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This manual is for GNU Aris, the logical proof program.
Copyright (C) 2012 Ian Dunn
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.
This manual is for GNU Aris, the logical proof program.
This is edition 0.1, for aris version 0.1
This manual is for GNU Aris, a sequential proof program, designed to assist anyone interested in solving logical proofs. Aris supports both propositional and predicate logic, as well as Boolean algebra and arithmetical logic in the form of abstract sequences. It uses a predefined set of both inference and equivalence rules, however gives the user options to use older proofs as lemmas, including Isabelle's Isar proofs.
This chapter describes the basic usage of GNU Aris.
When Aris is loaded up, you will see a few things. You will see the rules window and a proof window. The rules window contains the different rules. A rule will not be selected if a premise is in focus.
Aris has several connectives (see Connectives). When the keyboard command for the desired connective is activated, the desired connective will be inserted at the current cursor point, overwriting selected text.
Hitting Ctrl+J adds a conclusion to the proof. A conlusion is always added after the current line, or, if the line is a premise, then the conclusion is added after the last premise. If the current line is a conclusion, then it is highlighted in cyan. Hitting Ctrl+P adds a premise to the proof. A premise will always be added after the last premise. Pressing Ctrl+B adds a subproof to the proof. When in a subproof, a conclusion will always be added within the subproof. The first line of a subproof does not require a rule, but instead acts as a premise. Pressing Ctrl+D ends the current subproof. This creates a new conclusion just after the subproof.
Holing CTRL, and left-clicking on a sentence will select a sentence as a reference sentence. The current line's reference sentences are highlighted in violet. Only a line before the current line can be selected as a reference. In addition, if the sentence is in a different subproof than the current line, then the sentence can not be selected as a reference. Each rule requires a specific amount of references (see Rules Index).
Holding SHIFT and left-clicking on a sentence will select the sentence. The sentence will be highlighted in red-orange. Multiple sentences can be selected this way, however when another action is taken, all of them will be de-selected. Pressing CTRL+K will kill (cut) the selected lines, and CTRL+G will copy the selected lines. If no lines are selected, then the current line will be used.
The rules are divided into five categories: Inference, Equivalence, Predicate, Boolean, and Miscellaneous.
The premises of any of these rules can be in any order.
One of the basic rules of logic, modus ponens say that ‘if P happens, then Q must happen. P happened, so Q must happen’.
For example, if it is known that ‘If the dog begins to bark, then someone is at the door’, and it is also known that ‘the dog has begun to bark’, then modus ponens says that ‘someone must be at the door’.
Modus Ponens requires exactly two references.
What addition says is that something is already known, so it must be true that that something or something else, or something else, etc. must also be true.
For example, if it is known that ‘The sky is blue’, then addition says that it can be inferred that ‘The sky is blue, or the sky is yellow, or the sky is pink’, since only one of those statements has to be true.
Addition requires exactly one reference.
Simplification says that if it is known that P and Q and R, etc. is known to be true, then P is true.
For instance, if it is known that ‘It is cloudy, and it is raining’, then simplification allows the inference of ‘It is cloudy’ and ‘It is raining’.
Simplification requires exactly one reference.
What conjunction is saying is the exact opposite of simplification. If P is known, and Q is known, and R is know, etc. then P and Q and R, etc. is also known.
Take for example, that it is known that ‘I don't like green eggs and ham’, and ‘I would not eat them in a house’, and ‘I would not eat them with a mouse’. Conjunction allows us to infer that ‘I don't like green egss and ham, and I would not eat them in a house, and I would not eat them with a mouse.’.
Conjunction requires at least two references.
Also referred to as the chain rule, hypothetical syllogism states that if one knows that 'if P then Q', and 'if R then S', then one can infer 'if P then S'. For example, if it is known ‘if it is raining, then it is cloudy’, and ‘if it is cloudy, then it is not sunny’, and ‘if it is not sunny, then it is cold’, then hypothetical syllogism allows us to infer that ‘if it is raining, then it is cold’. This works with any number of conditional statements, as long as they all follow this pattern.
Hypothetical Syllogism requires at least two references.
Disjunctive syllogism is commonly used when disjunctions are present. It claims that if one knows that 'P or Q or R', and 'P is false', and 'R is false', then Q must be true. This works with any number of disjuncts.
A law of logic, excluded middle asserts that something is either true, or it is not true.
Excluded middle requires zero references.
Constructive Dilemma requires at least three references.
Equivalence rules operate on any valid part of the sentence, and work both ways. Each equivalence rule requires one reference.
Implication uses the definition of the conditional. It is also valid to claim something such as -(-P v Q) v (-R v S) <=> (P → Q) → (R → S), because implication is recursive.
DeMorgan's Laws.
A note to users, typically association is used as P ^ (Q ^ R) <=> (P ^ Q) ^ R. While Aris will allow you to prove that this is equivalent, association allows the removal of one pair of parentheses at a time. (P ^ Q) ^ (R ^ S) <=> P ^ Q ^ R ^ S is also valid in Aris, because association allows recursion, but only when removing several sets of parentheses or adding several sets of parentheses.
