[ΛΟΓΟΣ] De Logica Propositionali

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[ΛΟΓΟΣ] De Logica Propositionali

Postby Gaius Florius Lupus » Sat Jul 01, 2017 11:31 pm

Salvete, amici!

This time I would like to present another approach to deductive reasoning. The Stoics developed their own system of logic, which quite different from Aristotle's syllogistic logic, Propositional Logic. It was the Stoic philosopher Chrysippus of Soli, the third scholarch of the Stoa after Zeno of Citium and Cleanthes, who developed the first consistent system of propositional logic. This form of logic might look much more familiar to people of modern times, because it is used in programming languages.
It needs to be clarified that there are not two different logics in the world, which come to different conclusions or contradict each other. Of course there is only one logic that underlies the world, but there are different ways to put it in language or formulas. Aristotle's syllogistic logic was one, propositional logic is another. Logical problems can be solved by using either system, but sometimes one is easier than the other for a particular problem.

The symbols that I use in the following are not the original symbols of Chrysippus, but follow modern standardization. Many of the inference rules were also developed after Chrysippus. Nevertheless Chrysippus is known as the second founder of the Stoa (with Zeno being the first), which shows his importance. He brought a system into Zeno's teachings. In logic he developed propositional logic, in ethics he taught that unruly emotions (passions) should be eliminated, because they cloud one's rational judgement.

Propositions

Propositions are thought to be the atoms of logic. Atomic propositions are statements, which cannot be broken down into further partial statements without a loss of meaning.
Propositional logic is about valid inference rules between contingent propositions. This means that logical inference must preserve the truth value of the premises in the conclusion. If the premises are true, then the conclusion must also be true.

Symbols

Proposition: p, q, r, s
True V (verum)
False F (falsum)
Operators in order of precedence
Negation: ¬ not
Conjunction: ∧ and
Exclusive Disjunction: ⊕ either... or...
(Inclusive) Disjunction: ∨ or
Conditional: ⇒ if... then...
Biconditional: ≡ equivalent

The conditional operator ⇒ can be interpreted in several ways.

  • syllogistic
  • causal
  • conditional (logical consequence, entailment)
  • material implication

Understanding the meaning of the conditional operator is the essence of propositional logic.
p ⇒ q
p: antecendent
q: consequent
If p then q means the following.
p is a sufficient condition for q
q is a necessary condition for p

We find this operator also in programming languages like the IF - THEN command in BASIC. We also have AND (conjunctions) and OR (disjunction). This shows that propositional logic is the foundation of any kind of data processing.

Exempli gratia:
If Caesar crosses the Rubicon, it will trigger a civil war in Rome.
p = Caesar crosses the Rubicon
q = A civil war will be triggered in Rome.
Caesar crossing the Rubicon is a sufficient condition for the civil war. This means a civil war is always the consequence of Caesar crossing the Rubicon.
The civil war is a necessary condition for Caesar having crossed the Rubicon. This means if there is no civil war, then we can be sure that Caesar cannot have crossed the Rubicon. However a civil war alone does not mean that Caesar has crossed the Rubicon. It could also have been triggered by another event, like an uprising of the populares faction in Rome, because the optimates had declared Caesar an enemy of Rome.

Truth Table for the Logical Operators

truth table.jpg
truth table.jpg (14.06 KiB) Viewed 917 times


The table above shows what the logical operators mean for the truth value of the propositions.
Please note that the truth values for the conditional operator is somehow counter-intuitive.
Even if p is false, q can still be true.
Exempli gratia:
If Cicero joins the conspiracy, Caesar will be murdered.
This can still be a valid inference, even though we know that Cicero was not part of the conspiracy against Caesar. The inference would only be invalid, if Caesar were not killed, although Cicero had joined the conspiracy.

In the following I will provide a list of the most important inference rules of propositional logic.

