argument types

overview

arguments are structured collections of statements where premises provide support for a conclusion. the type of argument determines how strongly the premises support the conclusion and what kind of reasoning is involved.

this section covers the four main types of arguments in logical reasoning, each with distinct characteristics and applications in automated reasoning, science, and everyday thinking.

argument types

deductive arguments

premises provide necessary consequence - if premises are true, the conclusion must be true.

all humans are mortal    (premise)
socrates is human        (premise)
∴ socrates is mortal     (conclusion)

characteristics:

• validity: structure guarantees truth preservation
• soundness: valid + all premises actually true
• certainty: conclusions follow necessarily
• monotonic: adding premises cannot invalidate

learn more about deductive arguments →

inductive arguments

premises provide probabilistic support - make conclusion more likely but don’t guarantee it.

the sun has risen every day for 4.6 billion years    (premise)
the laws of physics remain constant                   (premise)
∴ the sun will probably rise tomorrow                 (conclusion)

characteristics:

• strength: degree of support (weak to strong)
• cogency: strong + premises probably true
• ampliative: conclusion goes beyond premises
• statistical: often based on patterns or frequencies

learn more about inductive arguments →

abductive arguments

premises suggest the best explanation for observed phenomena.

the ground is wet                        (observation)
if it rained, the ground would be wet    (premise)
∴ it probably rained                     (best explanation)

characteristics:

• inference to best explanation: among competing hypotheses
• explanatory power: accounts for observations
• defeasible: better explanations can defeat conclusions
• creative: generates new hypotheses

learn more about abductive arguments →

defeasible arguments

provide strong support that can be defeated by additional information.

birds typically fly      (default rule)
tweety is a bird        (premise)
∴ tweety probably flies (conclusion)

can be defeated by learning tweety is a penguin or has a broken wing.

characteristics:

• non-monotonic: new information can invalidate
• presumptive: based on defaults or generalizations
• revisable: conclusions are tentative
• exception-handling: admits counterexamples

learn more about defeasible arguments →

comparison matrix

validity vs soundness

understanding the distinction between validity and soundness is crucial for evaluating arguments:

validity (structure)

an argument is valid if the conclusion logically follows from the premises, regardless of whether the premises are actually true.

valid but unsound:
all cats are purple     (false premise)
fluffy is a cat        (true premise)
∴ fluffy is purple     (false conclusion, but validly derived)

soundness (structure + truth)

an argument is sound if it is both valid AND all premises are actually true.

sound argument:
all mammals are warm-blooded    (true premise)
whales are mammals             (true premise)
∴ whales are warm-blooded      (true conclusion)

only deductive arguments can be sound. inductive and abductive arguments are evaluated differently:

• inductive: strong/weak + cogent/uncogent
• abductive: plausible/implausible based on explanatory power
• defeasible: reasonable/unreasonable given current information

strength measures

different argument types use different evaluation criteria:

deductive arguments

• validity: all or nothing (valid or invalid)
• soundness: requires both validity and true premises

inductive arguments

• strength: continuous scale from weak to strong
• cogency: strong + probably true premises
• probability: can assign numerical confidence values

abductive arguments

• plausibility: relative to competing explanations
• explanatory virtues: simplicity, scope, fit with background knowledge
• likelihood: $P(\text{evidence}|\text{hypothesis})$

defeasible arguments

• reasonableness: given current information
• defeat conditions: what evidence would overturn the conclusion
• strength: how much additional evidence needed for defeat

practical applications

automated reasoning

• theorem provers: deductive arguments for formal verification
• machine learning: inductive arguments from data to models
• diagnostic systems: abductive arguments for best explanations
• expert systems: defeasible rules with exception handling

scientific reasoning

• mathematical proofs: deductive arguments
• hypothesis testing: inductive arguments from samples
• theory formation: abductive arguments explaining phenomena
• model revision: defeasible arguments updated by new evidence

everyday reasoning

• formal logic: deductive arguments in structured domains
• pattern recognition: inductive arguments from experience
• troubleshooting: abductive arguments for problem causes
• practical decisions: defeasible arguments with exceptions

legal reasoning

• statutory interpretation: deductive application of rules
• precedent analysis: inductive reasoning from case patterns
• fact finding: abductive reasoning about what happened
• burden of proof: defeasible presumptions and rebuttals

common confusions

argument vs explanation

• argument: premises support conclusion (evidence → claim)
• explanation: describes why something is true (cause → effect)

argument: "it's raining, so the ground will be wet"
explanation: "the ground is wet because it's raining"

correlation vs causation

• correlation: statistical association between variables
• causation: one variable actually causes changes in another

inductive arguments from correlation to causation are often weak without additional evidence.

necessary vs sufficient conditions

• necessary: must be present for conclusion (if not A, then not B)
• sufficient: enough by itself for conclusion (if A, then B)

confusion about these leads to fallacies like affirming the consequent.

implementation notes

when building reasoning systems:

deductive systems

• use formal logic (propositional, predicate)
• implement inference rules (modus ponens, etc.)
• ensure soundness and completeness
• handle computational complexity

inductive systems

• collect sufficient sample sizes
• handle biased data appropriately
• compute confidence intervals
• account for multiple inductive biases

abductive systems

• generate multiple hypotheses
• rank by explanatory power
• update as new evidence arrives
• balance simplicity vs accuracy

defeasible systems

• represent default rules with exceptions
• implement defeat mechanisms
• handle rule conflicts
• maintain consistency during updates

further reading

foundational texts

• aristotle’s “prior analytics” (deductive logic)
• hume’s “enquiry concerning human understanding” (inductive reasoning)
• peirce’s collected papers (abductive reasoning)
• reiter’s “default logic” (defeasible reasoning)

modern treatments

• bergmann, moor & nelson: “the logic book”
• salmon: “logic”
• josephson & josephson: “abductive inference”
• brewka, dix & konolige: “nonmonotonic reasoning”

computational approaches

• russell & norvig: “artificial intelligence: a modern approach”
• pearl: “probabilistic reasoning in intelligent systems”
• pollock: “cognitive carpentry”
• prakken & vreeswijk: “logics for defeasible argumentation”