Argumentation and Logic
Foundations of Sound Reasoning
When evaluating deductive arguments, we must ensure they meet several critical standards. An argument achieves soundness when it satisfies both formal and informal validity requirements, contains premises that correspond to reality, and presents premises whose plausibility exceeds that of their negations. Understanding the fundamental inference rules of propositional logic provides essential tools for constructing and analyzing such arguments.
Core Inference Rules
The following nine inference patterns form the foundation of valid deductive reasoning in propositional logic:
Rule #1: Modus Ponens
This rule allows us to affirm the consequent when we have both a conditional statement and its antecedent:
Example: If it's raining, then the ground is wet. It's raining. Therefore, the ground is wet.
Rule #2: Modus Tollens
By denying the consequent of a conditional, we can validly deny its antecedent:
Example: If Jesus rose from the dead, then the tomb would be empty. The tomb was not empty. Therefore, Jesus did not rise from the dead. (Note: This is a valid form, though Christians would dispute the second premise.)
Rule #3: Hypothetical Syllogism
When conditionals chain together, we can derive a direct connection between the first antecedent and the final consequent:
Example: If you study hard, you'll pass the exam. If you pass the exam, you'll graduate. Therefore, if you study hard, you'll graduate.
Rule #4: Conjunction
Separate truths can be combined into a single conjunctive statement:
Example: God is omnipotent. God is omniscient. Therefore, God is omnipotent and omniscient.
Rule #5: Simplification
From a conjunction, we may extract either of its component parts:
Example: The car is red and fast. Therefore, the car is red. (Or: Therefore, the car is fast.)
Rule #6: Absorption
A conditional statement implies a stronger conditional that includes both antecedent and consequent:
Example: If you pray, then God hears you. Therefore, if you pray, then you pray and God hears you.
Rule #7: Addition
Any true statement can be expanded into a disjunction by adding any other statement:
Example: The Bible exists. Therefore, the Bible exists or unicorns exist. (While this seems odd, it's logically valid because at least one disjunct is true.)
Rule #8: Disjunctive Syllogism
When faced with a disjunction, eliminating one disjunct establishes the other:
Example: Either the disciples stole the body or Jesus rose from the dead. The disciples did not steal the body. Therefore, Jesus rose from the dead.
Rule #9: Constructive Dilemma
This rule combines conditional reasoning with disjunctive reasoning:
Example: If it's a weekday, I go to work. If it's the weekend, I go to church. It's either a weekday or the weekend. Therefore, I either go to work or go to church.
Fundamental Logical Equivalences
Beyond the inference rules, mastering key logical equivalences enables more sophisticated reasoning and proof construction:
Double Negation:
P ≡ ¬¬P— Negating a statement twice returns to the original statementExample: "It is not the case that God does not exist" is equivalent to "God exists."
Idempotence:
P v P ≡ P— Repeating a disjunct adds no new informationExample: "Either I'm going to church or I'm going to church" just means "I'm going to church."
Material Implication:
P → Q ≡ ¬P v Q— Conditionals can be expressed as disjunctionsExample: "If it rains, the game is cancelled" is equivalent to "Either it doesn't rain or the game is cancelled."
Contraposition:
P → Q ≡ ¬Q → ¬P— A conditional and its contrapositive are logically equivalentExample: "If someone is a Christian, they believe in God" is equivalent to "If someone doesn't believe in God, they're not a Christian."
De Morgan's Laws
These powerful transformation rules allow us to interchange conjunctions and disjunctions through negation. The conversion process follows this systematic approach:
Negate each individual proposition
Swap the logical operator (
&becomesv, orvbecomes&)Enclose the result in parentheses and negate the entire expression
Resulting Equivalences:
¬(P & Q) ≡ ¬P v ¬Q— The negation of a conjunction equals the disjunction of negationsExample: "It's not the case that both John is honest and Mary is trustworthy" is equivalent to "Either John is not honest or Mary is not trustworthy (or both)."
¬(P v Q) ≡ ¬P & ¬Q— The negation of a disjunction equals the conjunction of negationsExample: "It's not the case that either I'll go to the store or I'll go to the park" is equivalent to "I won't go to the store and I won't go to the park."
Predicate Logic
Predicate logic extends propositional logic by enabling us to reason about properties of objects and relationships between them. This system introduces quantifiers that allow us to make claims about entire classes or specific members of those classes.