Definitions

Tautology
p ≡ (p ∨ p)
p ⊕ ¬p
p is equivalent to p is true or p is true
Either p or not p is true.
Explanation:
A tautology is a syllogism, which is always true independent from its values.
Law of Identity (Everything is identical with itself.)
Law of the Excluded Middle (Something is either true or false.)

Conjunction
p ∧ q ≡ (p ∧ q)
p is true and q is true is equivalent to p and q being true conjointly.
Explanation:
For a conjunction of two propositions to be true, both propositions must be true.

Inclusive Disjunction
p ∨ q ≡ (p ∨ q)
p is true or q is true is equivalent to p or q are true.
Explanation:
For a disjunction of two propositions to be true, at least one of the two propositions must be true.

Exclusive Disjunction (Mutual Exclusiveness)
p ⊕ q ≡ (p ∧ ¬q) ∨ (¬p ∧ q)
Either p is true or q is true is equivalent to p is true and q is not true or p is not true and q is true.
Explanation:
For an exclusive disjunction of two propositions to be true, one of the two propositions must be true and the other one false.

Implication
(p ⇒ q) ≡ ¬p ∨ q
If p then q is equivalent to either not p or q.
Explanation: (material implication)
The statement that one condition implies a second one means that either the implied condition is true or the first statement is false. It is only false, if the first statement is true and the implication is false.
The antecedent implies the consequent.

Equivalence (Biconditional)
(p ≡ q) ≡ [(p ⇒ q) ∧ (q ⇒ p)]
(p ≡ q) ≡ [(p ∧ q) ∨ (¬p ∧ ¬q)]
p is equivalent to q means if p is true then q is true and if q is true then p is true.
p is equivalent to q means either p is true and q is true or p is false and q is false.
Explanation:
Two propositions are equivalent means that the first proposition implies the second one and the second proposition implies the first one. (bidirectional material implication)
Two propositions are equivalent also means that either both of them are true or both of them are false.
Both propositions are sufficient and necessary conditions of each other.

This is where the word "only" comes into the picture. Equivalence can also be expressed as "if and only if". In C. Curtius Philo's example this would look like the following:
ONLY humans are rational.
Therefore pandas are not rational.
If and only if someone is a human, then he is rational. This means that the class "humans" is equivalent with the class "rational beings". If we assume that pandas are not humans, then pandas are also not rational, since both words would be equivalent.
(human ≡ rational being) ⇒ [(panda ⇒ ¬human) ⇒ (panda ⇒ ¬rational being)]
This is valid, because equivalence means here that the propositions "it is a human" and "it is a rational being" can be substituted for each other.

Inference Rules

Double Negation
p ≡ ¬(¬p)
p is equivalent to the negative of not p.
Explanation:
A proposition is true means it is not false.

Simplification
p ∧ q ⇒ p
If p and q are true, then p is true.
Explanation: (conditional interpretation)
If a conjunction of two conditions is true, each of them is also true alone.

Modus Ponens
[(p ⇒ q) ∧ p] ⇒ q
If p then q and p therefore q.
Explanation: (causal interpretation)
If one event leads to another one and the first one has taken place, then the latter one will occur too.
The first event is sufficient condition for the latter one.

Modus Tollens
[(p ⇒ q) ∧ ¬q] ⇒ ¬p
If p then q and not q, therefore not p.
Explanation: (causal interpretation)
If one event leads to another one and the latter one does not take place, then the first one has not occurred.
The latter event is necessary condition for the first one.

Modus Ponendo Tollens
[(p ⊕ q) ∧ p] ⇒ ¬q
If either p is true or q is true and p is true, then q is not true.
Explanation: (conditional interpretation)
If one of two mutually exclusive propositions is true, the other one must be false.

Modus Tollendo Ponens (Disjunctive Syllogism, Elimination)
[(p ∨ q) ∧ ¬p] ⇒ q
If p or q is true and p is not true, then q is true.
Explanation: (conditional interpretation)
If one proposition of a disjunction is false, the other one must be true.

