Spreadsheet Proof of Arguments 

by John D. Sullivan 

ARL-TR-2815 September 2002 

Approved for public release; distribution is unlimited. 
NOTICES 

Disclaimers 

The findings in this report are not to be construed as an official Department of the Army position 

unless so designated by other authorized documents. 

Citation of manufacturer’s or trade names does not constitute an official endorsement or approval 

of the use thereof. 

Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-2815 September 2002 

Spreadsheet Proof of Arguments 

John D. Sullivan 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
ii

Abstract 

This report describes the use of a spreadsheet to solve simple problems in prepositional logic. A 

spreadsheet can easily generate and recall truth tables, and with its built-in logical functions and 

connectives it can calculate the truth value of logic expressions. Expressions can also be 

premises and together with a conclusion constitute an argument, which can be shown valid with 

a truth table. 

This report covers the application of a spreadsheet to truth table generation, evaluation of logical 

expressions, recasting arguments into spreadsheet form, and demonstration of validity. 
iii

Acknowledgments 

The author thanks Timothy J. Mermagen of the Individual Performance Branch, Soldier 

Performance Division, Human Research and Engineering Division, U.S. Army Research 

Laboratory, for critically reading this report.
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Contents 

Acknowledgments iii

List of Figures vii

1. Introduction 1

2. Truth Table Generation 1

3. Spreadsheet Particulars 4

4. Development 5

4.1 Logical Expressions ........................................................................................................5 

4.1.1 A Very Simple Example......................................................................................5 

4.1.2 Simple Expression...............................................................................................6 

4.2 Implication Operation......................................................................................................7 

4.2.1 Truth Table for Implication.................................................................................7 

4.2.2 Paradoxes of Material Implication ......................................................................8 

4.3 Proof of Arguments.........................................................................................................8 

4.3.1 Controlling Rule ..................................................................................................8 

4.3.2 Configuring Arguments for Spreadsheets ...........................................................9 

4.3.3 Restyled Truth Table for an Argument .............................................................10 

4.4 Fallacies.........................................................................................................................11 

4.5 Examples of Arguments ................................................................................................13 

4.5.1 Three Symbol Argument ...................................................................................14 

4.5.2 Four Symbol Argument.....................................................................................16 

4.5.3 Five Symbol Argument .....................................................................................18 

5. Discussion 21

6. Conclusion 22

7. References 23

Report Documentation Page 25
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List of Figures 

Figure 1. 2-variable truth table........................................................................................................2 

Figure 2. 3-variable truth table........................................................................................................2 

Figure 3. 4-variable truth table........................................................................................................3 

Figure 4. A 2logic table. .................................................................................................................6 

Figure 5. Example of logic table.....................................................................................................6 

Figure 6. Truth table for implication...............................................................................................7 

Figure 7. Truth table. ......................................................................................................................9 

Figure 8. Improved truth table. .....................................................................................................10 

Figure 9. Restyled truth table........................................................................................................10 

Figure 10. Affirming the Consequent truth table..........................................................................11 

Figure 11. Truth table with tautology and contradiction columns................................................12 

Figure 12. Denying the Antecedent truth table.............................................................................12 

Figure 13. Three symbol argument truth table method.................................................................16 

Figure 14. Four symbol argument truth table method. .................................................................18 

Figure 15. Five symbol argument truth table method...................................................................20 
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1

1. Introduction 

From the standpoint of logic, the problem with any argument is to determine that it is valid. By 

abstracting an argument from (English) language to a symbolized language, eye clutter is much 

reduced and the tools of logic can be applied to investigating the validity of the argument. 

Symbolization takes skill separate from the method of solution used. Getting solutions to logic 

problems may be done by truth table or deductive methods. The truth table is the basic 

introductory method of logic. Deduction is a much shorter method of validating arguments and 

the only method for complicated arguments. 

Both methods have drawbacks. Truth tables have a conventional order, which the user may not 

know. Working with even small truth tables is tedious and error prone if done by hand, and it is 

impractical for most sizes. Within arguments, it is restricted to the least complex ones that have 

sentences of a simple variety—subject, verb, and object. The deductive method is much shorter, 

but is artful in nature and requires rule learning and skill in symbol manipulation. Still, it is by 

far the normal method of validating arguments. 

For logic problems that are amenable to the truth table method, a spreadsheet is a moderately 

easy and very accurate tool to achieve the solution. However, the spreadsheet approach is 

infeasible without having connectives like and, or, or not, to operate on symbolized sentences. 

But these connectives are logical operators that do reside in every spreadsheet program, for other 

reasons, and are here given an unintended new use. 

A method of generating truth tables via spreadsheets is shown. Proving validity of arguments 

with symbolic logic is recounted. Intertwined are examples of problems susceptible to the truth 

table method, such as finding the truth values of logic expressions and determining validity of 

simple arguments. The emphasis is on the spreadsheet use of truth tables, but for contrast the 

deductive method is also used. 

2. Truth Table Generation 

A logical variable takes on the values of true (T) or false (F). A truth table with several variables 

is an orderly list of all possible arrangements of T and F. The size of a table of n variables is 2n

rows by n columns. For instance, five variables means 25

 × 5 = 160 table entries, which is 

lengthy to write out and apply. A method suited for spreadsheet generation of a truth table that 

finds all the arrangements in order will be explained (Sullivan 2000). The customary appearance 

of the table is top row all T, bottom row all F; mixed rows are also in a conventional order. For 
2

2-Variable 

Truth Table 

p q 

T T 

T F 

F T 

F F 

Figure 1. 2-variable truth table. 

3-Variable 

Truth Table 

p q r 

T T T 

T T F 

T F T 

T F F 

F T T 

F T F 

F F T 

F F F 

Figure 2. 3-variable truth table. 

two variables p and q, the truth table is as shown in Figure 1. For three variables p, q, and r, the 

truth table is as shown in Figure 2. 

