Robert C. Koons and David V. Newman

© 1995

at the University of Texas at Austin.

Unauthorized use or duplication is prohibited.

- Introduction
- Getting Started
- Starting Socrates
- The Menus: An Overview
- Files: Opening, Saving, Closing
- Navigating through a Table
- Editing: Cut & Paste, Undo & Redo
- Logical Moves: The Entry, SC, PC, and MC Menus
- Annotations: Editing and Removing
- Appearance: The Font and Size Menus
- Sentential Logic
- Entering premises and conclusions
- Entering sentences of sentential logic
- The structure of the table: left and right sides
- The intuitive meanings of the connectives
- Inference moves in sentential logic
- Closing paths in sentential logic
- By contradiction
- By bridging open sub-tables

- Closing the table in sentential logic
- Open, complete tables in sentential logic
- Predicate Logic (with Identity)
- Entering sentences of predicate logic
- The intuitive meanings of the quantifiers
- Inference moves in predicate logic
- Closing paths and closing the table in predicate logic
- Identity
- Performing substitutions
- Use of the law of excluded middle
- Closing paths through denied self-identity

- Open, complete tables in predicate logic
- Modal Logic
- Entering sentences of modal logic
- Intuitive meanings of the box and diamond
- Inference rules in modal logic
- Closing paths and closing the table in modal logic
- Identity in modal logic

- Modal Conditional Logic
- Entering sentences of modal conditional logic
- The intuitive meanings of the modal conditionals
- Inference rules in modal conditional logic
- Closing paths and closing the table in conditional modal logic
- Complete, open tables

- Defeasible Logic
- Closing paths by anomaly elimination
- Varieties of defeasible logics

- Appendix: Keyboard Shortcuts

Socrates is a software application for the Macintosh computer developed at the University of Texas at Austin. It is intended to assist the learning of formal logic. It has been designed to be compatible with the semantic table approach taken by Jaakko Hintikka and James Bachman in their introductory text,

Socrates runs on the Apple Macintosh computer (Mac Plus and higher using System 6.0.5 or higher). To use Socrates, you should be familiar with the basic operation of the Macintosh, including clicking and dragging with the mouse, and using menus. The Socrates application includes the following files and folders:

- The Socrates application itself (whose icon is illustrated in Figure 1).
- A font suitcase containing the Terlingua, Salt Flat and Pittsburgh fonts.

Figure 1.

The fonts in the suitcase should be installed on the Macintosh that you are using. If you are using System 6, you can use the DA/Font mover to add the fonts to the System file, or you can use some other font utility to install these fonts. If you are using System 7, you may simply drag the fonts into the Fonts folder within the System Folder.

Find the icon for Socrates – a picture of a hand holding a pencil to a table divided into columns (illustrated above in Figure 1). Double-click on the icon, or click on the icon and select “Open” from the file menu. After a moment, the window for Socrates will appear on your screen. When it starts, Socrates will open an empty document, as shown below in Figure 2.

At the top of the screen, you will see the usual Apple, File, and Edit menus as well as six new menus: Entry, SC, PC, MC, Font, and FontSize. The menu bar is illustrated immediately below in Figure 2. You can pull down each menu to see what it contains.

Figure 2.

Most of the commands in the File menu are standard Macintosh File menu commands.

- New - Creates a new document.
- Open - Opens an existing document.
- Save - Saves the topmost document to a file. You may not overwrite any files that were not created by the Socrates program.
- Save As - Saves the topmost document to a file, prompting for a new file name.
- Save As MacDraw - Saves the topmost document as a PICT document, suitable for use with most drawing programs (e.g. MacDraw, SuperPaint, etc.).
- Close - Closes the topmost document. If you have changed the contents of the document, you will be asked if you want to save the document before closing it.
- Quit - Quits the application. You will be asked if you wish to save any changes to open documents.

One can navigate through a Socrates table in one of two ways:

- by means of the mouse, or
- by means of the keyboard arrows.

Most of the Edit menu commands are standard Macintosh Edit menu commands. However, the Undo and Redo commands implement infinite undo and redo.

- Undo - Undoes the most recent change made to the topmost document. Once a change is undone, subsequent use of Undo undoes the changes made previous to the most recent change, in reverse chronological order.
- Redo - Redoes the most recently undone change to the topmost document. Once a change is redone, subsequent use of Redo redoes the previous most recent undone changes, in reverse chronological order. However, when a new command is selected from the Move menu, Redo cannot be used.
- Copy - Copies the selected formula, or, if no formula is selected, copies the entire table as a picture that can be pasted into another application.
- Cut - Not enabled except for use with a desk accessory. However, the command-key equivalent (CMD-X) works in all dialogs where text is entered.
- Paste - Not enabled except for use with a desk accessory. However, the command-key equivalent (CMD-V)works in all dialogs where text is entered.
- Show Clipboard - Shows the contents of the clipboard in a window.

These menus allow you to build a semantic tableaux and evaluate the argument that it represents for the logical propertie of validity and other related properties. The commands in the SC, PC, and MC menus change the contents of the topmost document. These menus deal with the Sentential Calculus, the Predicate Calculus, and the Modal Calculus respectively. Most commands in the SC and PC menus are described in sections 2 and 3 below. In general, they allow you to simplify logical formulae or alter them so that they can be simplified by other commands. Those commands in the MC menu are described in later sections (for the most part, in sections 4, 5 and 6).The Entry menu contains the following items, which are discussed in the indicated sections:

Command | Section |
---|---|

Add Premise | 2.1 |

Add Conclusion | 2.1 |

Close Path | 2.6, 3.5.3, 4.4, 6.1 |

Close Table | 2.7, 4.4 |

Edit Title | 1.6 |

Edit Comment | 1.6 |

Remove Title | 1.6 |

Remove Comment | 1.6 |

List Unbridged Subtables | 2.7 |

The SC menu represents the logical moves that may be made on formulas in the language of the Sentential Calculus and the languages of the Predicate and Modal Calculi. It consists of the following items:

Command | Section |
---|---|

Decompose Conjunction | 2.5 |

Decompose Disjunction | 2.5 |

Decompose Conditional | 2.5 |

Decompose Biconditional | 2.5 |

Remove Double Denial | 2.5 |

Alter Denied Conjunction | 2.5 |

Alter Denied Disjunction | 2.5 |

Alter Denied Conditional | 2.5 |

Alter Denied Biconditional | 2.5 |

Modus Ponens | 2.5 |

Law Of Excluded Middle | 3.5.2 |

Moves that may be performed on arguments represented in the language of the Predicate Calculus or the Modal Calculus, but not the Sentential Calculus are in the PC menu. The PC menu consists of the following items:

Command | Section |
---|---|

Perform Substitution | 3.5.1 |

Instantiate Universal Variable | 3.3 |

Instantiate Existential Variable | 3.3 |

Alter Denied Quantifier | 3.3 |

The commands in the MC menu correspond to moves that apply to arguments expressed in the language of the Modal Calculus (including nonmonotonic expressions). These moves do not apply to the Sentential or the Predicate Calculi. The MC menu consists of the following items:

Command | Section |
---|---|

Decompose Necessity | 4.3 |

Decompose Possibility | 4.3 |

Decompose Box-Arrow | 5.3 |

Decompose Diamond-Arrow | 5.3 |

Alter Denied Modal | 4.3 |

Alter Denied Nonmonotonic | 5.3 |

Box-Definition | 5.3 |

Diamond-Definition | 5.3 |

Modal Law Of Excluded Middle | 4.5 |

Socrates documents may be annotated with a title and a comment. The title appears at the top of the document, above the logical table. The comment appears at the bottom of the document, below the table. Four menu items in the Entry menu allow you to change the title and comment of a document.

