40. Compound Statements and the Forms of Argumentation

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Lecture Notes

Main Topics #

The Definition and Nature of Statements #

• Statements are a species of speech that signify true or the false
• Speech is sound that signifies by custom rather than by nature
• Names are parts of speech that signify individually (no part of a name signifies by itself)
• Simple statements: affirming or denying something of something one
• Compound statements: combining two or more simple statements

The Three Main Kinds of Compound Statements #

1. Disjunctive (Either-Or) Statements

   • Form: “Either A or B or C” (exhaustive alternatives)
   • Truth depends on exhausting all possibilities
   • For affirmative argument: eliminate all but one possibility
   • For negative argument (e.g., proving the Father is not before the Son): must eliminate all possibilities

2. Conjunctive (And) Statements

   • Form: “A and B” combining two simple statements into one
   • Truth requires both component statements to be true
   • Less useful for reasoning than other compound forms
   • Can be shortened without losing meaning (e.g., “I am a philosopher and grandfather” = “I am a philosopher and a grandfather”)
   • Usually requires reduction back to component simple statements for valid reasoning

3. Conditional (If-Then) Statements

   • Form: “If A, then B”
   • Truth means: if the antecedent is true, the consequent must be true
   • Does not claim that either part is actually true, only that one follows from the other
   • Can be true even when both component statements are false

Truth in Compound vs. Simple Statements #

• Simple statements: true or false based on whether the thing asserted corresponds to reality
• Compound statements: true or false based on the logical relationship between components, not necessarily on reality
• Example: “If I’m a number seven, then I’m an odd number” is TRUE in form, though “I’m a number seven” is FALSE

Conditional Reasoning and Valid Forms #

Modus Ponens (Affirming the Antecedent)

• If A is so, then B is so
• A is so
• Therefore, B is so
• This form is necessarily valid

Denying the Antecedent

• If A is so, then B is so
• A is not so
• Does this necessarily mean B is not so?
• Answer: Only in cases where the matter is convertible (when A and B are equivalent)
• Example of convertible matter: “If this number is 2, then this number is half of 4. This number is not 2. Therefore, it is not half of 4.” (Valid because 2 and half-of-4 are equivalent)
• Example of non-convertible matter: “If this number is 2, then this number is less than 10. This number is not 2. Therefore, it is not less than 10.” (Invalid because other numbers less than 10 exist)

Affirming the Consequent

• If A is so, then B is so
• B is so
• Does this necessarily mean A is so?
• Answer: No, not from form alone
• Example: “If I am a man, then I am an animal. I am an animal. Therefore, I am a man.” (Invalid—I could be a dog)
• This can be disproven by contraposition: if B is not so, then A is not so (valid form)

Denying the Consequent

• If A is so, then B is so
• B is not so
• Therefore, A is not so
• This form is necessarily valid (by contraposition)

Conjunctions in Sacred Scripture #

Revelation 1:8 - “I am the Alpha and the Omega”

• Combines two simple statements into one compound statement
• God is the beginning and God is the end
• These function as two of the four causes (efficient/material cause and final cause)
• The combination is not arbitrary but illuminates God’s comprehensive causality
• It works better for liturgical/religious purposes than abstract statement

John 1:1 - “In the beginning was the Word, the Word was with God, and the Word was God”

• The addition of the third statement (“the Word was God”) resolves apparent contradiction with “the Word was toward God”
• If the Word was “toward” God, it cannot be God (principle of opposites)
• The third statement clarifies that the Word both is distinct from God (toward God) and is God (identical to God)
• This shows careful logical construction in sacred texts

Key Arguments #

Why Conjunctions Are Less Useful for Reasoning #

• Conjunctive statements assert two distinct things of one subject: “I am a philosopher” AND “I am a grandfather” are really two separate facts
• When reasoning with conjunctions, one must break them back down into component simple statements
• The conjunction itself does not yield new conclusions—reasoning proceeds from the simples

The Principle of Exhaustion in Disjunctive Statements #

• A disjunctive statement is true only if the alternatives are exhaustive
• Example: “A human being is either a man or a woman or a boy or a girl…” requires all possibilities to be listed
• If stating “A human is either a man or a woman,” this omits the possibility of boys and girls, making the statement potentially false