Universal Quantification
Statements claiming something about all members of a class take the logical structure of conditional statements. Using F and G as arbitrary predicates, we represent:
Affirmative Universal:
(x)(Fx → Gx)- "All F are G"Negative Universal:
(x)(Fx → ¬Gx)- "No F are G"
Existential Quantification
Claims about the existence of at least one member with certain properties typically employ conjunctive structure:
Affirmative Existential:
(∃x)(Bx & Wx)- "Some B are W"Negative Existential:
(∃x)(Bx & ¬Wx)- "Some B are not W"
Applying Inference Rules
To derive conclusions in predicate logic, we instantiate the quantified variables by substituting specific individuals, then apply the nine inference rules from propositional logic to the resulting statements.
Modal Logic
Modal logic represents an advanced logical system that analyzes necessity and possibility. The possible worlds framework provides a semantic interpretation for modal concepts:
Necessary truth: A proposition holds in every possible world
Possible truth: A proposition holds in at least one possible world
Critical Distinctions in Modality
Understanding modal reasoning requires distinguishing between two types of necessity:
Necessity de dicto: Necessity that applies to propositions or statements—a claim is necessarily true when it holds across all possible worlds
Necessity de re: Necessity that applies to objects and their properties—an object possesses a property essentially when it has that property in every world where it exists
Common Modal Reasoning Errors
Modal logic contains subtle pitfalls. The following inference patterns appear plausible but are actually invalid:
Fallacy 1: Distributing Necessity Over Disjunction
While a disjunction may be necessarily true, this doesn't mean one of the disjuncts must be necessarily true.
Example: "Necessarily, either it will rain tomorrow or it won't rain tomorrow" does NOT mean "Either it necessarily will rain tomorrow or it necessarily won't rain tomorrow." The first is true (a logical necessity), but the second is false (tomorrow's weather is contingent).
Fallacy 2: Combining Necessity with Disjunctive Syllogism
Eliminating one disjunct doesn't transfer necessity to the remaining disjunct.
Example: "Necessarily, either God exists or 2+2=5. But 2+2 doesn't equal 5. Therefore, God necessarily exists." This is invalid—even though the disjunction is necessarily true (because one disjunct is necessarily true), eliminating the false disjunct doesn't prove the remaining one is necessary.
Fallacy 3: Transferring Necessity from Conditional to Consequent
A necessarily true conditional doesn't make its consequent necessary when the antecedent is merely true.
Example: "Necessarily, if I exist, then I'm thinking (Descartes' cogito). I exist. Therefore, I necessarily think." This is invalid—while the conditional is necessarily true, my existence is contingent, so my thinking is also contingent, not necessary.
Counterfactual Logic
Counterfactual reasoning examines what would or might happen under hypothetical circumstances contrary to fact. This logical system analyzes subjunctive conditional statements—those expressing "would" or "might" relationships.
Invalid Patterns in Counterfactual Reasoning
Counterfactual logic diverges from standard propositional logic in important ways. The following inference patterns, while valid in classical logic, fail in counterfactual contexts:
Hypothetical syllogism: Chaining counterfactual conditionals is not generally valid
Contraposition: A counterfactual and its contrapositive are not equivalent
Strengthening the antecedent: Adding conditions to the antecedent can invalidate the inference
Valid Counterfactual Inference Patterns
Despite these restrictions, counterfactual logic supports several distinctive valid inference forms:
Valid Form 1: Weakening the Consequent
When Q would hold if P were true, and R would hold if both P and Q were true, then R would hold if P were true.
Example: "If I had studied harder, I would have passed. If I had studied harder and passed, I would have graduated. Therefore, if I had studied harder, I would have graduated."
Valid Form 2: Transitivity with Biconditional
When P and Q are counterfactually equivalent, and Q counterfactually implies R, then P counterfactually implies R.
Example: "If the power had gone out, the alarm wouldn't have sounded. If the alarm hadn't sounded, the power would have gone out. If the alarm hadn't sounded, the burglary would have succeeded. Therefore, if the power had gone out, the burglary would have succeeded."
Valid Form 3: Combining Counterfactual with Necessary Conditional
When Q would hold if P were true, and Q necessarily implies R, then R would hold if P were true.
Example: "If I had dropped the glass, it would have fallen. Necessarily, if something falls (without support), it accelerates downward. Therefore, if I had dropped the glass, it would have accelerated downward."
Informal Fallacies
While formal logic identifies structural errors in reasoning, informal fallacies involve mistakes in content, relevance, or clarity. Recognizing these common errors strengthens critical thinking:
Begging the Question (Circular Reasoning): Occurs when a premise depends on the conclusion for its justification, creating a circular argument where the conclusion is assumed in order to support itself
Example: "The Bible is true because it's the Word of God, and we know it's the Word of God because the Bible says so."