- Edit Title - This command allows you to enter a new title for the document or edit an existing title. Within the dialog box, the standard Macintosh editing command keys work, and one may type a carriage return by typing option-return. Typing a return closes the dialog box, saving the title (this is the same as clicking the “OK” button with the mouse). Typing command-period (CMD-.) closes the dialog box without saving the title (this is the same as clicking the “Cancel” button).
- Edit Comment - This command works just as the Edit Title command does, except that it allows you to edit the comment of a document rather than the title.
- Remove Title - This command removes the title from a document.
- Remove Comment - This command removes the comment from a document.

The appearance of a Socrates document may also be changed by changing the typeface and the size of the typeface used in the document. This is accomplished using the Font and FontSize menus, which change the font or font size of the entire document. The Font menu restricts you to those fonts that have the logical characters that Socrates requires organized in the way that Socrates requires.The FontSize menu allows you to directly select a number of standard font sizes, and it allows you to type in any other font size up to 1024 by using the Other… menu item. You can also change the font size up and down in one point increments by using the Larger and Smaller menu items. The current font size is indicated by a check mark beside that size, or by a number in parentheses next to the Other… menu item.

In order to use Socrates to analyze a piece of reasoning or an argument, you must enter the premises (the information available or assumed to be true) and the conclusion (the thesis or proposition which may or may not follow logically from the premises). To enter premises and conclusions, you must use the first two items on the Entry menu.

• Add Premise - Prompts for a formula, and adds it to the topmost document as a premise. The dialog in which the formula is entered is illustrated in Figure 3.

Figure 3.

The dialog includes four major elements. The formula entry area is the large box; the formula to be entered can be edited within this box. The buttons immediately below this area allow the user to enter logical characters into the formula at the insertion point; some of these characters are not easily accessible from the keyboard. The top row of characters and the first character on the bottom row are all that is required for problems in the Sentential and Predicate Calcului; the second row of characters are needed for analyses of arguments using modal and nonmonotonic logics. The special logical symbols for sentential logic can also be produced by the following key combinations:

Command | Key Combination |
---|---|

¬ | Option-shift-hyphen |

∨ | Option-v |

→ | Option-e, a |

↔ | Option-i, a |

• Add Conclusion - Prompts for a formula, and adds it to the topmost document as a conclusion. The dialog presented by this command is identical to that presented by the Add Premise menu item.

To add premises and conclusions successfully, you must enter grammatical formulas into the dialog box of the Add Premise or Add Conclusion commands. For that part of logic known as "sentential logic," the permissible or grammatical formulas for Socrates are governed by rules that codify and regularize the practice of the Hintikka/Bachman text. These rules can be summarized as follows:

- Any capital letter A, B, … Z, followed by any number of primes ('), is a well-formed sentence.
- If è and ê are well-formed sentences, so are: —è, (è&ê), (è_ê), (èáê), and (èâê).
- All of the well-formed sentences of sentential logic can be formed by repeated applications of rules 1 and 2.

Note that parentheses are added whenever one of the four "binary" connectives, &, _, á, and â, are used to combine two sentences into a new, compound sentence. This is the only time such parentheses can be added. Since Socrates possesses limited intelligence, it cannot guess what you really meant to say. Thus, you must be careful to follow the rules literally and precisely. You may not arbitrarily add or omit parentheses or other symbols. Thus, the following strings of characters are not well-formed: (—A) —(A)(A) ((A)_(B))A&B (A&B) á A In contrast, the following are well-formed sentences of sentential logic: ((A&B)_(BáA)) ——(—Aâ — (B_—A))————A (A_(A&(AâA))) There should always be exactly as many left parentheses in a sentence as there are right parentheses, and there should be exactly one pair of parentheses for each binary connective.

