10. What other deductive arguments are there?

What are truth tables?

This chapter is about "truth-functional" logic (also called "propositional / sentential logic"). This specifically concerns the application of logic principles to assertions and analogies. Truth tables are often used in this context. These tables often contain two letters: P and Q. These are also called claim variables and are a symbolic representation of premises and conclusions.

A claim, P, is true (T) or false (F). This is indicated by noting the letter P, putting a line under it and then noting the letters T and F below each other. By noting it this way, the possible truth values ​​for P are displayed. Sometimes numbers are used, where "true" = 1 and "false" = 0.

What types of truth tables are there?

1. Negation (~): in this case the opposite (~ P) of the claim is processed in the table. An example of such a claim is "Jamie is not at home." In this case, P is "Jamie is at home" and ~ P that Jamie is not at home.

The truth table of the conjunction NOT (truth table for negation) shows that whatever value P may have, its denial (~ P) is always the opposite:

Truth table of the conjunction NOT:

2. Conjunction (&): this is a claim that consists of two claims. These claims are called conjuncts. A conjunction is only true if the two claims that make up the general claim are true (so if P and Q are true). An example of a conjunction is; Jamie is home and Sophie is working. Jamie is P and Sophie is Q.

Truth table of the conjunction AND:

3. Disjunction (∨): this is also a claim that consists of two claims. However, these claims are called disjuncts. A disjunction is only false when both disjuncts are false. So they can both be true. Example; Either Jamie is at home or Sophie is at work. "

Truth table of the conjunction OR;

4. Conditional claim (→): this is a claim that also consists of two claims. Such a claim takes the form: "if…., Then ....". When P precedes Q (P → Q), P is called antecedent. Q is then the result ("consistent"). A conditional claim is only false if the antecedent is true and the consequence is false. Example; "If Sophie is working, then Jamie is at home."

When we add an extra letter, for example "P, Q and R", the number of possible combinations of T and F is doubled, and therefore the number of rows in the truth table is doubled.

The columns of the letters (example: "P, Q and R") that are used when filling in the column of a general claim (example: Q&R) are called reference columns ("reference columns").

A table gives us a "truth-functional analysis" of the original claim. It displays the truth values ​​of a general claim based on the truth values ​​of smaller parts of the claim. (For a clear picture of the upcoming explanation of truth tables, see the illustrations in the book and on the college sheets)

Equal claims / claims

We say that two claims are the same ("truth-functionally equivalent") when they have exactly the same truth table. In that case, the Ts and Fs in the column under one claim are arranged in the same way as the Ts and Fs in the other column.

Symbolizing a claim

The main purpose here is to produce a claim that is similar to the original claim, but where the truth-functional structure is represented. A number of problems can also arise. The most important thing about symbolizing is that the claim is well read and understood.

"If" versus "Only if"

The word "if" introduces the antecedent of a conditional claim. The phrase "only if" introduces the effect of a conditional claim.

Example;

IF; If I buy lunch for you, it's because you won the bet. "

ONLY IF; I buy lunch for you, but only if you win the bet.

What are necessary and adequate conditions?

Conditional claims are sometimes described on the basis of necessary ("necessary") conditions and conditions that are sufficient.

An example is: "The presence of oxygen is necessary for breathing. If we can breathe (A), then we must have oxygen (Z). The necessary condition then becomes the result of a conditional claim: A → Z.

A sufficient condition guarantees that something can exist if only a specific condition is met. For example, being born in America is enough to get an American passport. You don't have to do anything else for that. Adequate conditions are described, such as the antecedents of conditional claims. If Hanna was born in America (A), then Hanna has an American passport (B): A → B.

Even in the case of necessary and sufficient conditions, the difference between "if" and "only if" must be taken into account. The word "if" introduces the adequate condition. The phrase "only if" introduces the necessary condition.

Unless

The word "unless" is the same as the (v) used for disjunction. To know where a disjunction starts, we can look at where the word "or" ("either") or "if" ("if") occurs in the sentence.

What are truth-functional arguments?

A "truth-functional" argument can be valid and invalid. An argument is not valid when the premises are true, but the conclusion is false. An argument is valid when the premises on which the conclusion is based are true. A distinction is made between three valid argument patterns and the corresponding three invalid argument patterns. Important concepts for the valid argument patterns are:

• Modus ponens: The confirmatory way. The schedule for this is:
  • If P, then Q
  • P
  • So Q
• Modus Tollens: The negative way. The schedule for this is:
  
  • If P, then Q
  • Not Q
  • Not P
• Chain argument Contains multiple premise, with an extra variable. The schedule for this is:
  
  • If P, then Q
  • If Q, then R
  • So, if P, then R

Important concepts for the invalid argument patterns are those discussed earlier: confirming the consequence ("affirming the consequent"), denying the antecedent and the undivided middle.