Transitivity (Hypothetical Syllogism)
[(p ⇒ q) ∧ (q ⇒ r)] ⇒ (p ⇒ r)
If p then q and if p then r, therefore if p then r.
Explanation: (syllogistic interpretation)
AAA-1 syllogism: All P are Q and all Q are R, therefore all P are R.
This is where syllogistic logic and propositional logic meet. We can express any BARBARA (AAA-1) syllogism also as a transitivity in propositional logic.
Exempli gratia:
Major Premise: All humans are mortal.
Minor Premise: Aristotle is a human.
Conclusion: Aristotle is mortal.
p: Aristotle
q: human
r: mortal
IF Aristotle is human AND humans are mortal THEN Aristotle is mortal.
The same underlying logical principle just expressed in two different systems.

Constructive Dilemma
[(p ⇒ q) ∧ (r ⇒ s)] ∧ (p ∨ r) ⇒ (q ∨ s)
If p then q and if r then s and either p or r, therefore q or s.
Explanation: (causal interpretation)
If two initial events lead to distinct results and at least one of these initial events has occurred, then at least one of the results must have occurred too.

Destructive Dilemma

[(p ⇒ q) ∧ (r ⇒ s)] ∧ (¬q ∨ ¬s) ⇒ (¬p ∨ ¬r)
If p then q and if r then s and not q or not s, therefore not p or not r.
Explanation: (causal interpretation)
If two initial events lead to distinct results and at least one of the results has not occurred, then at least one of the initial events has not occurred either.

Composition
[(p ⇒ q) ∧ (p ⇒ r)] ⇒ (p ⇒ q ∧ r)
If p then q and if p then r, therefore if p then q and r.
Explanation: (conditional interpretation)
If two consequences follow from one condition, then if the condition is true, both consequences are certain.

De Morgan's Theorems
¬(p ∧ q) ≡ (¬p ∨ ¬q)
¬(p ∨ q) ≡ (¬p ∧ ¬q)
The negation of p and q is equivalent to either not p or not q.
The negation of p or q is equivalent to not p and not q.
Explanation:
The negation of a conjunction of two propositions means that at least one proposition is false.
The negation of a disjunction of two propositions means that both propositions are false.

Commutation
(p ∧ q) ≡ (q ∧ p)
(p ∨ q) ≡ (q ∨ p)
p and q is equivalent to q and p.
p or q is equivalent to q or p.
Explanation:
The order of a disjunction is irrelevant.
The order of a conjunction is irrelevant.

Association
[p ∨ (q ∨ r)] ≡ [(p ∨ q) ∨ r]
[p ∧ (q ∧ r)] ≡ [(p ∧ q) ∧ r]
p or (q or r) is equivalent to (p or q) or r.
p and (q and r) is equivalent to (p and q) and r.
Explanation:
In a conjunction or disjunction of more than two elements their order is irrelevant.

Distribution
[p ∧ (q ∨ r)] ≡ [(p ∧ q) ∨ (p ∧ r)]
[p ∨ (q ∧ r)] ≡ [(p ∨ q) ∧ (p ∨ r)]
p and (q or r) is equivalent to (p and q) or (p and r).
p or (q and r) is equivalent to (p or q) and (p or r).
Explanation:
Distribution is a rule of replacement for use in formal logic.
A conjunction of a proposition and a disjunction of two other propositions can be replaced by a disjunction of two conjunctions, each containing the first proposition and one of the other two propositions.
A disjunction of a proposition and a conjunction of two other propositions can be replaced by a conjunction of two disjunctions, each containing the first proposition and one of the other two propositions.

Transposition
(p ⇒ q) ≡ (¬q ⇒ ¬p)
If p then q is equivalent to if not q then not p.
Explanation: (causal interpretation)
If one event leads sufficiently to a result and the result does not occur, then the initial event has not occurred either.

Exportation
[(p ∧ q) ⇒ r] ≡ [p ⇒ (q ⇒ r)]
If p and q are true, then r is true is equivalent to the statement that if p is true, then q implies r.
Explanation: (causal interpretation)
If two causes are only together sufficient for an event to occur, then only if the first cause occurs, the second one can lead sufficiently to the event.