In Socrates, the main table is divided into two parts, a left-side, labeled "Premises and logical inferences", and a right-side, labeled "Conclusions". The two sides are divided by a double vertical line. In trying to discover whether a given logical inference or argument is correct, we will try to find out if it is possible for the premises to be all simultaneously true, while at least one of the conclusions is false. If this is possible, the argument is incorrect (or, at least, seriously incomplete). If this is impossible, the argument is correct (or "valid"). In Socrates, we try to construct a table that shows the argument is incorrect – if this proves to be impossible, then we can be sure that the argument is OK. Consequently, when we enter premises on the left-side of the table, we are in effect assuming that these premises are true. In contrast, when we enter a conclusion on the right-hand side, we are in effect assuming (for the sake of the analysis) that the conclusion is false. Thus, the very same sentence has quite different operational significance in Socrates, depending on whether it is entered on the left-hand side (the Truth side) or the right-hand side (the Falsity side). 2.4 The intuitive meanings of the connectives In sentential logic, we begin with simple statements (represented by capital letters), and we modify or combine sentences by means of five logical symbols. These symbols are called "logical connectives". In Socrates, these connectives are represented by the following five symbols: —, &, _, á, â. The first of these symbols, the hoe, is used to represent negation or denial. When attached to a sentence, it is essentially equivalent to adding the word 'not' to an English sentence, e.g., turning the sentence 'It is hot' into the sentence 'It is not hot'. The second symbol, the ampersand &, is used to represent the logical operation of conjunction. The sentence '(A&B)' means that both A and B are true. The wedge, _, represents the logical operation of disjunction. The sentence '(A_B)' means that either A or B or both are true. The arrow, á, represents what is known as the material conditional. It corresponds to the horseshoe (_) used in some other texts. The sentence '(AáB)' can mean that if A is true, then B is true. The material conditional corresponds to a special use of "if...then.." in English, a use that is quite common in mathematical contexts. The material conditional '(AáB)' is equivalent to the statement that either A is not true or B is true, i.e., that B is true unless A fails to be true. In other words, A is, in a mathematical sense, a sufficient condition – but not necessarily a necessary condition – for the truth of B. The sentence occurring between the first parenthesis and the horseshoe (to the left of the horseshoe) is called the antecedent of the conditional. The sentence occurring between the horseshoe and the right-most parenthesis is called the consequent of the conditional. The double-arrow, â, represents the biconditional. A biconditional is in effect the conjunction of two material conditionals, one reading from left to right and the other reading in the reverse direction. Thus, '(AâB)' is equivalent to the claim that both '(AáB)' and '(BáA)' are true. When a sentence of sentential logic is entered as a premise or for some other reason finds its way onto the left-hand side of the table, its operational significance on the table is identical to its intuitive meaning, since we are in this case assuming the sentence to be true. In contrast, when a sentence is added as a conclusion or otherwise finds its way onto the right-hand side of the table, we are assuming that the sentence is false, and the information we gain thereby is quite different from the ordinary meaning of the sentence. For example, if you enter 'A' as a premise, you are assuming that 'A' is true, while if you enter it is a conclusion, you are assuming exactly the reverse. Likewise, entering '—A' on the premise side commits you to assuming that '—A' is true and hence that 'A' is false, while entering '—A' on the conclusion side means that '—A' is false and so 'A' is true. Similarly, if a conjunction '(A&B)' is added to the conclusion side of the table, the information that is gained is disjunctive in force: if '(A&B)' is false, then either 'A' is false or 'B' is false, or both are false. If a disjunction '(A_B)' is added to the conclusion side, the information is essentially conjunctive: both 'A' and 'B' must be false. When a material conditional, say '(AáB)', is added to the premise side (the left-hand side), the information gained is disjunctive: either 'A' is false or 'B' is true. However, if the material conditional is added to the conclusion side, the information is conjunctive: 'A' must be true and 'B' must be false. A biconditional adds information that is simultaneously disjunctive and conjunctive, but the precise nature of this information depends on whether the biconditional occurs on the left or right side. If the biconditional '(AâB)' occurs on the premise side (and is assumed to be true), then either 'A' and 'B' are both true, or 'A' and B are both false. If the biconditional occurs on the conclusion side, then either 'A' is true and 'B' is false, or 'A' is false and 'B' is true. 2.5 Inference moves in sentential logic In Socrates, the process of logical analysis for sentential logic proceeds by unpacking the information that is implicit in compound sentences. One important skill to acquire is the ability to recognize the main connective of a sentence. If the first symbol on the left of a sentence is the hoe (—), then that negation sign is the main connective of the sentence. A sentence whose main connective is the negation sign is called a negation. The following sentences are negations: —A —(B_(C&D)) ———B ——(—Bá—(A_—D)) When the first symbol of a sentence is a left-parenthesis, then the main connective of the sentence is one of the binary connectives. After passing the first left parenthesis, look for a complete subsentence. Immediately following this subsentence will be one of the binary connectives – the main connective of the sentence – which in turn will be followed by a second complete sub-sentence and a final right-parenthesis. See if you can identify the main connectives of the following four sentences: ((BáC)&(D_E)) (D_(BâE))((C_A)áB) ((AáA)â(B&(Aá))) You should have recognized the main connectives as &, _, á, and â, respectively. The first sentence is called a conjunction, the second a disjunction, the third a conditional, and the fourth a biconditional. The two main sub-sentences of a conjunction are called its conjuncts. Likewise, the two main sub-sentences of a disjunction are called its disjuncts. Once you have recognized the main connective of a compound sentence on the table, you are ready to unpack that sentence by decomposing it. The relevant decomposition commands for the sentential calculus occur in the SC menu. A conjunction is decomposed by the Decompose Conjunction move, and so on for the other binary connectives. To apply one of these moves, use the mouse to click on a compound sentence, thereby selecting it, and then pull down the appropriate menu and release the mouse button when the appropriate move is hilighted. When a move is selected, new formulas will be added to the table in the paths of the table that contain the selected compound formula. The various decomposition moves are described below: • Decompose Conjunction - This command adds one or two new lines to a table depending on whether the selected formula is on the left or right side of the table. (Left ) If the selected formula is on the left side of the table, two new lines are added to the table, and thus two new formulas are added to each open path containing the selected formula. The first new line contains the first conjunct of the selected formula, and the second new line contains the second conjunct of the selected formula. This procedure embodies the idea that if a conjunction is true, both of the conjuncts are true. (Right) If the selected formula is on the right side of the table, a single line is added, and any open paths containing the selected formula are split into pairs of paths. The first conjunct of the selected formula is added to the first member of each pair of paths, and second conjunct of the selected formula is added to the second member of each pair of paths. This procedure embodies the idea that if a conjunction is false, then one or the other of the conjuncts must be false. • Decompose Disjunction - This command adds one or two new lines to the table depending on whether the selected formula is on the left or right side of the table. (Left) If the selected formula is on the left side of the table, a single line is added, and any open paths containing the selected formula are split into pairs of paths. The first disjunct of the selected formula is added to the first member of each pair of paths, and second disjunct of the selected formula is added to the second member of each pair of paths. This procedure embodies the idea that if a disjunction is true, then one of the two disjuncts must be true. (Right) If the selected formula is on the right side of the table, two new lines are added to the table, and thus two new formulas are added to each open path containing the selected formula. The first new line contains the first disjunct of the selected formula, and the second new line contains the second disjunct of the selected formula. This embodies the idea that if a disjunction is false, then both of the disjuncts are false. Note that the Decompose Disjunction command is similar to the Decompose Conjunction command with one major