A "truth-functional" argument can take countless forms. Nevertheless, we can still test the validity of such an argument. This is done through truth tables.

What are the rules of deduction?

Deduction is a useful means of proving, in particular, that an argument is valid instead of an argument being invalid. In this regard there are four groups of rules.

Group 1: Elemental valid argument patterns

Rule 1: Modus ponens (MP)

Also called 'affirming the antecedent': if there is a conditional claim between the premisses, and if the antecedent of this conditional claim occurs as another premise, then the consequence of the conditional claim from the two premises.

• If P, then Q
• P
• So Q

Rule 2: Modus tollens (MT)

Also called "denying the consequent": if one premise is a conditional claim of the reverse (negation) of the consequence of the conditional claim, then there is MT.

• If P, then Q
• Not Q
• Not P

Rule 3: Chain argument (CA)

This has the form: Premise 1: P → Q. Premise 2: Q → R. Conclusion: P → R.

• If P, then Q
• If Q, then R
• So, if P, then R

Rule 4: Disjunctive argument (DA)

This concerns the conditional claims, but written out according to the opposite (negation) of both P and Q (ie ~ P and ~ Q).

• P or Q
• Not P
• So Q

Rule 5: Simplification (SIM)

This has the form:

• P&Q → P P
• & Q → Q.

Rule 6: Conjunction (CONJ)

This rule takes the form:

• P
• Q
• So P&Q

Rule 7: Addition (ADD)

On the basis of this rule, two forms of deduction can be combined into a conjunction:

• P, conclusion: P \ / Q.
• Q, conclusion: P \ / Q.

Rule 8: Constructive dilemma (CD)

• P → Q.
• R → S.
• P \ / R. Conclusion: Q \ / S.

Rule 9: Destructive dilemma (DD)

1) P → Q. 2) R → S. 3) ~ Q \ / ~ S. Conclusion: ~ P \ / ~ R

Group 2: Truth-functional equivalents

Rule 10: Double opposition (DN)

•  P → (Q \ / R),
•  P → ~~ (Q \ / R).

Rule 11: Commutation (COM)

•  P → (Q \ / R),
•  P → (R \ / Q).

Rule 12: Implication (IMPL)

•  (P → Q) <-> ~ (P \ / Q).

Rule 13: Contraposition (CONTR)

•  (P → Q) <-> (~ Q → ~ P).

Rule 14: DeMorgan's Laws: (DEM)

• ~ (P&Q) <-> (~ P \ / ~ Q),
• ~ (P \ / Q) <-> (~ P & ~ Q).

Rule 15: Export (EXP)

•  [P → (Q → R)] <-> [(P&Q) → R].

Rule 16: Association (ASSOS)

• [P & (Q&R)] <-> [(P&Q) & R].
• [P \ / (Q \ / R)] <-> [(P \ / Q) \ / R].

Rule 17: Distribution (DIST)

• [P & (Q \ / R)] <-> [(P&Q) \ / (P&R)]
• [P \ / (Q & R)] <-> [(P \ / Q) & (P \ / R)].

Rule 18: Tautology (TAUT)

• P ∨ ~ P. A tautology is a sentence that is always true: it is raining, or it isn’t raining.

What is conditional evidence?

Conditional evidence is both a rule and a strategy to form a deduction. This proof is based on the following idea. Suppose we try to make a deduction for a conditional claim: P → Q. If we have formed this deduction, what have we actually proven? We have proven that if P is true, Q will also be true. In this case we can assume that P is true and on the basis of that we try to prove that Q must also be correct. If we can, so if we can prove Q after we have assumed that P is true, then we have proven that if P occurs, Q must also occur. There are, however, a number of important rules when it comes to conditional evidence. For example, conditional evidence can only be used to make a conditional claim and not to prove another claim. It is also true that if conditional evidence is used more than once in claims, they must then be approached exactly in the reverse order.

Summary

• There are four types of truth tables: conjunction, negation, conditional, and disjunction.
• Sentences (and therefore claims) can be indicated by means of letters in truth tables.
• We can determine whether an argument is valid based on a truth table. This can be done, for example, by deduction.