The 2- and 3-variable truth tables reveal a repeating pattern. The pattern is that the entire 

2-variable table is repeated twice (shaded portion) in the 3-variable table, once for p true and 

once for p false. Another way of describing the pattern is that there are 23

 = 8 rows of values for 

the 3-variable table. The first column headed p will get the first half of its rows true and the next 

half false. The 2-variable table values are supplied against the trues and against the falses, 

completing the table. The pattern with the 3-variable table will make the next higher table. The 

4-variable table has 16 rows (24) of values and the first column headed p will get half the rows 

true and half false. The entire 3-variable table is supplied against the trues (light shading) and 

also against the falses (dark shading), which completes the 4-variable table (Figure 3). 

 

2-variable truth table 

 

 2-variable truth table
3

4-Variable 

Truth Table 

p q r s 

T T T T 

T T T F 

T T F T 

T T F F 

T F T T 

T F T F 

T F F T 

T F F F 

F T T T 

F T T F 

F T F T 

F T F F 

F F T T 

F F T F 

F F F T 

F F F F 

Figure 3. 4-variable truth table. 

The tables are rapidly made in a spreadsheet or word processor because both programs carry the 

copy and paste commands, which allow the highlighting of the section that is carried forward to 

the next table. The leftmost column will always be trues for the upper half of rows and falses for 

the bottom half of rows. Thus generating truth tables is advanced one at a time. Each table 

should be saved as a file, if it is not within a document already, so that it does not have to be 

regenerated in sequence each time it is needed. Saving a truth table as a Corel WordPerfect 8 

file is done like saving any table. Click anywhere inside the table, then click on the menu names 

Edit|Select|Table|File|Save|Selected Text and supply a filename, e.g., “4var” for the 4-variable 

truth table. Making and saving in a spreadsheet will be explained next. 

 3-variable truth table

 

 3-variable truth table 
4

3. Spreadsheet Particulars 

The point of having a truth table is to be able to evaluate expressions and arguments by the truth 

table method. The tabled arrangements so far seen contain no logical values for computing; they 

are made with a word processor and are just for looks. The spreadsheet does not recognize T and 

F as anything but labels. The logic values entered in a cell are the numbers 1, 0, or alternately 

the functions @TRUE, @FALSE. Either entry prints on the screen as 1, 0, but the editing 

window will show the @-function if it were actually entered. For speedy typing, 1 and 0 are 

preferred, but @-functions may clarify intent in some situations. Truth tables for computing are 

made with logic values in the way described in section 2. That way will be restated ahead in the 

context of spreadsheets (Underdahl 1994). 

To know the best way to save and retrieve work, it is helpful to know a little more about the 

organization of spreadsheets. Spreadsheet programs are designed as notebooks of many tabbed 

pages, i.e., sheets. It is the whole notebook that is given a path name ending in a file name. To 

retrieve a particular page, the user selects the notebook name from the file menu and opens the 

sheet, say the one of truth tables, by clicking the mouse arrow on its tab. (Right clicking the tab 

allows the sheet to be named.) There is ample room on a sheet to fit all the truth tables needed. 

A needed table can be copied and pasted onto another sheet to start an actual problem. However, 

a better feature exists to retrieve a particular table. 

In the Corel Quattro Pro 8 spreadsheet program, the feature to save and retrieve a large block of 

data is called a block name. A repertoire of, say, six (variable) truth tables is enough to meet 

most textbook problems. After a truth table is first made, the technique is: highlight it, click on 

the menu names Insert|Name|Cells, type an obvious name for the block and click “Add.” For 

instance, the 2-variable truth table would be called by a brand new name “2logic,” and so on. If 

they are forgotten, all of the names of blocks can be reviewed in the above menus. To put a 

named block on a new sheet, the user just types its name in a cell (the upper left corner of the 

block starts there) and the entire block of data is printed on the screen. The editing window will 

show that the cell contains the formula @ARRAY(2logic). (The block can be quickly deleted by 

just deleting the named [left top corner] cell.) Very large tables can be highlighted by the 

spreadsheet itself: click menu names View|Toolbars|Data Manipulation and click the 

SpeedSelect button, then proceed as previously outlined to name the block. A truth table can be 

built from the lower one by the copy and paste route, or by inserts of the block name of the lower 

table. It is important to note the following: (1) Accidentally deleting the sheet of truth tables or 

the name of a block will eliminate all or one of the block names and all problems that were 

started by typing a block-named truth table will have only the all-false instance and (2) The 

block names pertain only to that file in which they were made and saved. 
5

The next higher table is made in the style that (T/F) truth tables were generated in section 2. For 

instance, by using copy and paste to create “3logic” skip over one cell from where the table is to 

start and type the previously named “2logic.” The leftmost column is created by typing the 

number 1 or @TRUE at the first row, copying it, highlighting under it down to the last row of 

2logic, and pasting, which fills the blank cells with logical trues (“1”). At the cell where the 

block should be repeated, “2logic” is typed again and entered. Completing the leftmost column 

begins by entering zero or @FALSE in the top cell, copying, highlighting down, and pasting to 

give the blank cells logical falses (“0”). With that, the logic table is complete and the block 

should be given its recognizable name. 

4. Development 

To apply the truth table to validating arguments, there first must be application to evaluating 

logical expressions. Then a rule for proving arguments must be given. Also the important 

logical connective of implication must be defined in truth table form. The method of showing 

invalidity in an argument (fallacies) will be given. Finally, some arguments in natural language 

will be symbolized and validated with truth tables. 