difference. Decompose Disjunction treats the left side of the table in the way that Decompose Conjunction treats the right side, and Decompose Disjunction treats the right side of the table in the way that Decompose Conjunction treats the left side. • Decompose Conditional - This command adds one or two new lines to the table depending on whether the selected formula is on the left or right side of the table. (Left) If the selected formula is on the left side of the table, a single line is added, and any open paths containing the selected formula are split into a pair of paths. The negation of the antecedent of the selected formula is added to the first member of each pair of paths, and the consequent of the selected formula is added to the second member of each pair of paths. This procedure is correct because whenever a conditional is true, either the antecedent is false, or the consequent is true (or both). (Right) If the selected formula is on the right side of the table, two new lines are added to the table, and thus two new formulas are added to each open path containing the selected formula. The first new line contains the negation of the antecedent of the selected formula, and the second new line contains the consequent of the selected formula. This is the correct procedure because whenever a material conditional is false, it must be the case that the antecedent is true and the conclusion is false. • Decompose Biconditional - This command adds two new lines to the table, and it splits the paths containing the selected formula depending on whether the selected formula is on the left or right side of the table. (Left) If the selected formula is on the left side of the table, this move adds two new lines to the table and splits the paths containing the selected formula. Both subformulas of the selected formula are added to the left member of each pair of paths, and the negations of both subformulas are added to the right member of each pair of paths. This is the right thing to do since a conditional is only true when either both subformulae are true or when both subformulae are false. (Right) If the selected formula is on the right side of the table, two lines are added and the paths containing the selected formula are split. In this case, each member of each pair of paths contains one subformula of the selected formula and the denial of the other subformula of the selected formula. This is the right thing to do since a conditional is false just in case the truth values of the two subformulae do not match. Note that the decomposition of a biconditional could have merely changed the biconditional into a conjunction of material conditionals. This would express the intuitive meaning of the biconditional described above (see Section 2.4). However, if this alternative form of the rule were used, subsequent use of the Decompose Conjunction and Decompose Conditional commands would have effectively the same results as the actual Decompose Biconditional command. Later, when you have learned about closing the paths of a table, you will be able to prove this for yourself by creating two tables, one containing a biconditional, and the other containing the conjunction of conditionals representing the same biconditional. When the main connective of a compound sentence is a negation, and the sentence to which the negation sign has been attached it itself a complex sentence (a negation, conjunction, disjunction, conditional or biconditional), it is possible to transform the sentence by one of the following five moves. • Remove Double Denial - This command adds a new line to the table. The formula on the new line is obtained by removing two initial negations from the selected formula. This is allowed since the negation of a negation is truth functionally equivalent to the original sentence. • Alter Denied Conjunction - This command adds a new line to the table. The new line contains a formula obtained from the selected formula by removing the initial negation, negating the conjuncts of the remaining formula, and changing the main connective of the remaining formula to a disjunction. This is an instance of what is sometimes called DeMorgan’s Law. • Alter Denied Disjunction - This command adds a new line to the table. The new line contains a formula obtained from the selected formula by removing the initial negation, negating the conjuncts of the remaining formula, and changing the main connective of the remaining formula to a conjunction. This is the second half of what is sometimes called DeMorgan’s Law. • Alter Denied Conditional - This command adds a new line to the table. The new line contains a formula obtained from the selected formula by removing the initial negation, negating the antecedent of the remaining formula, and changing the main connective of the remaining formula to a disjunction. This is effectively the equivalent of changing the conditional to its equivalent disjunction and then applying the Alter Denied Disjunction command. • Alter Denied Biconditional - This command adds a new line to the table. It allows the user to deal with a negated biconditional. The new formula added to the table by this command is a biconditional between the left subformula of the original formula and the negation of the right subformula of the original formula. • Modus Ponens - This command adds a new line to the table. It allows the user to decompose a conditional expression in a more compact fashion than the Decompose Conditional command. The new line added to the table contains the consequent of the selected conditional in each of the open paths containing the antecedent of the selected conditional. • Law of Excluded Middle - This command prompts the user for a formula, and adds a single new line to the table. All open paths containing the selected formula are split into two paths; the formula entered by the user is added to the left path in each pair of paths, and the negation of the formula entered by the user is added to the right path in each pair of paths. This is possible because we assume that any sentence is either true or false. Splitting the paths containing the selected formula and adding the formulas described above to the table is equivalent to adding a disjunction between the formula entered by the user and its negation to the assumptions, and then performing the Decompose Disjunction move on that disjunction. Thus, assuming that all such disjunctions are true, the Law of Excluded Middle move is justified. However, although this move is justified as a move in the sentential calculus, it is usually not needed. It is most convenient in dealing with the Predicate Calculus, introduced in Section 3. 2.6 Closing paths in sentential logic Socrates enables you to explore various combinations of assumptions and bits of information, to discover which of these are possible and which are impossible. A combination is shown to be impossible in Socrates in one of two ways: by closing a path or by closing the entire table. A path in Socrates consists of a series of sentences on the table. To find a path, start with a sentence at the bottom of the table. Every sentence in the same vertical column and any sentence in a higher column that contains that column as a subdivision is in one of the paths of the table. A path can be closed when the information and assumptions contained on the path cannot all be correct together, or when they cannot be combined with the information contained by any path on the other side of the table. In Socrates' treatment of sentential logic, this is represented in one of two ways: by contradiction within a path, and by bridging sub-tables. A path can be closed by contradiction when there are two contradictory sentences on that path. Two sentences are contradictory when one of the sentences is the negation of the other. For example, the following pairs are contradictory: A and —A—B and ——B(A&(BáC)) and —(A&(BáC)) A path can be closed by bridging sub-tables whenever it is possible to build a bridge from that path to every path on the opposite side of the table. You can build a bridge between a path on the left side of the table and a path on the right side whenever the very same sentence occurs on both paths. When two paths on opposite sides of the table can be bridged, this means that the assumptions the two paths make cannot be combined (recall that the left side of the table represents truth and that the right side of the table represents falsity). If a sentence A occurs on path L3 on the left side of the table and on path R2 on the right side, then we know that the information on these two paths are incompatible, since L3 says that A is true, while R2 (since it is on the Falsity side of the table) says that A is false. It is impossible for one and the same sentence to be both true and false at once (or so we assume in classical logic). When you pull down the SC menu and stop the cursor at the Close Path item, a sub-menu appears to the right. By moving the cursor to the right and releasing the mouse button while pointing to one of these sub-items, you select the designated move. The Close Path sub menu includes the following items: by Contradictionby Denied Self-Identityby Bridged Sub-Tablesby Anomaly Closing paths by denied self-identity is discussed in section 3.5.3, and closing paths by anomaly is discussed in section 6.1. These moves are not required in analyses of arguments expressed in the Sentential Calculus; they are only required in consideration of arguments expressed in the Predicate and Modal Calculi. 