Genetic Fallacy: Commits the error of rejecting or accepting a claim based solely on its origin or history, rather than evaluating the claim on its own merits
Example: "You can't trust that argument for God's existence—it came from a religious website."
Argument from Ignorance (Ad Ignorantiam): Incorrectly concludes that a proposition is false merely because it hasn't been proven true, or vice versa—confusing lack of evidence with evidence of absence
Example: "No one has proven that God doesn't exist, so God must exist." (Or conversely: "No one has proven God exists, so God doesn't exist.")
Equivocation: Exploits the ambiguity of language by shifting between different meanings of a word or phrase within a single argument, creating an illusion of validity
Example: "Faith can move mountains. I have faith in my doctor. Therefore, my faith in my doctor can move mountains." (The word "faith" shifts from religious trust to confidence.)
Amphiboly: Arises from grammatically ambiguous sentence structure that allows multiple interpretations, leading to misleading or invalid inferences
Example: "The pastor said on Sunday we should pray more." (Does this mean the pastor spoke on Sunday, or that we should pray more on Sundays?)
Composition: Fallaciously assumes that what is true of the parts must be true of the whole, ignoring emergent properties or collective characteristics that differ from individual components
Example: "Every member of the church choir sings beautifully. Therefore, when they sing together, they sound beautiful." (This ignores that they might not harmonize well together.)
Standards for Deductive Arguments
Evaluating deductive arguments requires attention to both truth and plausibility. While a sound deductive argument demands true premises, we need not possess absolute certainty about those premises. The key criterion is comparative plausibility: each premise should be more reasonable to accept than to reject.
This principle extends to the argument as a whole. When the combined premises (taken as a conjunction) are more plausible than their collective denial, the logical structure of deduction ensures that the conclusion likewise becomes more plausible than its negation. This relationship between premise plausibility and conclusion plausibility forms a cornerstone of rational argumentation.
Standards for Inductive Arguments
Inductive reasoning shares some requirements with deductive reasoning but differs in crucial ways. Like deductive arguments, strong inductive arguments require true premises that are more plausible than their negations, and they must avoid informal fallacies.
However, inductive arguments diverge fundamentally from deductive ones in their logical structure. Since true premises in an inductive argument provide support for, but do not guarantee, a true conclusion, the concept of formal validity does not apply. Instead, we evaluate inductive arguments by the degree of support the premises provide for the conclusion.
Bayesian Probability Analysis
When evaluating arguments that involve probabilistic reasoning, Bayes's theorem provides the mathematical framework for updating beliefs in light of new evidence.
Standard Form:
Components:
Pr(H|E)= Posterior probability: How probable is the hypothesis after considering the evidence?Pr(H)= Prior probability: How probable was the hypothesis before the evidence?Pr(E|H)= Likelihood: How probable is the evidence if the hypothesis is true?Pr(¬H)= Prior probability of the hypothesis being falsePr(E|¬H)= How probable is the evidence if the hypothesis is false?
Odds Form for Comparing Hypotheses:
When comparing two competing hypotheses, the odds formulation proves particularly useful:
Interpretation:
Left side: Posterior odds—the relative probability of the two hypotheses after observing the evidence
First right term: Prior odds—the relative probability of the two hypotheses before observing the evidence
Second right term: Likelihood ratio—the degree to which the evidence discriminates between the hypotheses
Inference to the Best Explanation
Another framework for understanding inductive reasoning treats it as abductive inference—selecting the most satisfactory explanation from available alternatives. This approach involves identifying which candidate explanation, if accurate, would provide the most compelling account of the observed phenomena.
Evaluating competing explanations requires applying multiple criteria systematically:
Explanatory scope: What range of phenomena does the explanation account for? Broader scope indicates stronger explanatory value.
Explanatory power: How thoroughly and precisely does it account for the observations? Deeper explanations that reveal underlying mechanisms are preferable.
Plausibility: Does the explanation cohere with established knowledge and well-confirmed theories? Explanations requiring fewer revisions to accepted beliefs are stronger.
Ad hoc nature: Does the explanation rely on arbitrary stipulations or special pleading tailored specifically to save the hypothesis? Explanations with independent motivation are superior.
Accord with accepted beliefs: How well does it integrate with the broader web of justified beliefs? Greater integration indicates higher quality.
Comparative superiority: When measured against rival explanations using these criteria, does it emerge as the strongest candidate? The best explanation must outperform its competitors, not merely satisfy minimum standards.