:Venn diagram|Draw a set diagram]]: Represent an intersection and explain why it corresponds to logical AND.
# [[English:De Morgan's laws|Verify an equivalence]]: Create two full truth tables showing that NOT P AND Q is not equivalent to NOT P OR Q; state the correct parenthesization.
# [[English:Fallacy|Audit a claim]]: Select an invented advertisement, identify a possible reasoning error, and rewrite its claim more carefully.

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=== Advanced ===
# [[English:First-order logic|Translate quantified statements]]: Express two claims using universal and existential quantifiers, then negate each one correctly.
# [[English:Modus ponens|Compare argument forms]]: Produce paired examples of modus ponens and affirming the consequent and explain their different validity.
# [[English:Boolean algebra|Design a decision rule]]: Specify a privacy-preserving program rule using AND, OR, and NOT; test every input combination.
# [[English:Critical thinking|Create a teaching resource]]: Produce a short openly licensed logic lesson with original examples, source acknowledgements, and an answer key.

{{:Open Task - Create a MOOC}}

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= Learning Check =
# [[English:Logical validity|Justify validity]]: Explain in your own words how an argument can be valid even when one premise is false; give an original example.
# [[English:Truth table|Assess a conditional]]: Build a truth table for a conditional, identify its false row, and explain why that row is decisive.
# [[English:First-order logic|Negate a generalization]]: Translate and negate the claim that every member of a set satisfies a property; explain the difference between some not and none.
# [[English:Fallacy|Evaluate persuasion]]: Analyze a fictional social-media argument, separating evidence, premises, conclusion, and any possible fallacy.
# [[English:Propositional calculus|Transfer to computing]]: Model a simple access rule with Boolean connectives and test whether an unintended case is admitted.
# [[English:Inductive reasoning|Compare kinds of inference]]: Explain how deduction differs from induction and abduction using the same real-world problem.

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= Learning evidence =
A successful learner can present an '''annotated argument map''' with clearly identified premises and conclusion; distinguish '''validity, soundness and persuasiveness'''; complete and interpret a '''truth table'''; translate and negate an elementary quantified statement; and explain an invalid inference using a '''counterexample'''. Strong evidence also includes a reasoned analysis of a fallacy, a Boolean decision rule tested against all cases, and correct source attribution for reused media.

Assess both the answer and the explanation: can you show ''why'' it follows? Recheck unfamiliar claims against trustworthy sources rather than assuming that a valid pattern establishes the truth of its premises.

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= OER and Wikipedia =
'''Open reference:''' The English Wikipedia overview covers formal and informal logic, argumentation, and related fields.
<iframe> https://en.m.wikipedia.org/wiki/Logic </iframe>

'''Further reference reading:'''
* [https://plato.stanford.edu/entries/logic-classical/ Stanford Encyclopedia of Philosophy: Classical Logic] — definitions of validity, deduction and logical consequence.
* [https://en.wikipedia.org/wiki/Propositional_calculus Wikipedia: Propositional calculus] — connectives and proof systems.
* [https://en.wikipedia.org/wiki/Truth_table Wikipedia: Truth table] — truth-value calculations.
* [https://en.wikipedia.org/wiki/First-order_logic Wikipedia: First-order logic] — quantifiers and predicates.

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== Media sources and reuse rights ==
The images, animation and audio are hosted on Wikimedia Commons. The linked file-description pages identify each creator and licence. The items have been embedded above, not reproduced as newly created media. Credit the author and retain licence links when reusing attribution-licensed material.
* [https://commons.wikimedia.org/wiki/File:Venn_diagram.png Venn diagram.png] — A bag of many boiled potatoes; [https://creativecommons.org/publicdomain/zero/1.0/ CC0 1.0].
* [https://commons.wikimedia.org/wiki/File:Modus_ponens.svg Modus ponens.svg] — Phlsph7; [https://creativecommons.org/publicdomain/zero/1.0/ CC0 1.0].
* [https://commons.wikimedia.org/wiki/File:Logical_connectives_table.svg Logical connectives table.svg] — Watchduck (Tilman Piesk); dedicated to the public domain by its creator.
* [https://commons.wikimedia.org/wiki/File:Venn-AND-sketch.png Venn-AND-sketch.png] — Marble machine; [https://creativecommons.org/licenses/by-sa/4.0/ CC BY-SA 4.0].
* [https://commons.wikimedia.org/wiki/File:Construction_of_a_logical_value_table.gif Construction of a logical value table.gif] — Stephan Kulla; [https://creativecommons.org/licenses/by-sa/3.0/ CC BY-SA 3.0] (also offered under GFDL).
* [https://commons.wikimedia.org/wiki/File:Existential_quantifier.svg Existential quantifier.svg] — Phlsph7; [https://creativecommons.org/licenses/by/4.0/ CC BY 4.0].
* [https://commons.wikimedia.org/wiki/File:Fallacy_of_division.ogg Fallacy of division.ogg] — JohnAnkerBow; [https://creativecommons.org/licenses/by-sa/4.0/ CC BY-SA 4.0].

'''Video sources:''' [https://www.youtube.com/watch?v=1A_CAkYt3GY What Is Philosophy?] and [https://www.youtube.com/watch?v=NKEhdsnKKHs How to Argue – Philosophical Reasoning] are published by CrashCourse on YouTube. They are embedded using YouTube playback; no permission to download, edit or relicense the videos is implied.

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= Linked learning areas =
These connections help you apply the same reasoning tools to different subjects.
{| align=center
{{:D-Tab}}
'''[[English:Logic|Logic]]'''
# [[English:Critical thinking|Critical thinking]]
# [[English:Deductive reasoning|Deductive reasoning]]
# [[English:Propositional calculus|Propositional calculus]]
# [[English:Truth table|Truth table]]
# [[English:First-order logic|First-order logic]]
# [[English:Fallacy|Fallacy]]
# [[English:Boolean algebra|Boolean algebra]]
# [[English:Philosophy of logic|Philosophy of logic]]
|}

[[Category:English]]
[[Category:Sprache:Englisch]]
[[Category:Locale:en-US]]
[[Category:Logic]]
[[Category:Philosophy]]
[[Category:Mathematics]]
[[Category:Computer science]]
[[Category:Critical thinking]]
[[Category:Grades 9-12]]
[[Category:Secondary education]]

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