2.6.1 By Contradiction • by Contradiction - This command closes the path containing the selected formula if possible. If a unique path is not indicated by the selected formula, the command does nothing. If a unique path is selected and it can be closed, it is marked by an x. A selected path can be closed if it contains two formulas which are contradictories. 2.6.2 By Bridging Open Sub-Tables • by Bridged Subtables - This menu item is an extension to Hintikka and Bachman. It closes a selected path and marks it with an x if that path bridges to all open paths on the other side of the table. Thus, this command allows the user to simplify a table by closing bridged subtables before the entire table is closed. 2.7 Closing the table in sentential logic If you have entered the premises and conclusions of a logically correct or valid argument, then the assumptions you have made should lead to an impossibility, since you have in effect assumed that all of the premises are true and that at least one of the conclusions is false, which cannot happen if the argument is indeed logically correct. The analysis of a logically correct argument in Socrates should terminate with a closed table. A table can be closed whenever either (1) all the paths on the table are closed, or (2) any open path on the left side can be bridged to any open path on the right side. Once you have reached the state of having produced a table that can be closed, you should apply the Close Table move on the Entry menu. • Close Table - This command adds a marker to the bottom of the table indicating that the table is closed if the table can be closed. A subtable of an interrogative table is a pair of paths including one path from the premise side of the table and one path from the conclusion side of the table. A subtable can be bridged if the premise path and the conclusion path both contain at least one formula that is the same. If every subtable in the table can be bridged, then the table can be marked as closed using the Close Table command. After a table is closed, no further operations may be performed upon it. If the table has no paths on the right side, this command will not close the table. If the table has no paths on the left side, but all right-side paths are closed (perhaps by contradiction), then this command will close the table, indicating that the ultimate conclusion is a theorem. A theorem is a sentence that must be true in any circumstances or given any set of assumptions. In many cases, you may believe that the table you are working on should close, but when you apply the Close Table move, the dialog box informs you that there are still unbridged sub-tables on the table. This means that your task is unfinished. It would be helpful to know which open paths on the left side cannot be bridged to which open paths on the right side. The List Unbridged Subtables move in the Entry menu allows you to learn just that. • List Unbridged Subtables - This command lists the unbridged subtables in an interrogative table in a separate window. The subtables are identified by a pair of numbers separated by a hyphen. The first number in a subtable identifier denotes the path on the left side of the table (the premise side) that is contained in the subtable. The second number in a subtable identifier denotes the path on the right side of the interrogative table (the conclusion side) that is contained in the subtable. To find the path corresponding to a particular number, count the paths at the bottom of the table in the appropriate side of the table from the left edge of that part of the table. 2.7 Open, complete tables in sentential logic If you are working with a logically incorrect argument, and you want to use Socrates to demonstrate that the argument is not sound, you will of course never succeed in closing the table, since that would mean that the argument was correct after all. How do you know that you have reached a point at which you have demonstrated that it is impossible to close the table you have constructed? This will have been demonstrated once you have made your table open but complete. In the sentential logic, a table is open and complete when: (1) there is at least one open sub-table, (2) every complex sentence on any open path has been decomposed, (3) every double denial has been removed, and (4) every path containing contradictory sentences has been closed. 3. Predicate Logic (with Identity) 3.1 Entering sentences of predicate logic Predicate logic with identity adds three new logical symbols to our language: Œ, ´, and =. The symbols Œ and ´ are quantifiers; they are the universal and existential quantifiers, respectively. In addition, we will use lower-case letters; the letters a,..,s shall be called constants, and the letters t, u, v, w, x, y, and z shall be called variables. Any lower-case letter may be followed by any number of prime signs ('), and the result will also be a constant or variable. The special logical symbols of predicate logic can be entered by using buttons on the Add Premise or Add Conclusion dialog boxes, or by using the following key combinations: Symbol Key Combination Œ Option-Shift-q ´ Option-Shift-r The rules for building well-formed sentences are identical to those for sentential logic, but with three additions: 1. Any capital letter A, B, .... Z, followed by any number of primes ('), is a well-formed sentence. 2. Any capital letter A, B,...Z, followed by any number of primes, and further followed by any number of constants, is a well-formed sentence. 3. If å and _ are constants, then å = _ is a well-formed sentence. 4. If è and ê are well-formed sentences, so are: —è, (è&ê), (è_ê), (èáê), and (èâê), so long as no variable occurs in both è and ê. 5. If è is a well-formed sentence, å is a constant, and _ is a variable that does not occur in è, then Œ_è[_/å] and ‰_è[_/å] are well-formed sentences, where è[_/å] is the result of replacing every occurrence of å in è with _. All of the well-formed sentences of predicate logic with identity can be formed by repeated applications of rules 1 – 5. The new sentence produced by rule 2 above correspond to simple subject-predicate sentences in English. For example, 'John is bald' might be represented as 'Bj', where the capital letter 'B' represents the predicate '__ is bald' and the constant 'j' stands for John. Similarly, to say that 'Mary is in Paris', we could use a sentence like 'Imp', where 'I' represents the two-place predicate '__ is in __', 'm' names Mary and 'p' names Paris. Unlike English, our logical language always puts the predicate first and follows it by a list of constants, representing the subject, direct object, indirect object, etc. Rule 3 allows us to form identity statements, of the kind that are often found in mathematics, like '——3 =3'. Identity statements are also useful in stating how many things of a certain kind there are. By attaching the negation sign to an identity statement, we can produce a statement of non-identity or distinctness. Rule 4 allows us to introduce variables and to attach one of the quantifiers, Œ and ´. Variables cannot occur in a well-formed sentence unless they are governed or "bound" by a quantifier associated with the very same variable. Thus, 'Fax' and '(´xFx _ Gx)' are not well-formed, since the variable 'x' could not have been added to these sentences in combination with adding a quantifier phrase 'Œx' or '´x'. In the second case, the main connective is a wedge, so the whole is well-formed only if 'Gx' is, but 'x' cannot occur in a well-formed sentence except when accompanied by an appropriate quantifier phrase. In our language, no variable (such as 'x', 'y' or 'z') can be quantified within the scope of another quantifier that quantifies over the same variable. Thus 'Œx´yAxy' is a legal (or well-formed) formula, but 'Œx´xAxx' and 'Œx(Fx_ŒyGxy)' are not. Adding quantifiers does not involve adding or omitting any parentheses. So, one should not write: '(Œx)Fx' or '´y(Ay)'. In contrast, the sentence '´y(Ay & By)' is well-formed, since the pair of parentheses is associated with the binary connective &, not with the quantifier. 3.2 The intuitive meanings of the quantifiers The universal quantifier, Œ, is used to say that some property is true of absolutely everything in the world. To say that all F's are G's, we use a combination of the universal quantifier and the material conditional: 'Œx(Fx á Gx)'. The existential quantifier is used to say that some property is true of at least one thing in the world. To say that some (at least one) F is a G, we use the existential quantifier and the conjunction sign: '´y(Fy & Gy)'. When a quantifier occurs on the right side of the table, its informational significance is reversed. If 'ŒxFx' occurs on the right side of the table, this means that the sentence 'ŒxFx' is false, so it is not the case that everything is F. This means that something must be non-F. So, the universal quantifier on the right has existential force. Similarly, the existential quantifier on the right side has universal force: if '´yGy' is false, then this means that nothing can be G, that is, that everything must be non-G. 3.3 Inference moves in predicate logic As with sentential logic, inference moves can be applied only to a sentence with an appropriate main connective. A quantifier is the main connective of a sentence when and only when it is the first (left-most) symbol of the sentence. Consequently, the move Instantiate Universal Variable can be applied to a selected sentence only when Œ is the first symbol of the sentence, and likewise Instantiate Existential Variable can be applied only when ´ is the first symbol. The following moves are contained in the PC menu: • Instantiate Universal Variable - This command instantiates the first quantified variable in a selected universally quantified expression to a specified