4.1 Logical Expressions 

A logical expression is any combination of the logical operators and, or, or not, which implies

acting on logical variables. The truth value of the expression depends on the values that are tried 

from the truth table. Usually the truth value is a mixture of T and F, and the expression is said to 

be contingent. For example, the truth value of the logical expression p and q (symbolized p · q) 

is true only if p and q are individually true, but is false for the other three arrangements of true 

and false. There are two special cases though, that involve implication, which are important in 

the proof of argument. An expression that evaluates as true for all table entries is a tautology; 

and one that evaluates as false for all table entries is a contradiction. 

4.1.1 A Very Simple Example 

We outline how to do a problem of two variables. On a new sheet “2logic” is typed into any cell, 

the truth table appears, and in the adjoining column is entered the 2-variable logical expression to 

be evaluated. The expression is copied down the rest of the rows, and the spreadsheet evaluates 

it. The 2logic table (Figure 4) reveals all possible values of the expression, here p · q. As the 

table confirms, if either p or q or both are false, the conjunction is false. 

The problem is begun at cell A1 by typing “2logic” which produces the block whose diagonal is 

A1..B4. The column identifications are typed in bold. Into cell C1 goes the spreadsheet 

rendition of the logical expression to be evaluated. So, in C1 (p · q) is typed as +A1#AND#B1; 
6

A B C 

p q p · q

1 1 1 1 

2 1 0 0 

3 0 1 0 

4 0 0 0 

Note: Cell C1 formula can be written 

as +A1#AND#B1 or @AND(A1,B1). 

Figure 4. A 2logic table. 

the + sign prevents the formula from being mistaken and printed as a label. An equivalent 

formula is @AND(A1,B1). Note that the variable’s cell location is used in the formula, and not 

the variable names p and q. The instruction cell and the empty column cells below that entry are 

highlighted and the Speedfill icon is clicked. The spreadsheet automatically adjusts the cell 

references, i.e., cell C2 holds +A2#AND#B2, and so on down to cell C4, and instantly computes 

and prints the values (ones and zeros) shown. This very simple example illustrates the truth table 

method of evaluating a logical expression. 

4.1.2 Simple Expression 

Some easy, successful examples of computing the logical value of expressions with a 

spreadsheet are shown for five examples involving two variables. The various connectives of the 

variables p and q are: p @ q means p and q, w means or, and ~ means not. The logic table that the 

spreadsheet uses is in columns A and B; five different expressions are in columns C–G 

(Figure 5). The computed results are easily mentally checked by the ordinary understanding of 

the logical results of combinations of true and false statements. Notice that all the expressions 

are contingencies. 

A B C D E F G 

p q p @ q p w q p @ q w q ~ p p @ q w ~ p

1 1 1 1 1 1 0 1 

2 1 0 0 1 0 0 0 

3 0 1 0 1 1 1 1 

4 0 0 0 0 0 1 1 

2logic table five logical expressions evaluated 

 Note: row 1 cell formulas are A1:@ARRAY(2logic) B1: 1 

 C1: +A1#AND#B1 or @AND(A1,B1) 

 D1: +A1#OR#B1 or @OR(A1,B1) 

 E1: +C1#OR#B1 or @OR(C1,B1) 

 F1: #NOT#A1 

 G1: +C1#OR#F1 or @OR(C1,F1 

Figure 5. Example of logic table. 
7

Because of a rule, compound expressions as in column G cannot be misread. The scope of a 

logical connective only extends to the next connective. So the dot includes q and stops there. 

If a different expression is meant, parentheses are used, e.g., p ⋅ (q ∨ ~ p). For simplicity, 

2-variable examples are shown, but a spreadsheet can compute the truth value of much more 

complicated expressions of numerous variables. 

4.2 Implication Operation 

Spreadsheets were never designed to perform symbolic logic, and they lack a built-in logical 

function for implication. Since many, many arguments contain the “if p, then q” construction of 

implication, the incomplete set of functions restricts symbolization. Fortunately, there is a 

“workaround” that fills in the set. The symbol of implication is a horseshoe between the p and q 

propositions, i.e., p ⊃ q. The propositions have names: p is the antecedent, and q is the 

consequent. 

4.2.1 Truth Table for Implication 

There is an insistence that, where p is true and q is false (row 2), the relation itself is false. If we 

decline to fix any more cases than that one, then column C shows the result. This resulting 

column for implication can be matched by column F for the expression ~ (p ⋅ ~ q) (Figure 6): 

A B C D E F G 

p q p ⊃ q ~ q (p . ~ q) ~ (p . ~ q) ~p ∨ q 

1 T T T F F T T 

2 T F F T T F F 

3 F T T F F T T 

4 F F T F F T T 

Figure 6. Truth table for implication. 

Expression F can be operated on to give the simpler expression shown in column G. Two 

substitutions transform the F to the G expression. Two needed expressions and their equivalents 

are 

 ~ (p ⋅ q) ≡ ~ p ∨ ~ q De Morgan’s law, and 

 ~ ~ q ≡ q Double Negation. 

These equivalents turn expression F into ~ (p ⋅ ~ q) ≡ (~ p ∨ ~ ~ q) ≡ (~ p ∨ q). The last 

expression becomes the working definition of implication: 

~ p ∨ q ≡ p ⊃ q 

When symbolizing implication p ⊃ q in a spreadsheet, we will write P.IMP.Q as the column 

heading. The symbol set in Quattro Pro 8 does not include the horseshoe. We usually write the 

cell formula as #NOT#cell with p#OR#cell with q. An alternative formula is @OR(#NOT#cell 

with p, cell with q). 
8

4.2.2 Paradoxes of Material Implication 

The Figure 6 truth table at row 1 has both propositions p and q true and the implication also true. 

Row 2 is fixed by sensible insistence that truth implying falsehood should be a false relation. 

Rows 3 and 4 are surprising in that they both have the implication true even though one or both 

of the propositions is false. The truth values of the parts have become irrelevant to the truth 

value of implication itself. Rows 3 and 4 have been called “paradoxes of material implication.” 