Importation
[p ⇒ (q ⇒ r)] ≡ [(p ∧ q) ⇒ r]
If p is true, then q implies r, is equivalent to if p and q are true, then r is true.
Explanation:
Reversal of exportation.

The validity of these rules means that the conclusions are a logical necessity (entailment) of the premises independent from their truth value. This is important, since the truth value of the premises may not always be known with certainty. It must therefore be remembered that the uncertainty of the premises is inherited by the conclusion. If all premises are known to be true and the conclusion is valid, then this is called soundness.

Propositional Fallacies

If the above rules are not properly applied, then it will result in a non sequitur, a logical fallacy. There are four typical formal fallacies that are frequently committed in propositional logic.

  • Affirming a Disjunct - The conclusion that one proposition of a disjunction must be false because the other proposition is true: p ∨ q; p is true; therefore q is false.
  • Affirming the Consequent - The claim that the antecedent in an indicative conditional is true because the consequent is true: p ⇒ q; q is true, therefore p is true. (Incorrect interpretation of Modus Ponens.)
  • Conjunction Fallacy - The assumption that an outcome simultaneously satisfying multiple conditions is more probable than an outcome satisfying a single one of them.
  • Denying the Antecedent - The claim that the consequent in an indicative conditional is false because the antecedent is false: p ⇒ q; ¬p ⇒ ¬q; p is false therefore q is false. (Incorrect interpretation of Modus Tollens.)

It would be absurd to expect that anyone would now learn all these rules mentioned in this post. It is probably even unrealistic to expect that anybody reads through all this text. But I hope I could demonstrate that logic operates with the same precision as mathematics. It has clearly defined rules, rules that are universally valid, not only for the human society, they are also used when programming computers. Because this is how any thinking process works, including the human brain (at least in the left hemisphere).
Those who are interested in the topic will find a Latin summary as PDF attachment to this post., where the rules can be looked up, if needed.

I think this is enough about deductive reasoning. We have seen that there are two important systems of deduction known already in classic times, the one of Aristotle (syllogistic logic) and the one of the Stoics (propositional logic), the latter one still being very important today in IT (information technology). However this does not conclude the topic of logic, but it was probably the most tedious and boring part.
If there is any doubt about the universal validity of propositional logic, feel free to provide an example where it fails in the real world. I would be happy to discuss it.

Keywords: atomic propositions; coherent propositions; sufficient condition; necessary condition; logical entailment/implication; truth value; disjunction; conjunction; equivalence

Optime valete!
C. Florius Lupus
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Re: [ΛΟΓΟΣ] De Logica Propositionali

Postby Publius Sextius Laevus » Sun Jul 02, 2017 5:06 pm

Salvete Lupe, et Omnes Logici Rationabilesque

How would the if-then-else if-else:
if (p) then
q=V
else if (r) then
s=V
else
t=V
end if

be expressed?

(p=>q) and ((not p and r)=>s) and ((not p and not r)=>t)

noting that q=V does not necessarily mean p=V.

Valete
Laevus
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Re: [ΛΟΓΟΣ] De Logica Propositionali

Postby Gaius Florius Lupus » Mon Jul 03, 2017 9:32 am

Salve Laeve,

You are using modern programming language, which is indeed also a form of deductive logic, mostly based on propositional logic, but still with a different syntax. In propositional logic there is no ELSE, ENDIF is indicated by brackets, and we deal strictly with propositions, not terms.

In your case "q=V", "s=V" and "t=V" are the propositions, q,s,t and V are terms.
We would have to redefine them as propositions the following way and replacing the lower case letters with upper case letters to indicate terms and vice versa:
q ≡ Q=V
s ≡ S= V
t ≡ T= V
Terms are now upper case letters and propositions lower case letters.

Then we can write it as following:
[p ⇒ q] ∧ [¬p ⇒ ((r ⇒ s) ∧ (¬r ⇒ t))]

Vale!
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