individual. The user is prompted to specify the variable and the individual, and all instances of the variable are replaced with the specified individual. The dialog, illustrated in Figure 4 below, allows copy, cut, and paste using CMD-C, CMD-X, and CMD-V. Figure 4. There are two restrictions on the operation of this command. (1) If the universal quantification occurs on the left side of the table, any constant may be specified to replace the quantified variable. (2) If the universal quantification occurs on the right side of the table, the specified constant must not occur on the same path or on any open, unbridged path on the left side of the table. • Instantiate Existential Variable - This command works in the same way that Instantiate Universal Variable does, except that it instantiates variables in existentially quantified formulas, and the restrictions are slightly different. (1) If the existential quantification occurs on the right side of the table, the specified constant must not occur on the same path or on any open, unbridged path on the left side of the table. (2) If the existential quantification occurs on the left side of the table, any constant may be specified to replace the quantified variable. • Alter Denied Quantifier - This command operates on a selected formula that begins with a denied quantifier (either existential or universal). It changes the quantifier from existential to universal or visa versa, removes the outer negation, and inserts a new negation between the quantified variables and the rest of the formula. This move is possible because the formula that the sentence produces is equivalent to the selected sentence (the two sentences mean the same thing). For example, saying 'Not everything is purple' is the same as saying 'There is something that is not purple.' Similarly, saying 'No one is mortal' is the same as saying 'Everyone is not mortal.' 3.4 Closing paths and closing the table in predicate logic The rules for closing paths and closing the table in the predicate logic include those presented above in the discussion of sentential logic. The use of identity involves one additional way of closing a path, (closure by denied self-identity) which is discussed in the next section. 3.5 Identity 3.5.1 Performing substitutions An identity statement, such as 'a=b', tells us that the individual named by 'a' is exactly the same individual named by 'b'. If the sentence 'a=b' occurs on the left side of the table in a path, this means that we can combine information in that path about 'a' with information in that same path about 'b'. This combination of information takes place by use of the Perform Substitution move in the PC menu. If the sentence 'a=b' occurs on a path on the right side of the table, this means that 'a=b' is false, and so 'a' and 'b' do not name the same individual. Consequently, no substitution is permitted. However, if the sentence '—a=b' occurs on the right side, this means that '—a=b' is false and so 'a=b' is true. Hence, the presence of '—a=b' in a path on the right side enable substitution of 'a' for 'b' or 'b' for 'a' in any sentence on that path. • Perform Substitution - This command allows the substitution of identical individuals in a formula. The user is prompted for two individuals, one to be replaced, and a second to be its replacement. See Figure 5. Figure 5. If any paths containing the selected formula also contain an identity between the entered individuals, the substitution will be performed. If the selected formula contains only one instance of the individual to be replaced, the substitution is performed automatically. Otherwise, the user is prompted to edit the selected formula so as to perform the required substitution. An improper substitution is not permitted. The tab key will switch the insertion point between the two boxes in which the individuals are to be entered. The dialog allows copy, cut, and paste using CMD-C, CMD-X, and CMD-V. 3.5.2 Use of the law of excluded middle Sometimes an identity statement, say 'a=b', occurs on the left side of the table, and the names 'a' and 'b' also occur in a path on the right side. In this case, closing the table may require moving the information about the identity of 'a' and 'b' from the left side of the table to the path on the right side. This can be accomplished by means of the Law of Excluded Middle move, contained in the SC menu. To bring the information to the path on the right side, simply select any sentence on that path and then choose the Law of Excluded Middle move from the SC menu. When prompted for a formula, enter 'a=b' (or whatever the identity statement is you want to move). As a result, two columns will be produced, one containing 'a=b' and the other containing '—a=b'. The first column can be bridged to the column on the left containing 'a=b'. The second column, since it contains '—a=b', enables you to use the Perform Substitution move on 'a' and 'b' in that path. Similarly, if the sentence '—a=b' occurs in a path on the right side and the constants 'a' and 'b' occur on a path on the left side, simply select the path on the left side, choose Law of Excluded Middle, and enter 'a=b'. This is the only case in which the Law of Excluded Middle move should be used. Using it for any other reason will not produce unsound results, but it will unnecessarily clutter up your table. 3.5.3 Closing paths through denied self-identity Everything is identical to itself. Consequently, if a path on the left side contains '—a=a', this path includes a logical impossibility and should be closed. Likewise, if a path on the right contains 'a=a', it should also be closed. In the Close Path item on the SC menu, there is an item on the sub-menu that allows you to close the path in these cases: • by Denied Self-Identity - This menu item closes a selected path on the premise side of the table if it contains a negated self-identity. It closes a selected path on the conclusion side of the table if it contains a self-identity. 3.6 Open, complete tables In the predicate logic, a table is open and complete when: (1) there is at least one open sub-table, (2) every conjunction, disjunction, conditional and biconditional on any open path has been decomposed, (3) every double denial has been removed, (4) every path containing contradictory sentences has been closed, (5) every quantificational sentence on an open path has been instantiated at least once, (6) every universal quantification on the left has been instantiated with every constant occurring on the same path or on any open, unbridged path on the right, (6) every existential quantification on the right has been instantiated with every constant occurring on the same path or on any open, unbridged path on the left, (7) every possible application of Perform Substitution has been performed on any open path containing an identity sentence on the left or a negated identity sentence on the right, and (8) whenever an identity statement occurs on the left or a negated identity statement on the right, the Law of Excluded Middle has been applied to every path on the other side of the table containing either of the constants involved. 4. Modal Logic 4.1 Entering sentences of modal logic The well-formed sentences of modal logic are those that can be composed by the following rules, which are identical to those for predicate logic, with the exception of the addition of rule 6: 6. If è is a well-formed sentence, so are ¡è and ™è. The special logical symbols may be entered by means of the dialog box buttons or by the following key combinations: Symbol Key Combination ¡ Option-1 ™ Option-2 4.2 Intuitive meanings of the box and diamond The box, ¡, is intended to express necessity, and the diamond, ™, possibility. The development of logical principles for these two notions goes back to the Greek philosopher Aristotle, and many important discoveries were made in antiquity and in the Middle Ages. Modal logic was revived in the 1930's by C. I. Lewis, and important contributions were made by the Princeton philosopher Saul Kripke in the late 1950's. When a necessity sentence (a sentence whose main connective is the ¡) occurs on the left side of the table, it states that it main subsentence is necessarily true, i.e., true in all possible situations or (to use the striking terminology of the German philosopher Leibniz) in all possible worlds. Thus, a ¡-sentence on the left has a kind of universal force, stating something about all possible worlds. A possibility sentence ™è on the left states that è is possibly true, true in some possible world. Thus, ™-sentences on the left have existential force, stating what is true in some possible world. When these formulas occur on the right side of the table, their significance is transformed. To add ¡è to the right side is to assume that ¡è is false, that è is not necessarily true, and that therefore è is false in some possible world (an existential implication). Similarly, adding ™è to the right means that ™è is false, that è is not possible true, and that è is false in every possible world (universal import). In Socrates, unpacking the information of modal sentences with existential import involves creating possible worlds in which the main subsentence is assumed to be true or false (depending on which side of the table it is on). Unpacking the information of modal sentences with universal import typically means deriving some new information about a possible world that has already been introduced to the table. 