Suber (1997) has discussed implication with examples of propositions that show the truth table 

definition has reasonableness. 

4.3 Proof of Arguments 

An argument is a number of statements called premises that end in a final statement called the 

conclusion. No lines of an argument have to be objectively true. What matters is that the 

premises entail the conclusion, or better, that the conclusion logically follows from the premises. 

If so, the argument is valid. There are two ways to prove an argument is valid: by deductions, 

which are new statements justified by rules of inference that lead to the conclusion or by a truth 

table showing it is not invalid. 

4.3.1 Controlling Rule 

Notice that deductions prove validity, and truth tables prove noninvalidity. An invalid argument 

cannot be shown so by deductions, but is readily detectable with a truth table. A truth table 

demonstration of validity uses a reverse approach, because the controlling rule for validity is 

essentially a negative one. The rule is: a valid argument cannot have true premises and a false 

conclusion (“truth can’t come from a lie”). We look only for that forbidden case in a truth table, 

and if it does not appear, the argument is valid. 

Let us prove a very simple argument both ways: 

1. p premise 1 

2. q premise 2 

∴ p ⋅ q conclusion 

Notice that the truth table in section 4.1.1 contained columns of these expressions, but there was 

no mention that the expressions could be united to make an argument. 

This argument is valid on sight by Conjunction, rule of inference no. 8 of Copi (1965). 

The truth table is as shown in Figure 7.
9

A B C 

p q p ⋅ q 

1 T T T 

2 T F F 

3 F T F 

4 F F F 

Figure 7. Truth table. 

The premises 1 and 2 are respective headings for columns A and B; the conclusion heads column 

C. Only in row 1 do both premises have the logic value of true. We do not need to further 

consider the other rows. Back in row 1, the conclusion is true, which is to say it is not false. So 

the forbidden case (true premises and a false conclusion) does not arise in this argument, and it is 

valid. 

4.3.2 Configuring Arguments for Spreadsheets 

A small truth table can be visually inspected for the invalidating case, but not a large one. The 

premises and conclusion are scattered, and there are too many rows. There are reconfigurations 

of an argument that make for automatic checking for validity. The idea is that in a valid 

argument nothing new is said by the conclusion—the premises imply it. Further, in the only 

meaningful case the premises are individually true, so conjoining them into one expression 

leaves the logic value as true. Let the conjoined premises be the antecedent of an implication, 

and let the conclusion be the consequent. A complicated argument is reduced this way to an 

expression of p ⊃ q. Recall the truth table for implication: 

A B C 

p q p ⊃ q 

1 T T T 

2 T F F 

3 F T T 

4 F F T 

where valid argument in this implication configuration will always show trues because case 2 is 

not present by definition of validity. In this configuration, the argument has been recast into a 

tautologous expression. In spreadsheet form, the valid argument (tautology form) has all ones in 

the implication column; the invalid argument holds a zero wherever row 2 (shaded) comes into 

play. 

Now consider the argument reconfigured as p ⋅ ~ q, where as before p is the conjoined premises 

and q is the conclusion. Therefore, the truth table is 
10

A B C 

p q p ⋅ ~ q 

1 T T F 

2 T F T 

3 F T F 

4 F F F 

A valid argument will always show falses in this configuration. The argument has been recast 

into a contradictory expression. In spreadsheet form, the valid argument (contradiction form) 

has all zeros in the p ⋅ ~ q column; the invalid argument holds a one wherever row 2 (shaded) 

comes into play. 

In spreadsheet evaluation of arguments, we will always make final columns where the argument 

is in tautology and contradiction form. (One or the other would suffice.) If it is valid, a glance 

down the columns will show that every row is one and zero, respectively. If it is invalid, each 

column will be spoiled by a discordant, opposite symbol. 

4.3.3 Restyled Truth Table for an Argument 

Take the same argument as before, but improve it by identifying the columns (Figure 8): 

Premise 1 Premise 2 Conclusion 

A B C 

p q p ⋅ q 

1 T T T 

2 T F F 

3 F T F 

4 F F F 

Figure 8. Improved truth table. 

See how it looks in a restyled truth table (Figure 9): 

Premise 1 Premise 2 Conclusion Tautology Contradiction 

A B C D E 

p q p ⋅ q p ⋅ q ⊃ p ⋅ q (p ⋅ q) ⋅ ~ (p ⋅ q) 

1 T T T T F 

2 T F F T F 

3 F T F T F 

4 F F F T F 

Figure 9. Restyled truth table. 
11

The first style (Figure 8) shows validity by inspection. Only in row 1 are the premises both true, 

and looking at the conclusion in column C, it is not false, so the controlling rule has not been 

violated. The second style (Figure 9) added columns D and E to reconfigure the argument as a 

tautology and a contradiction, respectively. Specifically, column D shows an implication having 

the conjoined premises as the antecedent and the conclusion as the consequent. That is the 

prescription for testing that an argument is a tautology, hence valid. If the argument is valid, the 

tautology necessarily will show a column of all trues (shaded area), and the contradiction will 

show the opposite. Column E conjoins the premises and the negation of the conclusion. Either 

column D or E suffices, but both are given to illustrate the method. Validity is easier to spot in 

the new style, and it is a style adaptable to spreadsheet calculation. 

4.4 Fallacies 

An invalid argument is called a fallacy. Some arguments, anciently known to be invalid, recur 

so often that they have been named. One called Affirming the Consequent goes 

Affirming the Consequent 

 1. p ⊃ q 

2. q 

∴ p (Invalid argument) 

Examining the argument with a truth table shows that it violates the controlling rule, and thus is 

invalid (Figure 10). 

A B C 

p q p ⊃ q 

1 T T T 

2 T F F 

3 F T T 

4 F F T 

Figure 10. Affirming the Consequent truth table. 