Practice questions

1. What is a proposition? What is the difference between a single and compound proposition? What is the role of conjunctions in this?
2. The two deduction rules associated with the conditional proposition "If ... then ..." are the Modus Ponens (MP) and Modus Tollens (MT). Give the truth table for "If ... then ...". Show the reasoning schemes for MP and MT. Use the truth table for "If ... then ..." to show why MP and MT are valid reasoning schemes. Also give 2 examples of invalid arguments.
3. What are conjunctions in proposition logic? What conjunctions are there? Give the truth table of two conjunctions.
4. What is fallacy? Why is verification also called "fallacy of the consequences"? Explain this on the basis of the reasoning scheme (syllogism) of verification and the accompanying truth table.
5. What four types of truth tables are there?
6. What does a "truth-functional analysis" mean?
7. By means of which tool can we investigate whether an argument is valid?
8. What does deduction mean?

Answers

1. A proposition is a statement that can be true or false, it cannot be further simplified (Paul is at home -> true or false). A single proposition is a single proposition. A compound proposition consists of 2 propositions that can be connected to each other in various ways by joining (e.g. A and B).
2. Truth tables help you investigate whether a formula is valid or can be fulfilled. They can also be used to find out whether a conclusion is valid and whether two formulas are logically equivalent. In truth tables the truth or untruth of a proposition can be indicated in different ways. One can simply write "true" or "false", but usually one writes T (for true, true) and F (for false, false). They also use the 1 for true and 0 for false.

The truth table for "if .. then" is as follows:

N.B. "If A, then B" is always 1, except if A = 1 and B = 0.

A is the antecedent and B the consequence. So "if A then B" is only false when the antecedent is true and the consequence false. If both A and B are false, the statement is still correct. For example: If Jan drives faster than 50 (A), Jan will be fined (B). If A and B are both false, so Jan does not drive faster than 50 and Jan is not fined, the claim is still true.

Modus ponens

Suppose: "if P then Q" = 1 and P = 1 then Q must always be 1. This is called the mode ponens, also called the wise or cut-off rule.

Modus Tollens

Suppose: "if P then Q" = 1 and Q = 0, then P must also be 0. This is also called the uplifting wise. The tollens mode is used for falsification.

Examples of reasoning:

Valid: Modus Ponens

• [1] If A, then B [1]

• If your rabbit eats wolfberries, it will get sick.

• [2] A [2] Your rabbit eats wolfberries.

• [3] So: B [3] So: he gets sick.

Valid: Modus Tollens

• [1] If A, then B [1]

• If your rabbit eats wolfberries, it will get sick

• [2] Not B [2] Your rabbit is not sick

• [3] So: Not A [3] So he did not eat any wolf-cherry.

Invalid: Confirming the consequent

• [1] If A, then B [1]

• If your rabbit eats wolfberries, it will get sick.

• [2] B [2] Your rabbit is sick.

• [3] So: A [3] So: he ate wolfberries.

Invalid: Denial of the antecedent

• [1] If A, then B [1]

• If your rabbit eats wolfberries, it will get sick.

• [2] Not A [2] Your rabbit does not eat wolfers

• [3] So: Not B [3] So: he doesn't get sick.

With the two invalid variants, A (the 'if' part) is seen as a necessary condition, while this is a sufficient condition. For example, look at the invalid confirmation of the consequences: there may be all sorts of other reasons why your rabbit is sick.

The propositions in a compound proposition are connected by conjunctions. These conjunctions are and, or and if ... then. The negation ('not') is also counted under the conjunctions.

Each conjunction or negation has a truth table that shows how the truth of a compound proposition can be derived from the sub-propositions. The truth table shows what the conjunction does when the two propositions are put together. Here, "true" = 1 and "false" = 0. For example: If "A" is true (1) and "B" is also (1), "A and B" is true (1). If "A" is true (1) and "B" is not (0), "A and B" is not true (0).

4. A fallacy is a proposition that cannot be verified on the basis of an argument, because this argument is missing, or because the argument does not apply to the proposition.

Suppose: "If P then Q" = 1 and Q = 1 then P can be both 1 and 0. You can see this in the first and second to last line of the truth table. Verification therefore provides no certainty and is also referred to as the fallacy of the consequence (fallacy of the consequent). Verification is used in testing and accepting hypotheses, but according to the proposition logic therefore gives no certainty. The hypothesis is correct with regard to the observation but can also be caused by something completely different.

5. Conjunction, negation, conditional and disjunction.
6. Such an analysis shows the truth values ​​of a general claim based on the truth values ​​of smaller parts of the claim.
7. We can determine whether an argument is valid based on a truth table.
8. Deduction is a useful means of proving, in particular, that an argument is valid instead of an argument being invalid