4.3 Inference rules in modal logic The following three rules are found in the MC menu: • Decompose Necessity - This command operates on a selected formula that begins with the box-character, or necessity. It prompts the user for a number between 0 and 99 using the dialog illustrated in Figure 6. Figure 6. Socrates uses this number to identify a possible world, and applies the formula to that world as described below. The world zero (0) is taken to represent the actual world. In the results of the application of this command, the set-membership symbol (é) indicates that a formula is true (or false, if on the conclusion side of the table) at the world denoted by the following number. If a formula is true or false at the actual world, the set-membership symbol (é) and the world number are left off. The sharp-symbol (#) indicates that the formula is normal at the world denoted by the following number. If the formula is normal at the actual world, the world number (i.e., '0') is omitted. (Left) If the sentence occurs on the left side, one is asked to choose a world: 0, 1, 2,...If world 0 (the actual world) is selected, the ¡ is simply stripped off. In the other cases, the ¡ is stripped off and the formula is followed by én (where n is the chosen world-number). (Right) If the sentence occurs on the right side, one is asked to designate a world, say n. The number selected must be (i) new to the table, or (ii) a world introduced as a è-normal world. If the original formula held in the actual world (i.e., it was not followed by 'ém'), then the ¡ is stripped off, and the remaining formula is followed by #én, where n is the chosen world number. If the original formula was followed by 'ém', for some number m (i.e., it read '¡è é m'), then the ¡ is still stripped off, and the remaining formula is followed by #mén, the '#m' indicating that the negation of the formula is normal at n (relative to the standards of normality in m). • Decompose Possibility - This command operates on selected formulas that begin with the diamond-character, or possibility. It operates in much the same way that Decompose Necessity does, except that the results produced are somewhat different, as described below. (Left) If the sentence occurs on the right side, one is asked to designate a world, say n. The number selected must be (i) new to the table, or (ii) a world introduced as a è-normal world. If the original formula held in the actual world (i.e., it was not followed by 'ém'), then the ™ is stripped off, and the remaining formula is followed by #én, where n is the chosen world number. If the original formula was followed by 'ém', for some number m (i.e., it read '™è é m'), then the ™ is still stripped off, and the remaining formula is followed by #mén, the '#m' indicating that the formula is normal at n (relative to the standards of normality in m). Three examples of applying the rule on the premise side: ™è ém ™è ™è #kém è #mén è #én è #mén (Right) If the sentence occurs on the left side, one is asked to choose a world: 0, 1, 2,... If 0 is selected, the ™ is simply stripped off. In the other cases, the ™ is stripped off and the formula is followed by én (where n is the chosen world-number). • Alter Denied Modal - If a selected formula begins with a negation and either a box or a diamond, then this command will remove the negation, change the modality (i.e. change the box to a diamond or visa versa), and insert a negation between the new modal character and the remainder of the formula. 4.4 Closing paths and closing the table in modal logic In closing a path by contradiction in modal logic, one must have contradictory sentences in that path that both hold in the same world, i.e., they must both hold in the actual world (no é-index) or they must both be followed by 'é n', for some one number n. Thus, 'A' and '—A é 1' are not contradictory, since the first says that 'A' is true in the actual world and the second says that 'A' is false in possible world 1. In contrast, 'Aé1' and ' —Aé1' are contradictory. Similarly, in closing a path by bridged subtables, or in closing the entire table in the same way, one must make sure that the same sentence, with the same é-index, occurs in both parts of the subtable (left and right). 4.5 Identity in modal logic Socrates assumes that whenever an identity statement holds in one world, it holds in all worlds. Consequently, the program ignores the world-index (é number), if any, attached to any identity statement when enabling Perform Substitution. At the same time, Socrates does preserve the world-index of any sentence upon which the substitution is performed. If you need to use the Law of Excluded Middle move to transfer information about identity from one side of the table to the other, but the identity statement in question happens to bear a world-index (é number), then you must use the command Modal Law of Excluded Middle instead. This will prompt you for a world number. For example, if 'a = b é 2' occurs on the left side of the table, you can use Modal Law of Excluded Middle to bring both 'a = b é 2' and '—a=b é2' to any path on the right side. 5. Modal Conditional Logic 5.1 Entering sentences of modal conditional logic The rules for building well-formed sentences of modal conditional logic are identical to those of modal logic, with the addition of the following rule: 7. If è and ê are well-formed sentences, so are (è ¡á ê) and (è ™á ê), so long as there is no variable that occurs in both è and ê. The special symbols '¡á' and '™á' cannot be entered from the keyboard in a single keystroke. You must use the buttons on the second row of the Add Premises and Add Conclusions dialog boxes, or you must first type the box or the diamond, then the arrow, in one of the following sequences. Symbol Key Combination ¡á Option-1, Option-e, a™á Option-2, Option-e, a 5.2 The intuitive meanings of the modal conditionals The modal conditional '¡á' can be used to express a variety of conditional statements in English (and other languages) that do not correspond closely to the material conditional. Within mathematics, no conditional other than the material conditional is needed (nor do modalities like necessity and possibility play any role). However, outside mathematics, we often need to express the existence of a real but imperfect connection between two states or features. For example, we often make generic statements, like 'Birds fly' or 'politicians are untrustworthy'. It would be incorrect to represent these generalizations as claiming that absolutely all birds fly, or that all politicians, without exception, are untrustworthy. At the same time, they clearly say more than just that some birds fly, or some politicians can't be trusted. They say that as a rule birds fly and politicians can't be trusted. One way to understand such statements is in terms of objective probability. A statement like 'Birds fly' could be understood as saying that the objective probability of a thing's flying, given only that it is a bird, is quite high, very close to 1. Another way of expressing the same thing (or something very closely related) is to say that a typical or normal bird flies. One way of thinking about typicality is this: a typical or normal bird has all the properties whose objective probability, conditional on being a bird, is infinitely close to 1 (differs from 1 by only an infinitesimal quantity). We can interpret the sentence '(A ¡á B)' as saying that the conditional probability of B on A is infinitely close to one. A quantified sentence like 'Œx(Ax ¡á Bx)' could be taken to mean that A's are typically B's. It could be used to translate a sentence like 'Most birds fly', where 'most' is intended to mean, not simply the majority, but very nearly 100%. Similarly, 'Œx(Ax ¡á —Bx)' could represent 'Few politicians are honest'. The diamond arrow could be used to say that something might very well have a certain property. We could use '(A ™á B)' to say that there is a significant, finite probability that B is true, conditional on A's being true. A quantified sentence like 'Œx(Ax ™á Bx)' could represent that many (not a few) A's are B's. When a box-arrow sentence occurs on the left side of the table, it has universal force. It says that something is true (the consequent) in every possible world in which the antecedent is normal. A diamond-arrow sentence on the left has existential force: it implies that there is some world in which the antecedent is normal and the consequent is true. On the right side, these relations are reversed. The box-arrow has existential force, implying that there is some world in which the antecedent is true and normal but the consequent is false. The diamond-arrow on the right side has universal force, implying that every world in which the antecedent is normal is a world in which the consequent is false. 5.3 Inference rules in modal conditional logic The following five moves are located in the MC menu: • Decompose Box-Arrow - This command operates on selected formulas whose main connective is the two-character sequence consisting of a box and an arrow. Such formulas, along with those containing main connectives consisting of a diamond and an arrow, are called nonmonotonic formulas. This command operates much like Decompose Necessity in prompting the user for a possible world, and then using that world to present a number of results to the user based on the other formulas present in the open paths containing the selected formula. These results are described below. On the Premise side: Let m be the world-index (if any) on the formula (è ¡á ê). I.e., we have selected: (è ¡á ê) ém [or, #kém]. For this rule there are three cases, depending on the nature of n, the world chosen: 1. The world 0 is selected. In this case, two columns are introduced. In the left column, Socrates puts two lines: è* and —ê*. In the right column, it puts the formula (è á ê). 