The premises 1 and 2 are the headings of columns C and B, respectively. The premises are both 

true only on rows 1 and 3. The conclusion, heading column A, is true on row 1, but is false on 

row 3. In other words, row 3 shows an instance of true premises and a false conclusion (shaded 

area), and that makes the argument itself invalid. The other rows (2 and 4) are irrelevant to the 

consideration, since the premises are not both true there. The invalid argument resembles a valid 

argument called Modus Ponens, listed in section 4.5. 

When the truth table adds tautology and contradiction columns (E and F) of section 4.3.2, the 

argument’s invalidity is observed differently, as shown in Figure 11. Since E and F do not 

contain just one truth value (row 3), the argument is shown to be invalid. Note that to save space 

the long expression in column D is called D (it is the conjoined premises) when used in columns 
12

A B C D E F 

p q p ⊃ q (p ⊃ q) ⋅ q D ⊃ p D ⋅ ~p 

1 T T T T T F 

2 T F F F T F 

3 F T T T F T 

4 F F T F T F 

Figure 11. Truth table with tautology and contradiction columns. 

E and F. This same space saving is done ahead when spreadsheet headings are made. There 

should not be any confusion that the column letter is merely standing for the expression heading 

the column. 

Another named fallacy is Denying the Antecedent (Figure 12). 

 Denying the Antecedent 

 1. p ⊃ q 

2. ~ p 

∴ ~ q (Invalid argument) 

A B C D E 

p q ~ p ~ q p ⊃ q 

1 T T F F T 

2 T F F T F 

3 F T T F T 

4 F F T T T 

Figure 12. Denying the Antecedent truth table. 

The premises 1 and 2 are the headings of columns E and C, respectively. The premises are both 

true only on rows 3 and 4. The conclusion, heading column D, is true on row 4, but is false on 

row 3. In other words, row 3 shows an instance of true premises and a false conclusion (shaded 

area), and that makes the argument itself invalid. The other rows (1 and 2) are irrelevant to the 

consideration, since the premises are not both true there. This invalid argument resembles a 

valid argument called Modus Tollens, listed in section 4.5. 

There is an oddity among fallacies. The fallacy known as the Slippery Slope, which is quite 

popular with television political pundits, is based on a valid argument called the Hypothetical 

Syllogism. The valid argument, is used in section 4.4.2, and goes 

 Hypothetical Syllogism 

1. p ⊃ q 

2. q ⊃ r 

∴ p ⊃ r (Valid argument) 

 
13

The Hypothetical Syllogism has the consequent of one implication becoming the antecedent of 

the next. The chain can go on and on, past the two premises shown here, but the conclusion is 

always an implication having the first antecedent and the last consequent of the chain. In 

popular talk, the valid argument transmogrifies into the fallacy. The full package is a warning 

that if we do this first semi-innocuous thing then it will lead to the next thing, which will lead to 

the next, etc., and finally we are given a consequence that everyone admits is dire. So, we ought 

never to do the first thing. The fallacy is the speaker’s omniscient presumption in stating that 

one thing always causes another, and that the next change is irresistible or inevitable, and that 

events can be forecast to an ordained end. Usually though the speaker is too lazy to think of a 

chain of worsening developments, and the speech shortens from fallacy to metaphor. “If we do 

that, we’re on a slippery slope.” So, a fallacy is converted from invalid argument to a “puts-astop-to-it” reason. So besides the oddity of a valid form being converted to a fallacy, a double 

oddity occurs, e.g., the fallacy itself is invoked to prove the argument. One more way this 

fallacy is shuffled off on people is the even shorter plaint, “If we do this, where will it end?” 

Very few invalid arguments are named. Most fallacies are not based on logic at all (informal 

fallacies), but are really techniques of propaganda, persuasion, advocacy, and grifting. The aim 

in using a fallacious argument is overcoming someone. A fallacious argument cannot for the 

most part be analyzed by a truth table. The number of catalogued fallacies is three dozen, and 

Web sites describing them can be found by a search engine looking for “fallacy.” 

4.5 Examples of Arguments 

The following subsections give three examples of verbal arguments. They are first translated 

into symbols, and then, to contrast the methods, are proven by the rules of inference, and then 

proven using the truth table method on a spreadsheet. The arguments are posed in very stilted 

English, with all premises explicit, and the “if...then” construction used for implication. The 

awkwardness comes from the strict argumentation, which is probably there for pedagogical 

reasons. Spoken English sometimes leaves out structure, and the auditor himself must supply 

lines that the speaker thinks are unnecessary for comprehension. Additionally many arguments 

are inherently complex. So for several reasons, symbolizing can get difficult. 

A proof proceeds by lengthening the symbolized argument with new lines justified by the rules 

of inference. These rules give equivalents that replace all or part of the premises and 

subarguments that combine lines to yield a new line. The proof ends when a line reaches the 

argument’s conclusion. In all, Copi (1965) provides 19 rules (10 equivalents plus 9 

subarguments) and two techniques, conditional proof and indirect proof. We will not use 

examples needing the two techniques. As for the rules themselves, they are recommended as 

practical; they are not independent, meaning that some could be disposed of and proofs still 

accomplished. Retaining more rules than are strictly needed, is useful. 
14

The subarguments used in the examples have the following forms. They can all be proven by a 

truth table: 

Modus Ponens Modus Tollens Destructive Dilemma 

1. p ⊃ q 1. p ⊃ q 1. (p ⊃ q) ⋅ (r ⊃ s) 

2. p 2. ~ q 2. ~ q ∨ ~ s 

∴ q ∴ ~ p ∴ ~ p ∨ ~ r 

Disjunctive Syllogism Hypothetical Syllogism 

1. p ∨ q 1. p ⊃ q 

2. ~ p 2. q ⊃ r 

∴ q ∴ p ⊃ r 

The only nonobvious equivalents used are De Morgan’s laws and Exportation. Both may be 

proven by truth tables: 