Just like addition and multiplication, conjunctions and disjunctions are commutative. This of course means that ‘I would like some pie and I would like some cake’ is the same as saying ‘I would like some cake and I would like some pie’.
Idempotence claims that ‘I like blue and I like blue’ is the same as saying ‘I like blue’.
Equivalence uses the definition of the biconditional. Claiming that ‘P if and only if Q’ is exactly the same as claiming ‘if P then Q’ and ‘if Q then P’. Equivalence is the only rule that works with biconditionals explicitly, and is thus used any time a biconditional is seen.
You probably learned in english class that saying ‘I would not like to disagree’ is the same thing as saying ‘I would like to agree’. That's what double negation claims.
For the convenience of the user, the equivalence rules work recursively. For example
This is an example of using implication recursively. Recursion only works if the rule is being used the same way. For example, removing multiple parentheses with association is fine, however adding and removing parentheses with association is not.
Commutatvitity and idempotence work differently than the others when it comes to recursion. If commutativity is applied to a connective, then no parts of that connective, or parts of those parts, and so on, can be used in commutativity. However, other parts from the sentence can be rearranged. The same goes for idempotence.
The predicate rules are the rules that work specifically with predicate logic.
Universal Generalization claims that if a property ‘P’ is true for some arbitrary object, then it is true for all objects. A symbol is arbitrary if nothing is known about, or rather if it was not introduced through a premise or using existential instantiation.
Universal Generalization claims that if a property ‘P’ is true for all objects, then it must be true for an object ‘a’.
Existential Generalization claims that if ‘P’ is true for some object, then there exists an object for which ‘P’ is true.
Existential Instantiation claims that if there exists an object for which property ‘P’ is true, then it can be claimed that some unused object has this property. In this case, ‘a’ becomes a placeholder for the object.
Bound Variable allows the user to substitute any bound variable for another bound variable, given that the second bound variable does not appear anywhere in the scope of the quantifier of the first bound variable. For example, if it is known that Vx(Vy(P(x) ^ P(y))), an invalid use of bound variable would be to state that Vx(Vx(P(x) ^ P(x))).
Identity asserts that any variable is identical to itself.
Free variable allows the user to substitute a free variable for another free variable, given that the two are identical.
Aris can be set to use 'boolean mode', a mode used for boolean algebra. In boolean mode, only equivalence rules and boolean rules can be used.
This handy little rule allows one to use proofs one has already done. The premises don't have to match exactly, but they must be of the same form. Aris will check for each symbol it recognizes (connectives, quantifiers, parentheses, comma, and identity). These symbols must match exactly. Aris will then check that the sentences match the correct form, or rather that they appear in the correct order.
For example, if you already did a proof of the form:
And want to reuse it, then your reference sentences must be in the form ‘A’ <-> ‘B’, and ‘A’. They do not have to be in that order, however. Then, your conclusion must be the second half of the biconditional.
This is where Isar interoperability comes in. Instead of selecting a previous Aris proof, a .thy file can be used. Aris will attempt to translate it into a form that it can use, using most of the keywords as references, and the lemmas and theorems as goals. These are the sentences that can be proved. For more information, see Isabelle/Isar.
Given a subproof with premise 'P' and conclusion (the LAST sentence) 'Q', one can infer from subproof 'P → Q'.
This introduces a new sequence given a function. The sequence introduced this way must not have been used, and the final argument of the given function must be the bound variable of the sentence.
This rule is how Aris implements mathematical induction. ‘P(z(x))’ is the base case, and the inductive step is ‘P(x)’ → ‘P(s(x))’.
Sequence Logic, often abbreviated 'seqlog', is an alternative arithmetical representation system from the standard Peano Axioms.
Seqlog uses the symbols 's' (the sucessor function), 'z' (the zero function), 'n' (the sequence of natural numbers), 'v' (the value function), '0' (the number zero), and '\0' (null object).
Sequence Logic, often abbreviated 'seqlog', uses the following six axioms:
The first axiom states that no sucessor is the zero object, or, to put it differently, that the zero object is the first object. The second axiom states that no two different objects have the same sucessor. Using these two axioms, a 'Universal Sequence' can be defined, in a way similar to how the Peano Axioms define the natural numbers. The third axiom is the definition of a sequence, stating that the value under a given sequence 'S' of every object 'x' can be determined by a function. The rule 'sq' introduces such a sequence (see sq).
The fourth axiom defines zero, and introduces the natural numbers. It states that the first element in the natural numbers is zero. The fifth axiom defines the null object, and states that the natural numbers are defined on all inputs. The sixth axiom states that the natural numbers are one-to-one, or that if two numbers are equal, then the objects that they represent are equal.
Mathematical induction requires a base case, and an inductive step. In Aris, this is used in conjunction (ha see cn) with seqlog. For seqlog, the induction scheme is:
In addition to everything else Aris can do, Aris can also use other proofs from other systems with the lemma rule (see lm).
Aris will scan an Isar proof, which is a proof done using Isabelle, and look for certain keywords.
Standard definition of a function in seqlog.
Lemmas and theorems are treated the same. Lemmas end up as the goals of the proofs that Aris creates, and are the actual sentences that can be deduced. It takes the 'if-then' form of each lemma.