2. The world n is new to the table, or the formula è#[m]én occurs somewhere on the premise side of the table, or —è#[m]én occurs somewhere on the conclusion side of the table. In any of these cases, two columns are introduced. In the left column, Socrates puts two lines: è#[m]én and êén. In the right column it puts: ¡—è[ém]. 3. In all other cases, the move produces two columns. In the left column, it produces: (Î &(èáê))én. In the right column, it produces the formula: (è ¡á —Î) [ém], where Î is the unique formula such that either: (i) the line Î#[k]én occurs somewhere on the premise-side of the table, or (ii) —Î#[k]én occurs somewhere on the conclusion-side of the table. [Every world is introduced at some unique point in the table, and lines labeled with #[j]én can therefore exactly once for each world n.] On the Conclusion side: One is asked to designate a world. The number selected must be new to the table. If the selected formula is (è ¡á ê) #[k]ém, the result is: on the first line, Socrates places the negation of the antecedent followed by the label #[m]én (i.e., —è#[m]én), and on the second line, êén. • Decompose Diamond-Arrow - This command operates on selected formulas whose main connective is the two-character sequence consisting of a diamond and an arrow. It operates in much the same way as Decompose Box-Arrow except that the results produced are somewhat different, as described below. On the Premise side: One is asked to designate a world, say n. The number selected must be new to the table. If the selected formula is (è ™á ê) [ém], the result is: on the first line, è#[m]én, and on the second line, êén. On the Conclusion side: Suppose one selects (è ™á ê) [ém]. For this rule there are three cases, depending on the world selected: 1. The world 0 is selected. In this case, two columns are introduced. In the left column, Socrates puts two lines: —è* and —ê*. In the right column, it puts the formula (è & ê). 2. The world n is new to the table, or the formula è#[m]én occurs somewhere on the premise side of the table, or —è#[m]én occurs somewhere on the conclusion side of the table. In any of these cases, two columns are introduced. In the left column, Socrates puts two lines: —è#[m]én and êén. In the right column it puts: ™è[ém]. 3. In all other cases, Socrates produces two columns. In the left column, it produces: (—Î _ (è&ê))én. In the right column, it produces the formula: (è ™á Î) [é m], where Î is the unique formula such that either (i) the line Î#[k]én occurs somewhere on the premise-side of the table, or (ii) —Î#[k]én occurs somewhere on the conclusion-side of the table. • Alter Denied Nonmonotonic - This command operates much like Alter Denied Biconditional, except that the selected formula must have as its main connective either box-arrow or diamond-arrow. In addition, like Alter Denied Modal, this move changes the main connective. In this case, it changes a box-arrow into a diamond-arrow, and vice versa. • Box-Definition - This command changes a selected formula whose main connective is a box-arrow and whose consequent is the negation of its antecedent. The result is the consequent of the original formula. • Diamond-Definition - This command changes a selected formula whose main connective is a diamond-arrow and whose consequent is the same as its antecedent. The result is the consequent of the original formula. 5.4 Closing paths and closing the table in conditional modal logic Paths and tables are closed in modal conditional logic in exactly the ways they are closed in modal logic. 5.5 Complete, open tables In modal and modal conditional logic, a table is open and complete when the eight conditions mentioned in section 3.6 are fulfilled and, in addition, the following conditions are met: (9) Every modal and modal conditional sentence in an open path has been decomposed at least once. (10) Every modal and modal conditional sentence in an open path with universal modal force (¡ and ¡á on the left, ™ and ™á on the right) have been decomposed with respect to every world-number occurring on the table. (11) Every sentence that occurs as the antecedent of any modal conditional on the table is normal at some world introduced on the table (this may involve using Decompose Box-Arrow on the left or Decompose Diamond-Arrow on the right, while selecting a new world-number – case 2). 6. Defeasible Logic 6.1 Closing paths by anomaly elimination An anomaly is a pair of sentences introduced on a path with the asterisk (*) label by use of the Decompose Box-Arrow move on the left or the Decompose Diamond-Arrow move on the right, while selecting world 0 (the actual world). The first sentence in the pair is called the "anomalous" sentence, and the second is called the "anomalizing" sentence. Both sentences are referred to as "constituents" of the anomaly. Since the generalizations represented by modal conditionals are leaky or exception-permitting, we cannot simply close every path on which an anomaly occurs. The actual world may well contain some real anomalies. However, it is reasonable to minimize the anomalies we attribute to the world. If a table is open and complete, and some subtables contain no anomalies while others do, we should then close the paths containing anomalies and tentatively conclude that the actual world is as it is described on one of the non-anomalous subtables. The Close Path sub menu contains the item by Anomaly. This command allows you to close any selected path, so long as that path contains at least one anomaly. Unlike the other commands of Socrates, the Close Path by Anomaly command allows you to transform the table in ways that may not be rationally defensible. It is up to you to decide when a given anomalous path can legitimately be closed. Be prepared to defend your action, if challenged. • by Anomaly - This menu item is required for modal and nonmonotonic logics. It closes a selected path if that path contains an anomalous formula. Anomalous formulas are indicated by asterisks, and are obtained only through the use of some commands in the MC menu. 6.2 Varieties of defeasible logics By adopting various rules for when the Close Path by Anomaly command may be used, we can work out a variety of systems of defeasible or "nonmonotonic" logic. A very conservative system (called system Z by Judea Pearl of UCLA) would close a path by anomaly only when: (i) the table is open and complete in modal conditional logic, and (ii) there is at least one open subtable that contains no anomalies. In that case, it seems quite reasonable to close any path containing an anomaly. Some defeasible logics are more daring. Suppose that every open subtable contains at least one anomaly, but that the anomalies on subtable 2-6 are a proper superset of the anomalies on table 1-4 (i.e., every anomaly on 1-4 also occurs on 2-6, and in addition there are other anomalies on 2-6). In some sense, subtable 2-6 is then strictly more anomalous than 1-4, and that may justify closing it. Alternatively, you might compare the sets of sentences that are anomalous on the two subtables (remember, a sentence is anomalous when it occurs as the first constituent of some anomaly). Finally, you may wish to consider which sentences explain which anomalies on a given subtable. Let's say that sentence è explains an anomaly <ê,Î> on subtable m-n if and only if: (1) the sentence (è ¡á ê) occurs in path m (on the left side) or the sentence (è ™á —ê) occurs in path n (on the right side), and (2) the sentence (è ¡á Î) occurs in path m (on the left side) or the sentence (è ™á —Î) occurs in path n. Suppose there is a formula è that is anomalous in sub-table m-n [the sub-table consisting of the m-th and n-th paths on the table], and there is another sub-table k-l such that for every anomaly <ê,Î> in k-l, è explains <ê,Î> in m-n. And, moreover, let us suppose that every moral fact recorded in k-l is also recorded in m-n. In such a case, we call è a super-anomaly on sub-table m-n. Any subtable containing a super-anomaly can be treated as bridged. If every sub-table containing path m contains a super-anomaly, then we should use the Close Path by Anomaly command to close path m. Since the modal conditional ¡á carries probabilistic information, it is possible to show that the defeasible logic defined above has the following property: if è/,...è˜ _ ê is nonmonotonically correct, then the probability P(ê/è/&...&è˜) is infinitely close to one. This means that if you have a high degree of confidence in the conjunction of the premises and you have no other relevant information, it is reasonable to have a high degree of confidence in the conclusion. A. Appendices A.1 Keyboard Shorcuts Socrates includes a number of keyboard shortcuts for various menu commands. Those that are specific to Socrates and are not shared with other Macintosh applications are listed in the table below. These shortcuts consist of using the command key (which has one or both of the following symbols on it: _ _) and one other key from the keyboard. Entry Menu Command Shortcut Add Premise… Command-; Add Conclusion… Command-’ Close Table Command-T Close Path by Contradiction Command-1 Close Path by Bridged Subtables Command-2 Close Path by Denied Self-Identity Command-3 Close Path by Anomaly Command-4 SC Menu Command Shortcut Decompose Conjunction Command-H Decompose Disjunction Command-J Decompose Conditional Command-K Decompose Biconditional Command-L Modus Ponens Command-M PC Menu Command ShortCut Perform Substitution… Command-I Instantiate Universal Variable… Command-U Instantiate Existential Variable… Command-E Alter Denied Quantification Command-D FontSize Menu Command Shortcut Larger Command-] Smaller Command-[

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