De Morgan’s laws Exportation 

 ~ (p ∨ q) ≡ ~ p ⋅ ~ q p ⊃ (q ⊃ r) ≡ (p ⋅ q) ⊃ r 

 ~ (p ⋅ q) ≡ ~ p ∨ ~ q 

4.5.1 Three Symbol Argument 

According to Copi (1965) problem II-4, p. 48: 

If he uses good bait then if the fish are biting then he catches the legal limit. He 

uses good bait but he does not catch the legal limit. Therefore, the fish are not 

biting. (G, B, C): 

1. G ⊃ (B ⊃ C) 

2. G ⋅ ~ C / ∴ ~ B 

3. (G ⋅ B) ⊃ C 1, Exportation 

4. ~ C ⋅ G 2, Commutation 

5. ~ C 4, Simplification 

6. ~ (G ⋅ B) 3, 5, Modus Tollens 

7. ~ G ∨ ~ B 6, De Morgan 

8. G 2, Simplification 

9. ~ B 7, 8, Disjunctive Syllogism 

Line 3 replaces line 1 with its equivalent by a rule called Exportation. Lines 4 and 5 play with 

line 2, i.e., if we have two things together then we have one of them (~ C) by itself. An ancient 

argument form known as Modus Tollens justifies line 6. This argument has the form 1. p ⊃ q 

2. ~ q /∴ ~ p. It is clearly seen that lines 3, 5, and 6 have the form of Modus Tollens. Visual 

recognition of argument forms is a standard procedure in applying the rules of inference. Line 7 

is equivalent to line 6 by one of De Morgan’s laws. Lines 7 and 8 form the premises of another 
15

argument form known as Disjunctive Syllogism, whose conclusion is line 9. But line 9 is also 

the conclusion of the argument in question, which is proven valid. (As stated in section 4.5, the 

three named subarguments can themselves be proven valid by truth tables, which we shall not 

bother to give.) 

Since Copi’s rules of inference are more than sufficient to prove arguments, more than one proof 

may sometimes be found. A shorter proof goes 

1. G ⊃ (B ⊃ C) 

2. G ⋅ ~ C / ∴ ~ B 

3. G 2, Simplification 

4. B ⊃ C 1, 3, Modus Ponens

5. ~ C ⋅ G 2, Commutation 

6. ~ C 5, Simplification 

7. ~ B 4, 6, Modus Tollens

The truth table method is shown in Figure 13. The spreadsheet page is titled with the problem 

number, and the block of work is located in a frame of rows (4−13) and columns (B−K). Cells 

B5..D5 are titled by the variables of the problem (G, B, C). In cell B6, typing the previously 

made and saved block name “3logic” generates the 3-variable table. Important columns that 

identify premises, conclusion, tautology, and contradiction are so labeled. Other columns are 

workups to the needed premises and conclusion. Column I is the conjunction of both premises, 

and it is the antecedent of the implication in column J. The consequent of the implication is the 

conclusion in column E. As section 4.1.2 states, this arrangement of a valid argument is a 

tautology. Column K is dispensable, and just shows the alternative arrangement that gives a 

contradiction. 

The necessary cell formulas are entered on row 6. The cell formulas in the first row are printed 

below the block of work. It is unnecessary to print the other formulas as the spreadsheet 

automatically adjusts the cell formula down each column. For instance, cell E6 contains the 

formula #NOT#C6 and beneath it is cell E7, which must contain the formula #NOT#C7. A 

common error in writing formulas is confusing the column letter with the column heading. For 

instance, in column E headed by ~ B the formula in cell E6 must refer to the contents in cell C6 

and be written as #NOT#C6, not as #NOT#B6. Another thing to notice is that formulas can be 

written in other ways. The implication in column G is straightforwardly written as material 

implication is symbolized, i.e., p ⊃ q ≡ ~ p ∨ q. But in column J, we choose to use the @ORform. Another kind of choice is displayed in column K where an @AND-form is preferred for 

readability over the form needing adjacent pound signs, i.e., +I6#AND##NOT#E6. 
16

Figure 13. Three symbol argument truth table method. 

Printing is adjusted as desired from File|Print|Page Setup to tabs controlling details, such as 

orientation, framework, centering, heading, and many others. Print Preview shows the 

adjustments on screen. When the setup is sufficiently close to desired, the tab Named Settings 

allows the setup to be saved and reused by name, e.g., II 4. Any further changes are saved with 

buttons Update and OK. 

4.5.2 Four Symbol Argument 

According to Copi (1965) problem II-2, p. 48: 

If the supply of silver remains constant and the use of silver increases then the 

price of silver rises. If an increase in the use of silver implies that the price of 

silver rises then there will be a windfall for speculators. The supply of silver 

remains constant. Therefore, there will be a windfall for speculators. 

(S, U, P, W): 
17

1. (S ⋅ U) ⊃ P 

2. (U ⊃ P) ⊃ W 

3. S / ∴W 

4. S ⊃ (U ⊃ P) 1, Exportation 

5. S ⊃ W 4, 2, Hypothetical Syllogism 

6. W 5, 3, Modus Ponens

Lines 4−6 are deductions made by replacing a premise with its equivalent or by combining two 

lines via a recognized argument. Line 4 replaces line 1 with its equivalent, and names the rule by 

which it is permitted. Lines 4 and 2 are in the form of Hypothetical Syllogism, whose 

conclusion is line 5. Lines 5 and 3 produce line 6, the conclusion of a different subargument

Modus Ponens. But line 6 is also the conclusion to be proved, so the argument is valid by means 

of (Copi’s) rules of inference. 

A proof doesn’t have to be unique. Two applications of Modus Ponens will also get to the 

conclusion. 