Matter vs. Form in Conditional Arguments #

• The validity of certain conditional forms (like “denying the antecedent”) depends on the MATTER (the actual content) not the FORM
• Form alone is never sufficient to prove these invalid conditional arguments valid
• This distinction is crucial for understanding when arguments appear valid but are not

Important Definitions #

• Simple Statement: Speech affirming or denying something of something one
• Compound Statement: Speech combining two or more simple statements
• Disjunctive Statement: Compound statement of the form “either A or B or C…” presenting exhaustive alternatives
• Conjunctive Statement: Compound statement of the form “A and B”
• Conditional Statement (If-Then): Compound statement of the form “If A, then B” where B follows from A
• Modus Ponens: The valid conditional form that affirms the antecedent to derive the consequent
• Convertible Matter: When two terms or statements are equivalent (e.g., “2” and “half of 4”)

Examples & Illustrations #

Compound Statements #

• “I am a philosopher and I am a grandfather” (conjunction requiring both to be true)
• “I am a philosopher and a grandfather” (shortened form, same meaning)
• “A human being is either a man or a woman or a boy or a girl” (disjunctive requiring exhaustive alternatives)
• “If I’m a number seven, then I’m an odd number” (true conditional with false antecedent)

Conditional Reasoning Examples #

Valid Form - Modus Ponens

• “If this number is 2, then this number is less than 10. This number is 2. Therefore it is less than 10.” (Valid)

Invalid Form - Affirming the Consequent

• “If I am a man, then I am an animal. I am an animal. Therefore I am a man.” (Invalid—could be a dog)

Matter-Dependent Form - Denying the Antecedent

• Convertible: “If this number is 2, then this number is half of 4. This number is not 2. Therefore it is not half of 4.” (Valid because terms are equivalent)
• Non-convertible: “If this number is 2, then this number is less than 10. This number is not 2. Therefore it is not less than 10.” (Invalid because other numbers satisfy the consequent)

Rhetorical and Political Examples #

• Richard Nixon’s “Checkers” speech (1952): emotional appeal using gift of dog to counter false narrative about slush fund
  • Demonstrates second means of persuasion: arousing emotions
  • Justified use of emotion against those already using emotional manipulation
• Nixon vs. Kennedy debate (1960): Kennedy campaign controlled room temperature to make Nixon sweat on camera
  • Demonstrates first means of persuasion: image/ethos projected by speaker
  • Showed importance of appearance in media age (TV vs. radio coverage gave different impressions)

Notable Quotes #

“You can make a true compound statement right out of two false simple statements. Sounds crazy, doesn’t it?”

“The compound statement has a different meaning, right? And maybe the different kinds of compound statement means something different, right?”

“[The if-then statement’s] not saying that it is yet true. You can even have it, you know, be true with them being false.”

“It’s not because of the form, right? So some days people will think it’s going to follow, right?”

“I don’t think the conjunctions use, you know, really to syllogize, right? Because you break it down and then you bring it back to this, right?”

“Now the other one is called the conjunctive or conjunction, right? I don’t know if you use that or it’s not a kind of junky word, but there you have an and in there, right?”

Questions Addressed #

How can a compound statement be true when both component simple statements are false? Because the conditional statement does not assert the truth of either component; it only asserts that if the antecedent were true, the consequent would follow. The form is about logical relationship, not correspondence with reality.

When does denying the antecedent validly conclude that the consequent is false? Only when the antecedent and consequent are convertible (equivalent). If A and B are equivalent, then denying A necessarily denies B. But if B can be true through other means, denying A does not deny B.

Why use conjunctions in sacred scripture if they’re less useful for reasoning? Conjunctions serve theological and liturgical purposes beyond logical reasoning. They reveal the unified causality of God (beginning and end) and can help resolve apparent contradictions through strategic combination of statements.

How should emotional appeals be understood in persuasion? Emotional appeal is a legitimate means of persuasion (second of three) according to Aristotle’s Rhetoric, alongside establishing credibility/character and logical proof. However, it can be misused and requires care in application.

Connections to Thomas Aquinas #

• Berquist references Thomas Aquinas’s Summa Contra Gentiles and its tripartite structure:
  • Book I: God in himself
  • Books II-III: God as beginning (maker/efficient cause) and as end (final cause/moving things toward himself)
  • Book IV: Truths knowable by faith alone
• This structure parallels the theological significance of conjunctive statements like “I am the beginning and the end,” which unifies two distinct kinds of divine causality