1. (S ⋅ U) ⊃ P 

2. (U ⊃ P) ⊃ W 

3. S / ∴W 

4. S ⊃ (U ⊃ P) 1, Exportation 

5. U ⊃ P 4, 3, Modus Ponens 

6. W 2, 5, Modus Ponens

The truth table method is shown in Figure 14. The spreadsheet page is titled with the problem 

number and the block of work is located in a frame of rows (3−20) and columns (B−L). Cells 

B4..E4 are titled by the variables of the problem (S, U, P, W). In cell B5 typing “4logic” 

generates the 4-variable table. Important columns that identify premises, conclusion, tautology, 

and contradiction are so labeled. Other columns are workups to the needed premises and 

conclusion. Column J is the conjunction of all three premises, and the antecedent of the 

implication in column K. The consequent of the implication is the conclusion in column E. As 

section 4.1.2 states, this arrangement (in column K) of a valid argument is a tautology. 

Column L is dispensable, and just shows the alternative arrangement that gives a contradiction. 

The necessary cell formulas are entered on row 5. Filling the rest of a column can be done two 

ways. For large tables, a method in section 4.4.3 is better. For small tables the following 

method works well. When the row is completed, one practice is to highlight a column from 

row 5 to row 20, and hit Speedfill. The cells are automatically filled with the formula adjusted 

for the cell location. It is good practice to put the spreadsheet in manual recalculation mode, 

which prevents it operating every time a cell is changed. The sequence needed is: 

Format|Notebook Recalculate Tab|Manual. Pressing key F9 or a calculator icon on the bottom 

bar of the screen triggers the calculation. Forgetting to do this step can lead to hunting for a 

formula mistake. 
18

Figure 14. Four symbol argument truth table method. 

After key F9 is pressed, the block of work shows the printed values of Figure 14. A glance at 

column K shows that the tautology has all ones, as it must for a valid argument (or if preferred, 

column L has all zeros). 

4.5.3 Five Symbol Argument 

According to Copi (1965) problem II-7, p. 48: 

If he attracts the farm vote then he will carry the rural areas, and if he attracts the 

labor vote then he will carry the urban centers. If he carries both the urban 

centers and the rural areas then he is certain to be elected. He is not certain to be 

elected. Therefore, either he does not attract the farm vote or he does not attract 

the labor vote. (F, R, L, U, C): 
19

1. (F ⊃ R) ⋅ (L ⊃ U) 

2. (U ⋅ R) ⊃ C 

3. ~ C / ∴ ~ F ∨ ~ L 

4. ~ (U ⋅ R) 2, 3, Modus Tollens

5. ~ U ∨ ~ R 4, De Morgan 

6. ~ R ∨ ~ U 5, Commutation 

7. ~ F ∨ ~ L 1, 6, Destructive Dilemma 

Lines 2 and 3 have the form of the premises of Modus Tollens, so we can infer the conclusion of 

that argument which here takes the form of line 4. Line 5 is the equivalent of line 4 by one of 

De Morgan’s laws. Line 6 simply switches the order of terms in line 5. Lines 1 and 6 have the 

form of the premises of another argument called the Destructive Dilemma, so we can infer the 

conclusion of that argument which here takes the form of line 7. But line 7 is also the conclusion 

of the argument in question, which is proved valid. 

The truth table method is described next. The spreadsheet page (Figure 15) is titled with the 

problem number and the block of work is located in a frame of rows (3−36) and columns (B−P). 

Cells B4..F4 are titled by the variables of the problem (F, R, L, U, C). In cell B5 typing “5logic” 

generates the 5-variable table. Important columns that identify premises, conclusion, tautology, 

and contradiction are so labeled. Other columns are workups to the needed premises and 

conclusion. Column M is the conjunction of all three premises, and the antecedent of the 

implication in column O. The consequent of the implication is the conclusion in column N. As 

section 4.1.2 states, this arrangement (in column O) of a valid argument is a tautology. 

Column P is dispensable, and just shows the alternative arrangement that gives a contradiction. 

The necessary cell formulas are entered on row 5. The cell formulas in the first row are printed 

below the block of work. Filling the long columns can be done differently than using Speedfill, 

as was done in section 4.4.1. The method uses array formulas and works mainly on the first 

row with manual recalculation off. Array formulas specify that arithmetic or logical operations 

be performed throughout an entire block. For instance, in cell G5 we want the implication 

operation performed like this: F ⊃ R or #NOT#B5#OR#C5. But since we really want the 

column filled down to row 36, we specify a range of cells by the double dot notation 

#NOT#B5..B36#OR#C5..C36. The spreadsheet supplies the @ARRAY prefix if we don’t care 

to type it ourselves. The formula is executed pairwise from B5 and C5 together through to B36 

and C36. In other words, after the formula is correctly entered the spreadsheet properly fills 

column G from row 5 down to row 36. The advantage of the array formula is that we don’t have 

to scroll down a long distance. Columns H..L of row 5 have been filled via the array command. 

In cell M5 we come to the step of combining all the premises into one expression. Since there 

are three premises, it is shorter to use an @AND command with commas, rather than repeating 

the #AND# command. At M5 though we have reverted to Speedfill to fill the work block. 
20

Figure 15. Five symbol argument truth table method. 
21

At cells M5, N5, O5, and P5, we write the formulas without the array command and see that the 

top row is properly one or zero in the tautology and contradiction cell, respectively. (If they 

weren’t, we could go back and correct a mistake in the formulas.) Then with one highlight of the 

block (M5..P36), Speedfill creates the filling. 

5. Discussion 

Limitations of this material and fields related to it are discussed next. 

Logic outcomes (true/false) are recognized by spreadsheets in order to permit simple 

comparisons and cause something else to be done. The function used for these tests is an 

@IF-function, which usually employs inequality tests to reach a logic outcome. The explicit 

logic operators and, or, not are seldom used, but their opportune presence leads to a symbolic 

logic capability in spreadsheets. 

To symbolize more complicated sentences than seen in the examples of section 4.5, additional 

rules have to be developed. The greatest extension is caused by quantification logic, which is 

needed to deal with sets of objects and arguments about them. For instance, symbolizing 

All dogs are mammals and Not all mammals are dogs requires new notation that looks like 

calculus and not like the bare symbols in the examples. Relations extend quantification even 

more and allow symbolizing sentences like Grant and Lincoln were acquainted. Copi (1965) 

explains quantifiers in chapter 4 and relations in chapter 5. Truth tables are inapplicable to 

deciding validity of arguments involving those logics. Furthermore this report does not use other 

definitions of implication (strict implication), which try to avoid the paradoxes of material 

implication, section 4.2.2. Also while we only admit two truth values (true and false), multivalued logic and fuzzy logic are accepted elsewhere. To sum up, there are these restrictions on 

the use of truth tables: 

• No sentences but simple declaratives or conditionals (implication). 

• No other than material implication is used in the report. 

• No quantifier logic (nouns modified by words like all, some, only, every, each). 

• No relational logic (phrases like x belongs to y, x is acquainted with y, x is related to y). 

• No multi-value logic; logical variables can have more than two values.

Aside from logic, truth tables make an appearance in probability and electrical engineering. 

They are a visual aid to solving some basic probability problems. In probability, coin-flipping 

problems are encountered. The possible outcomes of a single coin flip are heads and tails, which 

correspond to the true and false values of a logical variable. The four arrangements of heads and 
22

tails from flipping two coins correspond to the two-variable table, and so on. Flipping just n = 3 

coins can create confusion as there are 8 arrangements (2n) of heads and tails (rows of a truth 

table) and 4 outcomes (n + 1), i.e., no head, one head, two heads, three heads. Generating this 

situation with the “3var” truth table allows one to answer numerous small questions about the 

probability of various numbers of heads (or tails) appearing. For instance, counting how many 

arrangements there are of each outcome leads to the probability of each outcome. 

Misunderstandings in conditional probability are cleared up with a table. The visual solution of a 

truth table aids understanding of the shorter solution by a formula (Runyon and Haber 1971). 

In electrical engineering, the logic circuit course introduces a truth table and meshes it with 

Boolean algebra to create formulas that are interpreted as switching circuits. Their truth table is 

inverted: top row is all zeros (all switches open); the bottom row is all ones (all switches closed) 

(Whitesitt 1995). It is also shown that to each truth table corresponds a Boolean expression and 

vice versa. So a switch circuit represents a truth table and a Boolean expression and vice versa. 

The simplest examples of this are p ⋅ q, which stands for two switches in series, and p ∨ q, which 

stands for two switches in parallel. Sloane (1996) provides an example of going from a truth 

table to a Boolean expression. As the Boolean expressions tend to be long sums, simplification 

of an expression with the axioms of Boolean algebra has to be mastered. The last step 

corresponds to finding a simpler (cheaper to build) circuit that obeys the truth table. 

6. Conclusion 

Though not by design, spreadsheets can be applied to solve some problems in symbolic logic. 

The spreadsheet can evaluate a logical expression, mixtures of logical connectives and symbols. 

It can generate a truth table, an exhaustive array of true and false values for a useful number of 

variables. It can display a truth table with headings of logical expressions. The expressions can 

be premises and conclusion of an argument. The truth table can be inspected to show that the 

argument is valid. Validity occurs if there is no row of the truth table where the premises 

evaluate as true and the conclusion evaluates as false. If the argument is reconfigured as an 

implication, the spreadsheet can automatically searched for the invalidating case. The 

reconfiguration makes a valid argument into a tautologous expression. To do this, premises are 

joined by the and connective, and the conjunction become the if-proposition of an implication. 

The conclusion is made the then-proposition of the implication. Only for a valid argument does 

the implication column of the spreadsheet show all ones.
23

7. References 

Copi, I. M. Symbolic Logic. Second edition, New York: Macmillan, 1965. 

Runyon, R. P., and A. Haber. Fundamentals of Behavioral Statistics. Second edition, Reading, 

MA: Addison-Wesley, p. 161, 1971. 

Sloane, T. “Extracting a Boolean Function From a Truth Table.” <http://www.cs.jcu.edu.au/ftp 

/web/teaching/Subjects/cp1200/1996/org/node24.html>, James Cook University, 

Queensland, Australia, 1996. 

Suber, P. “Paradoxes of Material Implication.”<http://www.earlham.edu/~peters/courses/log 

/mat-imp.htm>, Earlham College, Richmond, IN, 1997. 

Sullivan, J. D. “Spreadsheet Generation of a Truth Table.” ARL-MR-497, U.S. Army Research 

 Laboratory, Aberdeen Proving Ground, MD, September 2000. 

Underdahl, B. Using Quattro Pro 6 for Windows, Special Edition. Indianapolis, IN: Que Corp.,

 1994. 

Whitesitt, J. E. Boolean Algebra and Its Applications. Mineola, NY: Dover, 1995. 
24

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25

truth table, logic, spreadsheet, Quattro Pro 29

This report describes the use of a spreadsheet to solve simple problems in prepositional logic. A spreadsheet can easily

generate and recall truth tables, and with its built-in logical functions and connectives it can calculate the truth value of

logic expressions. Expressions can also be premises and together with a conclusion constitute an argument, which can

be shown valid with a truth table. 

This report covers the application of a spreadsheet to truth table generation, evaluation of logical expressions, recasting

arguments into spreadsheet form, and demonstration of validity. 

ARL-TR-2815

U.S. Army Research Laboratory

ATTN: AMSRL-WM-TB

Aberdeen Proving Ground, MD 21005-5066

John D. Sullivan

Spreadsheet Proof of Arguments 28T8FC

September 2002 Final, January 2000–June 2002

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